Quantum entanglement

5236:(local operations and classical communication). These operations do not allow the production of entangled states between systems A and B. But if A and B are provided with a supply of entangled states, then these, together with LOCC operations can enable a larger class of transformations. For example, an interaction between a qubit of A and a qubit of B can be realized by first teleporting A's qubit to B, then letting it interact with B's qubit (which is now a LOCC operation, since both qubits are in B's lab) and then teleporting the qubit back to A. Two maximally entangled states of two qubits are used up in this process. Thus entangled states are a resource that enables the realization of quantum interactions (or of quantum channels) in a setting where only LOCC are available, but they are consumed in the process. There are other applications where entanglement can be seen as a resource, e.g., private communication or distinguishing quantum states. 8500:: "For EPR that would be an unthinkable 'spooky action at a distance'. To avoid such action at a distance they have to attribute, to the space-time regions in question, real properties in advance of observation, correlated properties, which predetermine the outcomes of these particular observations. Since these real properties, fixed in advance of observation, are not contained in quantum formalism, that formalism for EPR is incomplete. It may be correct, as far as it goes, but the usual quantum formalism cannot be the whole story." And again on p. 144 Bell says: "Einstein had no difficulty accepting that affairs in different places could be correlated. What he could not accept was that an intervention at one place could influence, immediately, affairs at the other." Downloaded 5 July 2011 from 8295: 5665: 1361:, the result of a spin measurement on one of the particles is a collapse (of wave function) into a state in which each particle has a definite spin (either up or down) along the axis of measurement. The outcome is taken to be random, with each possibility having a probability of 50%. However, if both spins are measured along the same axis, they are found to be anti-correlated. This means that the random outcome of the measurement made on one particle seems to have been transmitted to the other, so that it can make the "right choice" when it too is measured. 1428:, whose values effectively determine, right from the moment of separation, what the outcomes of the spin measurements are going to be. This would mean that each particle carries all the required information with it, and nothing needs to be transmitted from one particle to the other at the time of measurement. Einstein and others (see the previous section) originally believed this was the only way out of the paradox, and the accepted quantum mechanical description (with a random measurement outcome) must be incomplete. 9531: 15429: 5248:(see below) can be used, that assign a numerical value to each quantum state. However, it is often interesting to settle for a coarser way to compare quantum states. This gives rise to different classification schemes. Most entanglement classes are defined based on whether states can be converted to other states using LOCC or a subclass of these operations. The smaller the set of allowed operations, the finer the classification. Important examples are: 1081: 7951:, Yin et al. report setting a new quantum entanglement distance record of 1,203 km, demonstrating the survival of a two-photon pair and a violation of a Bell inequality, reaching a CHSH valuation of 2.37 ± 0.09, under strict Einstein locality conditions, from the Micius satellite to bases in Lijian, Yunnan and Delingha, Quinhai, increasing the efficiency of transmission over prior fiberoptic experiments by an order of magnitude. 1344:

system considered here seems to acquire a probability distribution for the outcome of a measurement of the spin along any axis of the other particle upon measurement of the first particle. This probability distribution is in general different from what it would be without measurement of the first particle. This may certainly be perceived as surprising in the case of spatially separated entangled particles.

33: 8884: 4654: 1464:. This is contrary to what is found in classical physics, where any number of properties can be measured simultaneously with arbitrary accuracy. It has been proven mathematically that compatible measurements cannot show Bell-inequality-violating correlations, and thus entanglement is a fundamentally non-classical phenomenon. 6432:, the more uncertainty (number of microstates) the system should possess, the larger the entropy. For example, the entropy of any pure state is zero, which is unsurprising since there is no uncertainty about a system in a pure state. The entropy of any of the two subsystems of the entangled state discussed above is 1415:(In fact similar paradoxes can arise even without entanglement: the position of a single particle is spread out over space, and two widely separated detectors attempting to detect the particle in two different places must instantaneously attain appropriate correlation, so that they do not both detect the particle.) 960:. For example, if a pair of entangled particles is generated such that their total spin is known to be zero, and one particle is found to have clockwise spin on a first axis, then the spin of the other particle, measured on the same axis, is found to be anticlockwise. However, this behavior gives rise to seemingly 1231:

Aspect writes that "... no experiment ... can be said to be totally loophole-free," but he says the experiments "remove the last doubts that we should renounce" local hidden variables, and refers to examples of remaining loopholes as being "far fetched" and "foreign to the usual way of reasoning in physics."

5256:. This is the finest of the usually considered classes. Two states in the same LU class have the same value for entanglement measures and the same value as a resource in the distant-labs setting. There is an infinite number of different LU classes (even in the simplest case of two qubits in a pure state). 5647:

that contain sufficient correlations for A to modify ("steer") by local measurements the conditional reduced state of B in such a way, that A can prove to B that the state they possess is indeed entangled, and finally (3) those entangled states that are neither non-local nor steerable. All three sets

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into at least one pure entangled state. States that have this property are called distillable. These states are the most useful quantum states since, given enough of them, they can be transformed (with local operations) into any entangled state and hence allow for all possible uses. It came initially

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Yin, Juan; Yu-Huai Li; Sheng-Kai Liao; Meng Yang; Yuan Cao; Liang Zhang; Ji-Gang Ren; Wen-Qi Cai; Wei-Yue Liu; Shuang-Lin Li; Rong Shu; Yong-Mei Huang; Lei Deng; Li Li; Qiang Zhang; Nai-Le Liu; Yu-Ao Chen; Chao-Yang Lu; Xiang-Bin Wang; Feihu Xu; Jian-Yu Wang; Cheng-Zhi Peng; Artur K. Ekert; Jian-Wei

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provides both a necessary and a sufficient criterion for separability, and thus—inadvertently—for detecting entanglement. However, for the general case, the criterion is merely a necessary one for separability, as the problem becomes NP-hard when generalized. Other separability criteria include (but

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Entropy provides one tool that can be used to quantify entanglement, although other entanglement measures exist. If the overall system is pure, the entropy of one subsystem can be used to measure its degree of entanglement with the other subsystems. For bipartite pure states, the von Neumann entropy

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In quantum information theory, entangled states are considered a 'resource', i.e., something costly to produce and that allows implementing valuable transformations. The setting in which this perspective is most evident is that of "distant labs", i.e., two quantum systems labeled "A" and "B" on each

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The paradox is that a measurement made on either of the particles apparently collapses the state of the entire entangled system—and does so instantaneously, before any information about the measurement result could have been communicated to the other particle (assuming that information cannot travel

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In August 2014, Brazilian researcher Gabriela Barreto Lemos and team were able to "take pictures" of objects using photons that had not interacted with the subjects, but were entangled with photons that did interact with such objects. Lemos, from the University of Vienna, is confident that this new

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would certainly be required to do so, based on conservation of angular momentum in classical and quantum mechanics alike. The difference is that a classical system has definite values for all the observables all along, while the quantum system does not. In a sense to be discussed below, the quantum

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by which it was possible to question the validity of the results. However, in 2015 an experiment was performed that simultaneously closed both the detection and locality loopholes, and was heralded as "loophole-free"; this experiment ruled out a large class of local realism theories with certainty.

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in their paper "Geometrical aspects of entanglement". Leinaas et al. offer a numerical approach, iteratively refining an estimated separable state towards the target state to be tested, and checking if the target state can indeed be reached. An implementation of the algorithm (including a built-in

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fail, however, when measurements of the spin of entangled particles along different axes are considered. If a large number of pairs of such measurements are made (on a large number of pairs of entangled particles), then statistically, if the local realist or hidden variables view were correct, the

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and colleague I. Shaknov in 1949, and was published on New Year's Day in 1950. The result specifically proved the quantum correlations of a pair of photons. In experiments in 2012 and 2013, polarization correlation was created between photons that never coexisted in time. The authors claimed that

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The mentioned proofs about the existence of local models assume that there is only one copy of the quantum state available at a time. If the particles are allowed to perform local measurements on many copies of such states, then many apparently local states (e.g., the qubit Werner states) can no

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An entangled system can be defined to be one whose quantum state cannot be factored as a product of states of its local constituents; that is to say, they are not individual particles but are an inseparable whole. In entanglement, one constituent cannot be fully described without considering the

4435: 1320:, and as a result, the measurement outcomes of one daughter particle must be highly correlated with the measurement outcomes of the other daughter particle (so that the total momenta, angular momenta, energy, and so forth remains roughly the same before and after this process). For instance, a 1015:

to the remote point, affecting the outcome at the second location. However, so-called "loophole-free" Bell tests have since been performed where the locations were sufficiently separated that communications at the speed of light would have taken longer—in one case, 10,000 times longer—than the

3604:. The apparatus might produce electrons that are all in the same state; in this case, the electrons received by the observer are then a pure ensemble. However, the apparatus could produce electrons in different states. For example, it could produce two populations of electrons: one with state 1222:

of the angle between the polarizer settings and decreased exponentially with time lag between emitted photons. Kocher's apparatus, equipped with better polarizers, was used by Freedman and Clauser who could confirm the cosine-squared dependence and use it to demonstrate a violation of Bell's

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An early experimental breakthrough was due to Carl Kocher, who already in 1967 presented an apparatus in which two photons successively emitted from a calcium atom were shown to be entangled – the first case of entangled visible light. The two photons passed diametrically positioned parallel

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is often portrayed as being equivalent to entanglement. While this is true for pure bipartite quantum states, in general entanglement is only necessary for non-local correlations, but there exist mixed entangled states that do not produce such correlations. A well-known example is the

7174: 7033: 12314:

Pirandola, S.; U. L. Andersen; L. Banchi; M. Berta; D. Bunandar; R. Colbeck; D. Englund; T. Gehring; C. Lupo; C. Ottaviani; J. L. Pereira; M. Razavi; J. Shamsul Shaari; M. Tomamichel; V. C. Usenko; G. Vallone; P. Villoresi; P. Wallden (2020). "Advances in quantum cryptography".

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The outcome of Alice's measurement is random. Alice cannot decide which state to collapse the composite system into, and therefore cannot transmit information to Bob by acting on her system. Causality is thus preserved, in this particular scheme. For the general argument, see

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In continuous variable systems, the Peres-Horodecki criterion also applies. Specifically, Simon formulated a particular version of the Peres-Horodecki criterion in terms of the second-order moments of canonical operators and showed that it is necessary and sufficient for

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In the literature "non-locality" is sometimes used to characterize concepts that differ from the non-existence of a local hidden variable model, e.g., whether states can be distinguished by local measurements and which can occur also for non-entangled states; see, e.g.,

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Living organisms (green sulphur bacteria) have been studied as mediators to create quantum entanglement between otherwise non-interacting light modes, showing high entanglement between light and bacterial modes, and to some extent, even entanglement within the bacteria.

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have shown in practice that Bell's inequality is not satisfied. However, prior to 2015, all of these experiments had loophole problems that were considered the most important by the community of physicists. When measurements of the entangled particles are made in moving

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can be performed, but which do not interact with each other quantum mechanically. The only interaction allowed is the exchange of classical information, which combined with the most general local quantum operations gives rise to the class of operations called

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For the appropriately chosen measures of entanglement, Bell, GHZ, and NOON states are maximally entangled while spin squeezed and twin Fock states are only partially entangled. The partially entangled states are generally easier to prepare experimentally.

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respectively. In other words, a state is separable if it is a probability distribution over uncorrelated states, or product states. By writing the density matrices as sums of pure ensembles and expanding, we may assume without loss of generality that

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may succeed with different probability, they are no longer equally valuable. E.g., for two pure qubits there are only two SLOCC classes: the entangled states (which contains both the (maximally entangled) Bell states and weakly entangled states like

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of the whole state is zero (as it is for any pure state), the entropy of the subsystems is greater than zero. In this sense, the systems are "entangled". This has specific empirical ramifications for interferometry. The above example is one of four

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Shadbolt, P. J.; Verde, M. R.; Peruzzo, A.; Politi, A.; Laing, A.; Lobino, M.; Matthews, J. C. F.; Thompson, M. G.; O'Brien, J. L. (2012). "Generating, manipulating and measuring entanglement and mixture with a reconfigurable photonic circuit".

7827:, and those based on uncertainty relations. See Ref. for a review of separability criteria in discrete-variable systems and Ref. for a review on techniques and challenges in experimental entanglement certification in discrete-variable systems. 1905:

Entanglement of a state shared by two particles is necessary, but not sufficient for that state to be non-local. Entanglement is more commonly viewed as an algebraic concept, noted for being a prerequisite to non-locality as well as to

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Aad, G.; Abbott, B.; Abeling, K.; Abicht, N. J.; Abidi, S. H.; Aboulhorma, A.; Abramowicz, H.; Abreu, H.; Abulaiti, Y.; Acharya, B. S.; Bourdarios, C. Adam; Adamczyk, L.; Addepalli, S. V.; Addison, M. J.; Adelman, J. (September 2024).

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of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between

4649:{\displaystyle \rho _{A}\ {\stackrel {\mathrm {def} }{=}}\ \sum _{j}^{N_{B}}\left(I_{A}\otimes \langle j|_{B}\right)\left(|\Psi \rangle \langle \Psi |\right)\left(I_{A}\otimes |j\rangle _{B}\right)={\hbox{Tr}}_{B}\;\rho _{T}.} 1930:

The following subsections are for those with a good working knowledge of the formal, mathematical description of quantum mechanics, including familiarity with the formalism and theoretical framework developed in the articles:

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Since 2016, various companies, for example IBM and Microsoft, have created quantum computers that allowed developers and tech enthusiasts to freely experiment with concepts of quantum mechanics including quantum entanglement.

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is the standard measure of entanglement for pure states (but no longer a measure of entanglement for mixed states). For mixed states, there are some entanglement measures in the literature and no single one is standard.

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If the former occurs, then any subsequent measurement performed by Bob, in the same basis, will always return 1. If the latter occurs, (Alice measures 1) then Bob's measurement will return 0 with certainty. Thus, system

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If two states can be transformed into each other by local operations including measurements with probability larger than 0, they are said to be in the same 'SLOCC class' ("stochastic LOCC"). Qualitatively, two states

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A possible resolution to the paradox is to assume that quantum theory is incomplete, and the result of measurements depends on predetermined "hidden variables". The state of the particles being measured contains some

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satisfying certain restrictions on the uncertainty of spin measurements, which are necessarily entangled. Spin squeezed states are good candidates for enhancing precision measurements using quantum entanglement.

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As mentioned above, a state of a quantum system is given by a unit vector in a Hilbert space. More generally, if one has less information about the system, then one calls it an 'ensemble' and describes it by a

7944:" after the ancient Chinese philosopher. The satellite was intended to demonstrate the feasibility of quantum communication between Earth and space, and test quantum entanglement over unprecedented distances. 7039: 7979:

measured entanglement at the highest energy so far, a rare intersection between quantum information and high energy physics based on theoretical work first proposed in 2021. The experiment was carried by the

6901: 2688: 2604: 1324:-zero particle could decay into a pair of spin-1/2 particles. Since the total spin before and after this decay must be zero (conservation of angular momentum), whenever the first particle is measured to be 1218:

polarizers with higher probability than classically predicted but with correlations in quantitative agreement with quantum mechanical calculations. He also showed that the correlation varied as the squared

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Later, however, the counterintuitive predictions of quantum mechanics were verified in tests where polarization or spin of entangled particles were measured at separate locations, statistically violating

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Schrödinger shortly thereafter published a seminal paper defining and discussing the notion of "entanglement." In the paper, he recognized the importance of the concept, and stated: "I would not call

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is first. Therefore, the correlation between the two measurements cannot be explained as one measurement determining the other: different observers would disagree about the role of cause and effect.

5602: 1894:, but can always be described using local hidden variables. Moreover, it was shown that, for arbitrary numbers of particles, there exist states that are genuinely entangled but admit a local model. 4801: 4337: 3015: 10770:

Bennett, Charles H.; DiVincenzo, David P.; Fuchs, Christopher A.; Mor, Tal; Rains, Eric; Shor, Peter W.; Smolin, John A.; Wootters, William K. (1999). "Quantum nonlocality without entanglement".

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between two pairs of entangled photons after measuring the polarization of one photon of the early pair, and that it proves that quantum non-locality applies not only to space but also to time.

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lines of thought." Like Einstein, Schrödinger was dissatisfied with the concept of entanglement, because it seemed to violate the speed limit on the transmission of information implicit in the

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The plot of von Neumann entropy Vs Eigenvalue for a bipartite 2-level pure state. When the eigenvalue has value 0.5, von Neumann entropy is at a maximum, corresponding to maximum entanglement.

3000: 2931: 1372:, it is not possible for any information to travel between two such measuring events. It is not even possible to say which of the measurements came first. For two spacelike separated events 6449:

of reduced states is the unique measure of entanglement in the sense that it is the only function on the family of states that satisfies certain axioms required of an entanglement measure.

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Ma, Xiao-song; Zotter, Stefan; Kofler, Johannes; Ursin, Rupert; Jennewein, Thomas; Brukner, Časlav; Zeilinger, Anton (26 April 2012). "Experimental delayed-choice entanglement swapping".

5218:, where it has full rank. It was proved that in the thermodynamic limit, the spectrum of the reduced density matrix of a large block of spins is an exact geometric sequence in this case. 3365: 3309: 6494: 5438: 12785:

Gurvits, L., Classical deterministic complexity of Edmonds' problem and quantum entanglement, in Proceedings of the 35th ACM Symposium on Theory of Computing, ACM Press, New York, 2003.

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A different entanglement classification is based on what the quantum correlations present in a state allow A and B to do: one distinguishes three subsets of entangled states: (1) the

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published a paper on the counterintuitive predictions that quantum mechanics makes for pairs of objects prepared together in a particular way. In this study, the three formulated the

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Bhaskara VS, Panigrahi PK (2017). "Generalized concurrence measure for faithful quantification of multiparticle pure state entanglement using Lagrange's identity and wedge product".

4932: 1145:) "to describe the correlations between two particles that interact and then separate, as in the EPR experiment." However, Schrödinger had discussed the phenomenon as early as 1932. 5549: 5509: 5392: 5352: 3674: 3635: 13879:

Measurements of polarization and spin correlation and observation of entanglement in top quark pairs using lepton+jets events from proton-proton collisions at $ \sqrt{s}$ = 13 TeV

8209: 3995: 3587: 8104: 8025: 5727: 2800: 2019: 7594: 7338: 7207: 4709: 2256: 2208: 2092: 2057: 7809: 7877:-mode Gaussian states (see Ref. for a seemingly different but essentially equivalent approach). It was later found that Simon's condition is also necessary and sufficient for 8065: 5604:

is possible. A very important (and very coarse) classification is based on the property whether it is possible to transform an arbitrarily large number of copies of a state

1687: 4227: 4195: 4154: 4122: 1659: 5466: 1198:, and showed that quantum theory predicts violations of this limit for certain entangled systems. His inequality is experimentally testable, and there have been numerous 7627:

Entanglement is usually created by direct interactions between subatomic particles. These interactions can take numerous forms. One of the most commonly used methods is

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of wave functions, but it is not the same Hilbert space as in the nonrelativistic case. This Hamiltonian no longer determines the evolution of the system because the

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Gerhardt, I.; Liu, Q.; Lamas-Linares, A.; Skaar, J.; Scarani, V.; Makarov, V.; Kurtsiefer, C. (2011), "Experimentally faking the violation of Bell's inequalities",

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can become entangled through various types of interactions. For some ways in which entanglement may be achieved for experimental purposes, see the section below on

7929:-mode Gaussian states. Simon's condition can be generalized by taking into account the higher order moments of canonical operators or by using entropic measures. 7685: 7390: 7364: 1805:, ceases to be valid. This property is known as timelessness. Various attempts to incorporate time in a fully quantum framework have been made, starting with the 1733: 5062:

This demonstrates that, as expected, the reduced density matrix for an entangled pure ensemble is a mixed ensemble. Also not surprisingly, the density matrix of

3589:. When a mixed state has rank 1, it therefore describes a 'pure ensemble'. When there is less than total information about the state of a quantum system we need 2750: 2457: 5314:

in the same SLOCC class are equally powerful (since I can transform one into the other and then do whatever it allows me to do), but since the transformations

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in particular, but produced relatively little other published work. Despite the interest, the weak point in EPR's argument was not discovered until 1964, when

8325:"Black Holes May Hide a Mind-Bending Secret About Our Universe – Take gravity, add quantum mechanics, stir. What do you get? Just maybe, a holographic cosmos" 8106:), so the spin information is transferred without much loss to the leptonic decays products that will be caught by the detector. Entaglement was tested with 14243:""Schrödinger's Bacterium" Could Be a Quantum Biology Milestone – A recent experiment may have placed living organisms in a state of quantum entanglement" 9397: 9882:

Megidish, E.; Halevy, A.; Shacham, T.; Dvir, T.; Dovrat, L.; Eisenberg, H. S. (2013). "Entanglement Swapping between Photons that have Never Coexisted".

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Lo Franco, Rosario; Compagno, Giuseppe (14 June 2018). "Indistinguishability of Elementary Systems as a Resource for Quantum Information Processing".

6351: 1187:, as applied to the kind of hidden variables interpretation hoped for by EPR, was mathematically inconsistent with the predictions of quantum theory. 9558: 1453:

reference frames, in which each measurement (in its own relativistic time frame) occurs before the other, the measurement results remain correlated.

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inequality for a set of fixed angles. All these experiments have shown agreement with quantum mechanics rather than the principle of local realism.

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Wineland, D. J.; Bollinger, J. J.; Itano, W. M.; Moore, F. L.; Heinzen, D. J. (1992). "Spin squeezing and reduced quantum noise in spectroscopy".

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The fundamental issue about measuring spin along different axes is that these measurements cannot have definite values at the same time―they are

1332:. (This is called the spin anti-correlated case; and if the prior probabilities for measuring each spin are equal, the pair is said to be in the 1816:

of time was also proposed as arising from quantum correlations between an evolving system and a reference quantum clock system, the concept of

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Physicist John Bell depicts the Einstein camp in this debate in his article entitled "Bertlmann's socks and the nature of reality", p. 143 of

7674:. Two independently prepared, identical particles may also be entangled if their wave functions merely spatially overlap, at least partially. 5856: 14874: 8130:

and a disparate distant spin system of a cloud of atoms. Later work complemented this work by quantum-entangling two mechanical oscillators.

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Bell's work raised the possibility of using these super-strong correlations as a resource for communication. It led to the 1984 discovery of

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The EPR paper generated significant interest among physicists, which inspired much discussion about the foundations of quantum mechanics and

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of that particle and changes the original quantum state. With entangled particles, such measurements affect the entangled system as a whole.

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Lee, K. C.; Sprague, M. R.; Sussman, B. J.; Nunn, J.; et al. (2 December 2011). "Entangling macroscopic diamonds at room temperature".

14242: 13679: 7169:{\displaystyle |\Psi ^{\pm }\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |1\rangle _{B}\pm |1\rangle _{A}\otimes |0\rangle _{B}).} 5735: 4232:

In general, finding out whether or not a mixed state is entangled is considered difficult. The general bipartite case has been shown to be

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Walborn, S.; Taketani, B.; Salles, A.; Toscano, F.; de Matos Filho, R. (2009). "Entropic Entanglement Criteria for Continuous Variables".

7028:{\displaystyle |\Phi ^{\pm }\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}\otimes |0\rangle _{B}\pm |1\rangle _{A}\otimes |1\rangle _{B})} 5203:

Reduced density matrices were explicitly calculated in different spin chains with unique ground state. An example is the one-dimensional

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quantum imaging technique could find application where low light imaging is imperative, in fields such as biological or medical imaging.

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into two arms leading to a beam splitter. They are the sum of multiple of NOON states, and can be used to achieve the Heisenberg limit.

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Hensen, B.; et al. (21 October 2015). "Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres".

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Hofmann, Holger F.; Takeuchi, Shigeki (22 September 2003). "Violation of local uncertainty relations as a signature of entanglement".

6807: 2609: 1273:"for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science". 11671:

Wiseman, H. M.; Jones, S. J.; Doherty, A. C. (2007). "Steering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen Paradox".

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Augusiak, R.; Demianowicz, M.; Tura, J.; Acín, A. (2015). "Entanglement and nonlocality are inequivalent for any number of parties".

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are unit vectors, and in the infinite-dimensional case, we would take the closure of such states in the trace norm. We can interpret

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direction. Generally, this is a mixed ensemble, as there can be any number of populations, each corresponding to a different state.

14723: 9943: 5640:, which produce correlations that cannot be explained by a local hidden variable model and thus violate a Bell inequality, (2) the 2100: 1540: 148: 13095:

Friis, Nicolai; Vitagliano, Giuseppe; Malik, Mehul; Huber, Marcus (2019). "Entanglement certification from theory to experiment".

9965: 7551:{\displaystyle |\psi _{\text{NOON}}\rangle ={\frac {|N\rangle _{a}|0\rangle _{b}+|{0}\rangle _{a}|{N}\rangle _{b}}{\sqrt {2}}},\,} 5069: 15409: 8653: 6872: 5656:

In this section, the entropy of a mixed state is discussed as well as how it can be viewed as a measure of quantum entanglement.

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arises as an emergent phenomenon of the quantum degrees of freedom that are entangled and live in the boundary of the spacetime.

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Vértesi, Tamás; Brunner, Nicolas (2014). "Disproving the Peres conjecture by showing Bell nonlocality from bound entanglement".

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It is a classical result that the Shannon entropy achieves its maximum at, and only at, the uniform probability distribution {1/

15093: 14666: 14641:"Spooky Actions at a Distance?": Oppenheimer Lecture, Prof. David Mermin (Cornell University) Univ. California, Berkeley, 2008. 8269: 7628: 6770:

Entanglement measures quantify the amount of entanglement in a (often viewed as a bipartite) quantum state. As aforementioned,

377: 36: 4916:{\displaystyle {\tfrac {1}{\sqrt {2}}}\left(|0\rangle _{A}\otimes |1\rangle _{B}-|1\rangle _{A}\otimes |0\rangle _{B}\right),} 4352: 3130:{\displaystyle {\tfrac {1}{\sqrt {2}}}\left(|0\rangle _{A}\otimes |1\rangle _{B}-|1\rangle _{A}\otimes |0\rangle _{B}\right).} 1258:. Although BB84 does not use entanglement, Ekert's protocol uses the violation of a Bell's inequality as a proof of security. 1121:

description of physical reality given by wave functions is not complete." However, the three scientists did not coin the word

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Yin, Juan; Cao, Yuan; Yong, Hai-Lin; Ren, Ji-Gang; et al. (2013). "Bounding the speed of 'spooky action at a distance".

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Extending the definition of separability from the pure case, we say that a mixed state is separable if it can be written as

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states. However, it remains an open question whether all entangled states become non-local given sufficiently many copies.

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Duan, Lu-Ming; Giedke, G.; Cirac, J. I.; Zoller, P. (2000). "Inseparability Criterion for Continuous Variable Systems".

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Georgiev, Danko D.; Gudder, Stanley P. (2022). "Sensitivity of entanglement measures in bipartite pure quantum states".

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Brunner, Nicolas; Cavalcanti, Daniel; Pironio, Stefano; Scarani, Valerio; Wehner, Stephanie (2014). "Bell nonlocality".

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The electron shells of multi-electron atoms always consist of entangled electrons. The correct ionization energy can be

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in the sense of statistical mechanics (comparing the two definitions in the present context, it is customary to set the

1368:, hence, any causal effect connecting the events would have to travel faster than light. According to the principles of 7937: 6300: 5955: 2943: 2874: 1291:, of products of states of local constituents; it is entangled if this sum cannot be written as a single product term. 1012: 7670:

It is also possible to create entanglement between quantum systems that never directly interacted, through the use of

1339:

The above result may or may not be perceived as surprising. A classical system would display the same property, and a

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Thomas, Rodrigo A.; Parniak, Michał; Østfeldt, Christoffer; Møller, Christoffer B.; et al. (21 September 2020).

13848:

Observation of quantum entanglement in top quark pair production in proton-proton collisions at $ \sqrt{s}$ = 13 TeV

11917: 9992: 9541: 9517: 9212:"Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's Inequalities" 8179: 8127: 6813:

Most (but not all) of these entanglement measures reduce for pure states to entanglement entropy, and are difficult (

143: 14646: 13602: 10593:

Werner, R. F. (1989). "Quantum States with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model".

6557:{\displaystyle {\begin{bmatrix}{\frac {1}{n}}&&\\&\ddots &\\&&{\frac {1}{n}}\end{bmatrix}}.} 3707: 2805: 14806: 10999:

Horodecki M, Horodecki P, Horodecki R (1996). "Separability of mixed states: necessary and sufficient conditions".

9286:

Hanson, Ronald (2015). "Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres".

7671: 6212:{\displaystyle S(\rho )=-{\hbox{Tr}}\left(\rho \log _{2}{\rho }\right)=-\sum _{i}\lambda _{i}\log _{2}\lambda _{i}} 1568: 1482: 232: 14325: 8294: 3317: 3261: 1364:

The distance and timing of the measurements can be chosen so as to make the interval between the two measurements

15364: 15073: 15068: 14791: 12369: 12043:

Bouwmeester, Dik; Pan, Jian-Wei; Mattle, Klaus; Eibl, Manfred; Weinfurter, Harald & Zeilinger, Anton (1997).

8324: 8199: 8169: 8107: 7933: 6792: 6753: 5398: 1536: 1040: 890: 339: 319: 187: 14527:

Swain SN, Bhaskara VS, Panigrahi PK (2022). "Generalized entanglement measure for continuous-variable systems".

7988:

of level of significance, the top quark is the heaviest known particle and therefore has a very short lifetime (

5195:

In general, a bipartite pure state ρ is entangled if and only if its reduced states are mixed rather than pure.

5052:{\displaystyle \rho _{A}={\tfrac {1}{2}}\left(|0\rangle _{A}\langle 0|_{A}+|1\rangle _{A}\langle 1|_{A}\right).} 2462: 15376: 15048: 12629:

Hardy, Lucien (1992). "Quantum mechanics, local realistic theories, and Lorentz-invariant realistic theories".

12471:

Holland, M. J; Burnett, K (1993). "Interferometric detection of optical phase shifts at the Heisenberg limit".

11170:

Franchini, F.; Its, A. R.; Korepin, V. E.; Takhtajan, L. A. (2010). "Spectrum of the density matrix of a large

9394: 8520: 5511:) one can define classes based on the possibility of multi-copy transformations. E.g., there are examples when 5252:

If two states can be transformed into each other by a local unitary operation, they are said to be in the same

3205: 1500: 579: 309: 9636:

Santos, E. (2004). "The failure to perform a loophole-free test of Bell's Inequality supports local realism".

6229: 5207:: the ground state can be divided into a block and an environment. The reduced density matrix of the block is 15354: 15131: 15053: 14836: 14706: 14022:

Lépinay, Laure Mercier de; Ockeloen-Korppi, Caspar F.; Woolley, Matthew J.; Sillanpää, Mika A. (7 May 2021).

11553:

Gour, G.; Wallach, N. R. (2013). "Classification of Multipartite Entanglement of All Finite Dimensionality".

8234: 8111: 7815: 7640: 5208: 1690: 619: 357: 257: 5514: 5474: 5357: 5317: 3685:

Following the definition above, for a bipartite composite system, mixed states are just density matrices on

3646: 3607: 1535:. In standard quantum theories time acts as an independent background through which states evolve, with the 1357:) and hence assured the "proper" outcome of the measurement of the other part of the entangled pair. In the 15088: 15022: 15017: 14988: 14701: 11614:

Horodecki, M.; Horodecki, P.; Horodecki, R. (1998). "Mixed-state entanglement and distillation: Is there a

9097:

The Born-Einstein Letters; Correspondence between Albert Einstein and Max and Hedwig Born from 1916 to 1955

8279: 7295:{\displaystyle |\mathrm {GHZ} \rangle ={\frac {|0\rangle ^{\otimes M}+|1\rangle ^{\otimes M}}{\sqrt {2}}},} 3596:

Experimentally, a mixed ensemble might be realized as follows. Consider a "black box" apparatus that spits

3410: 1436: 555: 550: 521: 372: 153: 11330:

Horodecki, Ryszard; Horodecki, Pawel; Horodecki, Michal; Horodecki, Karol (2009). "Quantum entanglement".

8923: 3557: 1914:, whereas non-locality is defined according to experimental statistics and is much more involved with the 15156: 15063: 8070: 7991: 6571: 5686: 3937: 2755: 1981: 1559: 1169: 1027:

dispute that there is any "effect" at all. However, all interpretations agree that entanglement produces

589: 334: 324: 13907: 12954:

Gühne, Otfried; Lewenstein, Maciej (24 August 2004). "Entropic uncertainty relations and entanglement".

7564: 7308: 4662: 2220: 2172: 2062: 2027: 15432: 15194: 15002: 14973: 10948:

Gurvits, L. (2003). "Classical deterministic complexity of Edmonds' Problem and quantum entanglement".

10872: 7775: 6842: 5215: 1176: 883: 535: 506: 12269:

Renner, R.; Gisin, N.; Kraus, B. (2005). "An information-theoretic security proof for QKD protocols".

12103:

Jozsa, Richard; Linden, Noah (2002). "On the role of entanglement in quantum computational speed-up".

7682:

A density matrix ρ is called separable if it can be written as a convex sum of product states, namely

1039:

is impossible. Thus, despite popular thought to the contrary, quantum entanglement cannot be used for

15216: 15083: 15007: 14968: 14925: 14899: 14856: 14749: 12044: 8214: 8204: 8184: 7984:

detector measuring the spin of top-quark pair production and the effect was observed with more than 5

7960: 7844: 6787: 6782: 6585: 6292: 5933: 4245: 1899: 1555: 1358: 540: 501: 454: 429: 352: 212: 12135: 9389:

C. H. Bennett and G. Brassard. "Quantum cryptography: Public key distribution and coin tossing". In

8034: 2159:

States of the composite system that can be represented in this form are called separable states, or

1664: 15283: 15263: 15253: 15243: 15199: 14774: 13384:

Shchukin, E.; Vogel, W. (2005). "Inseparability Criteria for Continuous Bipartite Quantum States".

12672:

Georgiev, Danko; Cohen, Eliahu (2022). "Entanglement measures for two-particle quantum histories".

11031: 9115: 8249: 8244: 7814:

For 2-Qubit and Qubit-Qutrit systems (2 × 2 and 2 × 3 respectively) the simple

7408: 6867: 6797: 6567:

For mixed states, the reduced von Neumann entropy is not the only reasonable entanglement measure.

4200: 4168: 4127: 4095: 3393: 1626: 1547: 1457: 1235: 624: 13567:

Huang, Yichen (October 2013). "Entanglement Detection: Complexity and Shannon Entropic Criteria".

13007:

Huang, Yichen (29 July 2010). "Entanglement criteria via concave-function uncertainty relations".

11733: 14978: 14904: 7180: 6771: 5443: 4342:

As indicated above, in general there is no way to associate a pure state to the component system

1227: 793: 511: 419: 15392: 14940: 14102:

Kotler, Shlomi; Peterson, Gabriel A.; Shojaee, Ezad; Lecocq, Florent; et al. (7 May 2021).

11383:

Grassl, M.; Rötteler, M.; Beth, T. (1998). "Computing local invariants of quantum-bit systems".

10519:

Swingle, Brian; Van Raamsdonk, Mark (12 May 2014). "Universality of Gravity from Entanglement".

9475: 5664: 1727: 1696: 469: 367: 133: 15303: 15078: 15058: 14983: 14851: 12130: 11026: 9792:

Wu, C. 's.; Shaknov, I. (1950). "The Angular Correlation of Scattered Annihilation Radiation".

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turns out to be nothing more than the operator with the same eigenvectors, but the eigenvalues

5290: 5263: 2167: 1829: 1820:

is introduced as a quantifier of the actual distinguishable evolution undergone by the system.

1551: 1425: 1340: 1075: 1023:, the effect of one measurement occurs instantly. Other interpretations which do not recognize 965: 798: 516: 344: 314: 277: 14894: 13148:

Leinaas, Jon Magne; Myrheim, Jan; Ovrum, Eirik (2006). "Geometrical aspects of entanglement".

9735:

Some of the history of both referenced Zbinden, et al. experiments is provided in Gilder, L.,

5846:

is a probability distribution over an ensemble, this leads naturally to the definition of the

1035:

between the entangled particles can be exploited, but that any transmission of information at

424: 15328: 14841: 14821: 8274: 8254: 7972: 7932:

On August 16, 2016, the world's first quantum communications satellite was launched from the

7656: 6802: 5244:

Not all quantum states are equally valuable as a resource. To quantify this value, different

1907: 1864: 1461: 1445: 1288: 1199: 1184: 1055: 1024: 930:: entanglement is a primary feature of quantum mechanics not present in classical mechanics. 267: 252: 39:

process can split photons into type II photon pairs with mutually perpendicular polarization.

14634: 13807: 13201:

Simon, R. (2000). "Peres-Horodecki Separability Criterion for Continuous Variable Systems".

9251:

Kocher, C. A. (1971). "Time correlations in the detection of successively emitted photons".

8411:

Schrödinger, Erwin (1935). "Discussion of probability relations between separated systems".

7631:

to generate a pair of photons entangled in polarization. Other methods include the use of a

5625:

as a surprise that not all entangled states are distillable, those that are not are called '

4229:

are themselves pure ensembles. A state is then said to be entangled if it is not separable.

1798:{\displaystyle {\hat {H}}|\psi \rangle =i\hbar {\partial \over {\partial t}}|\psi \rangle } 964:

effects: any measurement of a particle's properties results in an apparent and irreversible

15359: 15288: 15233: 14963: 14744: 14546: 14499: 14452: 14292: 14205: 14125: 14045: 13525: 13464: 13403: 13342: 13281: 13220: 13167: 13061: 13016: 12973: 12912: 12859: 12744: 12691: 12638: 12587: 12534: 12480: 12437: 12384: 12334: 12227: 12175: 12122: 12069: 12009: 11865: 11812: 11772: 11690: 11637: 11572: 11519: 11458: 11402: 11349: 11291: 11238: 11128: 11081: 11018: 10915: 10833: 10789: 10726: 10657: 10604: 10559: 10477: 10416: 10363: 10306: 10241: 10184: 10131: 10092: 9901: 9848: 9803: 9758: 9702: 9645: 9595: 9425: 9362: 9305: 9260: 9223: 9182: 9065: 8995: 8842: 8773: 8573: 8463: 8420: 8380: 8189: 6863: 6830: 5607: 4741: 4714: 3530: 1932: 1460:

in the sense that these measurements' maximum simultaneous precision is constrained by the

1441: 1191: 1161: 1156:

characteristic trait of quantum mechanics, the one that enforces its entire departure from

1011:. In earlier tests, it could not be ruled out that the result at one point could have been 996: 953: 933: 584: 496: 179: 10458:

Lee, Jae-Weon; Kim, Hyeong-Chan; Lee, Jungjai (2013). "Gravity from quantum information".

10165:

Moreva, Ekaterina (2014). "Time from quantum entanglement: an experimental illustration".

6884:

There are several canonical entangled states that appear often in theory and experiments.

1130: 984: 828: 8: 15344: 15313: 15258: 15238: 15146: 15103: 14958: 14884: 14811: 14801: 14713: 14330: 14247: 12893:

Gühne, Otfried (18 March 2004). "Characterizing Entanglement via Uncertainty Relations".

11753:

Plenio, Martin B.; Virmani, Shashank (2007). "An introduction to entanglement measures".

8174: 7824: 7644: 7369: 7343: 5847: 5635: 4407: 3601: 3153: 3140:

If the composite system is in this state, it is impossible to attribute to either system

2386:{\displaystyle |\psi \rangle _{AB}=\sum _{i,j}c_{ij}|i\rangle _{A}\otimes |j\rangle _{B}} 1915: 1853: 1558:, the quantum version of the Hamiltonian constraint using metric variables, leads to the 1329: 1325: 1303: 1157: 937: 683: 491: 409: 237: 217: 169: 14550: 14503: 14456: 14360: 14296: 14209: 14129: 14049: 13529: 13468: 13445:

Hillery, Mark; Zubairy, M.Suhail (2006). "Entanglement Conditions for Two-Mode States".

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Fan, H; Korepin V; Roychowdhury V (2004). "Entanglement in a Valence-Bond Solid State".

11085: 11022: 10919: 10837: 10793: 10730: 10661: 10608: 10563: 10481: 10420: 10367: 10310: 10245: 10188: 10135: 10096: 9905: 9852: 9807: 9762: 9706: 9649: 9599: 9429: 9391:

Proceedings of IEEE International Conference on Computers, Systems and Signal Processing

9366: 9309: 9264: 9227: 9186: 9149: 9132: 9101:

Hobson, M. P.; et al. (1998). "Quantum Entanglement and Communication Complexity".

9069: 8999: 8846: 8777: 8577: 8467: 8424: 8384: 7663:. In the earliest tests of Bell's theorem, the entangled particles were generated using 15404: 15273: 15171: 14879: 14826: 14718: 14562: 14536: 14515: 14489: 14468: 14442: 14282: 14223: 14195: 14157: 14115: 14077: 14035: 13971: 13945: 13754: 13641: 13584: 13549: 13515: 13488: 13454: 13427: 13393: 13366: 13332: 13305: 13271: 13244: 13210: 13183: 13157: 13130: 13104: 13077: 13051: 12989: 12963: 12936: 12902: 12875: 12849: 12822: 12804: 12768: 12734: 12707: 12681: 12611: 12577: 12550: 12524: 12350: 12324: 12296: 12278: 12251: 12199: 12148: 12112: 12085: 12059: 12025: 11999: 11954: 11881: 11855: 11828: 11802: 11762: 11714: 11680: 11653: 11627: 11596: 11562: 11535: 11509: 11482: 11448: 11418: 11392: 11365: 11339: 11307: 11281: 11254: 11228: 11201: 11183: 11152: 11118: 11044: 11008: 10981: 10953: 10931: 10905: 10805: 10779: 10750: 10716: 10689: 10647: 10575: 10549: 10520: 10501: 10467: 10440: 10406: 10397:

Van Raamsdonk, Mark (19 June 2010). "Building up spacetime with quantum entanglement".

10379: 10353: 10322: 10296: 10265: 10231: 10200: 10174: 10010: 9969: 9925: 9891: 9864: 9838: 9774: 9718: 9692: 9661: 9619: 9585: 9457: 9353: 9329: 9295: 9055: 9052:

The Structure of Bipartite Quantum States – Insights from Group Theory and Cryptography

9019: 8950: 8874: 8832: 8631: 8618:

Matson, John (13 August 2012). "Quantum teleportation achieved over record distances".

8597: 8563: 8479: 8436: 8329: 8164: 7985: 7976: 7660: 6575: 5673: 5626: 3997:, and the vectors are unit vectors. This is self-adjoint and positive and has trace 1. 2693: 2400: 1833: 1532: 1450: 1369: 1310: 1118: 1114: 1090: 1032: 404: 329: 262: 174: 14343: 14184:"Entanglement between living bacteria and quantized light witnessed by Rabi splitting" 14103: 14023: 13823: 11942: 11040: 10341: 10284: 10219: 8938: 7635:

to confine and mix photons, photons emitted from decay cascade of the bi-exciton in a

15414: 15323: 15293: 15221: 15184: 15179: 15161: 15126: 15116: 14831: 14796: 14779: 14682: 14598: 14579: 14566: 14472: 14417: 14395: 14347: 14227: 14161: 14149: 14141: 14081: 14069: 14061: 13975: 13963: 13827: 13736: 13716: 13697: 13661: 13541: 13480: 13419: 13358: 13297: 13236: 13187: 13134: 13122: 13081: 12993: 12928: 12760: 12711: 12654: 12603: 12496: 12453: 12410: 12354: 12300: 12255: 12243: 12191: 12029: 11972: 11923: 11913: 11706: 11657: 11588: 11474: 11311: 11258: 11144: 10971: 10935: 10876: 10849: 10742: 10681: 10620: 10595: 10579: 10505: 10493: 10444: 10432: 10326: 10204: 10147: 10059: 10014: 9917: 9868: 9778: 9665: 9611: 9537: 9513: 9449: 9441: 9333: 9321: 9272: 9023: 9011: 8968: 8878: 8866: 8711: 8686: 8635: 8601: 8589: 8516: 8483: 8440: 8224: 8219: 7831: 7191: 6876:

security of QKD is shown exploiting entanglement between the communication partners.

6849: 6223:

Since an event of probability 0 should not contribute to the entropy, and given that

5228: 1911: 1528: 1495:

can be used to carry entangled states. The first loophole-free Bell test was held by

1489: 1354: 1317: 1180: 1059: 1036: 1008: 1000: 941: 923: 838: 813: 753: 748: 648: 614: 594: 192: 51: 14519: 13933: 13553: 13492: 13431: 13370: 13248: 12879: 12772: 12615: 12554: 12218:

Pan (2020). "Entanglement-based secure quantum cryptography over 1,120 kilometres".

12203: 12152: 11718: 11539: 11486: 11422: 11369: 11156: 11048: 10809: 10693: 10383: 10269: 10119: 10080: 9929: 9722: 9623: 9461: 4244:

cases, a necessary and sufficient criterion for separability is given by the famous

1194:, regarding the strength of correlations that can be produced in any theory obeying 15141: 15136: 14993: 14889: 14554: 14507: 14460: 14339: 14300: 14213: 14133: 14053: 13955: 13883: 13852: 13819: 13728: 13651: 13588: 13576: 13537: 13533: 13472: 13411: 13350: 13309: 13289: 13228: 13175: 13114: 13069: 13024: 12981: 12940: 12920: 12867: 12814: 12795:

Gharibian, Sevag (2010). "Strong NP-Hardness of the Quantum Separability Problem".

12756: 12752: 12699: 12646: 12595: 12542: 12488: 12445: 12400: 12392: 12342: 12288: 12235: 12183: 12140: 12105:

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

12089: 12077: 12017: 11964: 11885: 11873: 11832: 11820: 11698: 11645: 11600: 11584: 11580: 11527: 11470: 11466: 11410: 11357: 11299: 11246: 11205: 11193: 11136: 11089: 11036: 10985: 10963: 10923: 10841: 10797: 10754: 10734: 10673: 10669: 10665: 10612: 10567: 10485: 10424: 10371: 10314: 10257: 10249: 10192: 10139: 10100: 10000: 9913: 9909: 9856: 9811: 9766: 9710: 9681:"Experimental test of nonlocal quantum correlations in relativistic configurations" 9653: 9607: 9603: 9433: 9370: 9313: 9268: 9231: 9190: 9144: 9003: 8960: 8858: 8850: 8823: 8781: 8623: 8585: 8581: 8471: 8428: 8388: 7981: 6856: 6589: 5642: 3418: 1306:

through interaction with the environment; for example, when a measurement is made.

843: 833: 823: 723: 703: 688: 658: 526: 414: 14464: 14376: 13476: 13415: 13073: 12924: 12826: 12599: 11702: 11140: 6759: 6665:

Indeed, without this property, the von Neumann entropy would not be well-defined.

917:

being generated, interacting, or sharing spatial proximity in such a way that the

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Nielsen, M. A. (1999). "Conditions for a Class of Entanglement Transformations".

11250: 9794: 9401: 8356: 8284: 8264: 8239: 8229: 8139: 7820: 5680: 1841: 1492: 1477: 1295: 1270: 1247: 1219: 1203: 1134: 1098: 972: 927: 868: 738: 718: 464: 304: 14558: 14369: 13781:"LHC experiments at CERN observe quantum entanglement at the highest energy yet" 13354: 13293: 13232: 12703: 12650: 12492: 11877: 11649: 10824:

Jaeger G, Shimony A, Vaidman L (1995). "Two Interferometric Complementarities".

8369:"Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" 6418:{\displaystyle \rho \log _{2}\rho =\int \lambda \log _{2}\lambda dP_{\lambda }.} 15349: 15318: 15308: 14930: 14920: 14754: 14647:"Quantum Entanglement versus Classical Correlation" (Interactive demonstration) 14218: 14183: 13732: 13179: 13028: 12985: 12871: 12292: 12187: 11968: 11531: 10375: 10318: 10253: 10196: 9714: 9437: 9195: 8939:"On-demand entanglement of molecules in a reconfigurable optical tweezer array" 8785: 8761: 8360: 8320: 8300: 8259: 8115: 7664: 7404: 3406: 1973: 1391: 1328:

on some axis, the other, when measured on the same axis, is always found to be

1321: 1313: 1102: 976: 949: 803: 763: 743: 713: 693: 643: 609: 459: 449: 242: 14640: 14619: 14511: 14305: 14270: 14269:

Krisnanda, T.; Marletto, C.; Vedral, V.; Paternostro, M.; Paterek, T. (2018).

13959: 13887: 13856: 13118: 12568:

Akopian, N. (2006). "Entangled Photon Pairs from Semiconductor Quantum Dots".

12449: 12239: 11824: 11361: 11303: 11197: 11094: 10571: 10428: 10005: 9680: 9657: 9236: 9211: 8733:"No, We Still Can't Use Quantum Entanglement To Communicate Faster Than Light" 8627: 8475: 8454:

Schrödinger, Erwin (1936). "Probability relations between separated systems".

8432: 6817:) to compute for mixed states as the dimension of the entangled system grows. 991:. Einstein and others considered such behavior impossible, as it violated the 956:

performed on entangled particles can, in some cases, be found to be perfectly

15447: 15268: 15121: 15012: 14846: 14816: 14769: 14145: 14065: 13967: 13831: 13740: 13580: 13126: 12546: 12396: 11976: 11790: 11436: 11414: 10801: 10616: 10497: 10436: 10143: 10104: 9445: 8972: 8678: 8028: 7632: 6870:(QKD) under standard assumptions does not require entanglement. However, the 4415: 3499:{\displaystyle \rho =\sum _{i}w_{i}|\alpha _{i}\rangle \langle \alpha _{i}|,} 3417:

when the state space is infinite-dimensional, and has trace 1. Again, by the

2160: 1956: 1723: 1496: 1333: 1316:

into an entangled pair of other particles. The decay events obey the various

1287:

other(s). The state of a composite system is always expressible as a sum, or

1195: 1126: 1063: 992: 918: 863: 858: 788: 758: 728: 599: 545: 272: 247: 125:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle } 14137: 14057: 13656: 13629: 11927: 9395:

http://researcher.watson.ibm.com/researcher/files/us-bennetc/BB84highest.pdf

9170: 9007: 8964: 8805: 7765:{\displaystyle {\rho =\sum _{j}p_{j}\rho _{j}^{(A)}\otimes \rho _{j}^{(B)}}} 6845:. With the aid of entanglement, otherwise impossible tasks may be achieved. 1467: 1054:, top quarks, molecules and even small diamonds. The use of entanglement in 15151: 14764: 14759: 14409: 14351: 14153: 14073: 13717:"Observation of quantum entanglement with top quarks at the ATLAS detector" 13665: 13545: 13484: 13423: 13362: 13301: 13240: 12932: 12764: 12658: 12607: 12500: 12247: 12195: 12144: 11990:

Huang, Yichen (21 March 2014). "Computing quantum discord is NP-complete".

11941:

Wang, Yu-Xin; Mu, Liang-Zhu; Vedral, Vlatko; Fan, Heng (17 February 2016).

11846:

Hill, S; Wootters, W. K. (1997). "Entanglement of a Pair of Quantum Bits".

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Proceedings of the thirty-fifth annual ACM symposium on Theory of computing

10845: 10746: 10685: 10151: 10081:"Evolution without evolution: Dynamics described by stationary observables" 9921: 9815: 9615: 9453: 9325: 9015: 8870: 8593: 8393: 8368: 8364: 7811:

probabilities. By definition, a state is entangled if it is not separable.

7648: 1898:

longer be described by a local model. This is, in particular, true for all

1858: 1266: 1262: 1251: 1211: 1207: 1106: 980: 853: 848: 783: 768: 733: 227: 12818: 12457: 12414: 10967: 10853: 10624: 9375: 9348: 8138:

In October 2018, physicists reported producing quantum entanglement using

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Cirel'son, B. S. (1980). "Quantum generalizations of Bell's inequality".

9697: 9480: 9413: 9060: 7835: 7636: 7184: 5471:

Instead of considering transformations of single copies of a state (like

3414: 3202:. If in the entangled state given above Alice makes a measurement in the 3158: 1255: 1085: 1028: 988: 818: 773: 708: 663: 13628:

Yin, Juan; Cao, Yuan; Li, Yu-Huai; Liao, Sheng-Kai; et al. (2017).

12313: 12166:

Ekert, Artur K. (1991). "Quantum cryptography based on Bell's theorem".

11734:"Information-theoretic interpretation of quantum error-correcting codes" 10261: 9317: 8854: 7190:

of the Hilbert space of the two qubits. They play a fundamental role in

6570:

As an aside, the information-theoretic definition is closely related to

5922:{\displaystyle S(\rho )=-{\hbox{Tr}}\left(\rho \log _{2}{\rho }\right).} 13680:"China's quantum satellite achieves 'spooky action' at record distance" 12405: 12346: 11613: 11062: 10738: 10218:

Giovannetti, Vittorio; Lloyd, Seth; Maccone, Lorenzo (26 August 2015).

9770: 7612: 7420: 6892: 6087:{\displaystyle \log _{2}(\lambda _{1}),\cdots ,\log _{2}(\lambda _{n})} 5204: 4257: 4075:{\displaystyle \rho =\sum _{i}w_{i}\rho _{i}^{A}\otimes \rho _{i}^{B},} 3149: 957: 808: 778: 698: 673: 668: 653: 14576:

Entanglement, Information, and the Interpretation of Quantum Mechanics

14021: 13934:"Entanglement between distant macroscopic mechanical and spin systems" 13323:

Werner, R. F.; Wolf, M. M. (2001). "Bound Entangled Gaussian States".

11793:(2002). "The role of relative entropy in quantum information theory". 10927: 10677: 9860: 8862: 6287:

is adopted. This extends to the infinite-dimensional case as well: if

15043: 14739: 14595:

Quantum Mechanics: Its Early Development and the Road to Entanglement

8937:

Holland, Connor M.; Lu, Yukai; Cheuk, Lawrence W. (8 December 2023).

8194: 7179:

These four pure states are all maximally entangled (according to the

5832:{\displaystyle H(p_{1},\cdots ,p_{n})=-\sum _{i}p_{i}\log _{2}p_{i}.} 4156:'s are themselves mixed states (density operators) on the subsystems 3254:, there are two possible outcomes, occurring with equal probability: 2166:

Not all states are separable states (and thus product states). Fix a

1837: 1813: 1365: 299: 14651: 12216: 11329: 5211:

to a projector to a degenerate ground state of another Hamiltonian.

3388:

are spatially separated. This is the foundation of the EPR paradox.

14541: 14494: 14287: 14200: 14120: 14040: 13950: 13877: 13846: 13646: 13109: 12739: 12686: 12329: 12064: 11959: 11286: 11233: 10896:

Laloe, Franck (2001), "Do We Really Understand Quantum Mechanics",

10358: 10301: 10283:

Boette, A.; Rossignoli, R.; Gigena, N.; Cerezo, M. (27 June 2016).

10236: 9300: 8955: 8837: 8143: 8126:

In 2020, researchers reported the quantum entanglement between the

7652: 6855:

Most researchers believe that entanglement is necessary to realize

4346:. However, it still is possible to associate a density matrix. Let 3597: 3376:

has been altered by Alice performing a local measurement on system

1051: 945: 914: 678: 14447: 14268: 13520: 13056: 12809: 12529: 12081: 12004: 11567: 11453: 11219:

Chitambar, Eric; Gour, Gilad (2019). "Quantum resource theories".

11188: 10721: 10652: 10554: 10525: 10472: 10411: 10179: 9896: 9843: 9590: 8762:"Polarization Correlation of Photons Emitted in an Atomic Cascade" 8568: 3516:

are positive-valued probabilities (they sum up to 1), the vectors

3161:, which are (maximally) entangled pure states (pure states of the 1080: 14392:

The Quantum Handshake: Entanglement, Nonlocality and Transactions

13755:"ATLAS achieves highest-energy detection of quantum entanglement" 11439:(2010). "Local unitary equivalence of multipartite pure states". 8899:"Sorry, Einstein. Quantum Study Suggests 'Spooky Action' Is Real" 8683:

The road to reality: a complete guide to the laws of the universe

6814: 6429: 4233: 2683:{\textstyle |\phi \rangle _{B}=\sum _{j}c_{j}^{B}|j\rangle _{B}.} 961: 13808:"Entanglement and quantum tomography with top quarks at the LHC" 13630:"Satellite-based entanglement distribution over 1200 kilometers" 8807:

Polarization Correlation of Photons Emitted in an Atomic Cascade

8210:

Normally distributed and uncorrelated does not imply independent

2599:{\textstyle |\psi \rangle _{A}=\sum _{i}c_{i}^{A}|i\rangle _{A}} 11909: 11074:

Mathematical Proceedings of the Cambridge Philosophical Society

10539: 9559:"Cosmic Test Bolsters Einstein's "Spooky Action at a Distance"" 8456:

Mathematical Proceedings of the Cambridge Philosophical Society

8413:

Mathematical Proceedings of the Cambridge Philosophical Society

8147: 7941: 6888: 5952:

acts on a finite-dimensional Hilbert space and has eigenvalues

1047: 1046:

Quantum entanglement has been demonstrated experimentally with

14104:"Direct observation of deterministic macroscopic entanglement" 14024:"Quantum mechanics–free subsystem with mechanical oscillators" 11382: 8142:, particularly between photosynthetic molecules within living 6738:{\displaystyle U(t)=\exp \left({\frac {-iHt}{\hbar }}\right),} 5184:{\displaystyle \rho _{A}=|\psi \rangle _{A}\langle \psi |_{A}} 3177:

space, but which cannot be separated into pure states of each

2868:

If a state is inseparable, it is called an 'entangled state'.

1514: 32: 14326:"Was Einstein Wrong?: A Quantum Threat to Special Relativity" 13908:"Quantum entanglement realized between distant large objects" 13042:

Gühne, Otfried; Tóth, Géza (2009). "Entanglement detection".

9575: 8133: 7416: 7393: 2149:{\displaystyle |\psi \rangle _{A}\otimes |\phi \rangle _{B}.} 1550:

which relates directly with matter and moreover requires the

14643:

Non-mathematical popular lecture on YouTube, posted Mar 2008

13505: 11169: 10282: 5115:{\displaystyle |\psi \rangle _{A}\otimes |\phi \rangle _{B}} 1488:

In three independent experiments in 2013, it was shown that

987:

shortly thereafter, describing what came to be known as the

14182:

Marletto, C.; Coles, D. M.; Farrow, T.; Vedral, V. (2018).

14101: 13931: 12427: 10998: 10637: 8512: 7596:

except the basis kets 0 and 1 have been replaced with "the

5233: 3314:

Alice measures 1, and the state of the system collapses to

3258:

Alice measures 0, and the state of the system collapses to

1239: 13806:

Afik, Yoav; de Nova, Juan Ramón Muñoz (3 September 2021).

13088: 12513: 11108: 8114:

and subsequently the effect has been confirmed too in the

6833:

is sometimes seen as an analogue of quantum entanglement.

6672:

could be the time evolution operator of the system, i.e.,

14358: 14181: 13094: 10769: 9881: 9210:

Aspect, Alain; Grangier, Philippe; Roger, Gérard (1982).

7655:, can be created by partial overlap of the corresponding 1476:(entanglement) was successfully corroborated in a lab by 1468:

Notable experimental results proving quantum entanglement

13997:"Vibrating drumheads are entangled quantum mechanically" 13603:"China launches world's first quantum science satellite" 12042: 10814:

This non-standard use of the term is not discussed here.

10036: 9416:(1991). "Quantum cryptography based on Bell's theorem". 9349:"Closing the Door on Einstein and Bohr's Quantum Debate" 9054:(PhD thesis). University of Cambridge. pp. vi, iv. 8654:"Quantum entanglement shows that reality can't be local" 4395:{\displaystyle \rho _{T}=|\Psi \rangle \;\langle \Psi |} 1190:

Specifically, Bell demonstrated an upper limit, seen in

1125:, nor did they generalize the special properties of the 14629:

Entanglement experiment with photon pairs – interactive

10217: 5597:{\displaystyle \rho _{1}\otimes \rho _{1}\to \rho _{2}} 4256:

The idea of a reduced density matrix was introduced by

3600:

towards an observer. The electrons' Hilbert spaces are

995:

view of causality (Einstein referring to it as "spooky

14526: 14271:"Probing quantum features of photosynthetic organisms" 13713: 11552: 9536:(First ed.). The Cosmic Compendium. p. 100. 8121: 6848:

Among the best-known applications of entanglement are

6503: 6123: 5879: 4950: 4806: 4620: 3940: 3020: 2612: 2531: 1503:

in 2015, confirming the violation of Bell inequality.

1302:. Entanglement is broken when the entangled particles 13261: 9171:"Experimental Test of Local Hidden-Variable Theories" 8985: 8355: 8073: 8037: 8027:) being the only quark that decays before undergoing 7994: 7909: 7883: 7857: 7778: 7688: 7611:

for bosonic modes, which can be created by feeding a

7567: 7432: 7372: 7346: 7311: 7210: 7042: 6904: 6681: 6604: 6497: 6354: 6303: 6232: 6103: 6016: 5958: 5859: 5738: 5689: 5610: 5557: 5517: 5477: 5446: 5401: 5360: 5320: 5293: 5266: 5131: 5072: 4935: 4804: 4744: 4717: 4665: 4438: 4355: 4332:{\displaystyle |\Psi \rangle \in H_{A}\otimes H_{B}.} 4288: 4203: 4171: 4130: 4098: 4009: 3710: 3649: 3610: 3560: 3533: 3430: 3320: 3264: 3208: 3018: 2946: 2877: 2808: 2758: 2696: 2465: 2403: 2290: 2223: 2175: 2103: 2065: 2030: 1984: 1867: 1736: 1699: 1667: 1629: 1571: 1299: 63: 10823: 10518: 8290: 7903:-mode Gaussian states, but no longer sufficient for 7830:

A numerical approach to the problem is suggested by

7677: 5440:) and the separable ones (i.e., product states like 1519:

There is a fundamental conflict, referred to as the

13147: 11670: 10285:"System-time entanglement in a discrete-time model" 10079:Page, Don N.; Wootters, William K. (15 June 1983). 10054:Sakurai, J. J.; Napolitano, Jim J. (14 July 2010). 9828: 9209: 8705: 7622: 6655:{\displaystyle S(\rho )=S\left(U\rho U^{*}\right).} 3554:is the proportion of the ensemble whose states are 1972:. The Hilbert space of the composite system is the 1261:In 2022, the Nobel Prize in Physics was awarded to 1164:. Einstein later famously derided entanglement as " 971:Such phenomena were the subject of a 1935 paper by 14431:Gühne O, Tóth G (2009). "Entanglement detection". 9678: 9084:Philosophy of Quantum Information and Entanglement 8128:motion of a millimeter-sized mechanical oscillator 8098: 8059: 8019: 7921: 7895: 7869: 7803: 7764: 7588: 7550: 7384: 7358: 7332: 7294: 7168: 7027: 6737: 6654: 6556: 6436:(which can be shown to be the maximum entropy for 6417: 6334: 6272: 6211: 6086: 5990: 5921: 5831: 5721: 5616: 5596: 5543: 5503: 5460: 5432: 5386: 5346: 5306: 5279: 5214:The reduced density matrix also was evaluated for 5183: 5114: 5051: 4915: 4772:is sometimes called the reduced density matrix of 4757: 4730: 4703: 4648: 4394: 4331: 4221: 4189: 4148: 4116: 4074: 3989: 3916: 3668: 3629: 3581: 3546: 3498: 3359: 3303: 3242: 3129: 2994: 2925: 2860: 2794: 2744: 2682: 2598: 2517: 2451: 2385: 2250: 2202: 2148: 2086: 2051: 2013: 1886: 1797: 1714: 1681: 1653: 1612: 1431: 124: 14479: 13875: 13844: 12724: 10053: 9993:"Entangled photons make a picture from a paradox" 8406: 8404: 6335:{\displaystyle \rho =\int \lambda dP_{\lambda },} 5991:{\displaystyle \lambda _{1},\cdots ,\lambda _{n}} 5239: 5198: 2995:{\displaystyle \{|0\rangle _{B},|1\rangle _{B}\}} 2926:{\displaystyle \{|0\rangle _{A},|1\rangle _{A}\}} 15445: 12268: 11325: 11323: 11321: 10340:Boette, A.; Rossignoli, R. (12 September 2018). 10339: 9095:Letter from Einstein to Max Born, 3 March 1947; 7963:only by consideration of electron entanglement. 7392:. GHz states are occasionally extended to 6234: 1847: 14240: 12953: 12839: 11940: 11067:"Note on exchange phenomena in the Thomas atom" 10533: 9168: 6762:also can be used as a measure of entanglement. 6443: 5936:to calculate a non-polynomial function such as 5683:, is associated to a probability distribution, 2397:This state is separable if there exist vectors 1925: 1066:is an active area of research and development. 999:") and argued that the accepted formulation of 13444: 12470: 11271: 10867:Nielsen, Michael A.; Chuang, Isaac L. (2000). 10706: 9966:"Loophole-free Bell test | Ronald Hanson" 9169:Freedman, Stuart J.; Clauser, John F. (1972). 8936: 8505:Speakable and Unspeakable in Quantum Mechanics 8498:Speakable and unspeakable in quantum mechanics 8447: 8401: 8351: 8349: 8347: 7954: 6756:of the system. Here the entropy is unchanged. 3917:{\displaystyle \rho =\sum _{i}w_{i}\left\left} 2861:{\displaystyle c_{ij}\neq c_{i}^{A}c_{j}^{B}.} 1472:The first experiment that verified Einstein's 1183:proved that one of their key assumptions, the 14667: 14323: 14175: 13383: 12671: 11752: 11318: 11218: 11212: 10866: 10396: 9944:"Classical carrier could create entanglement" 9822: 9748: 8924:"Quantum entanglement observed in top quarks" 8759: 8699: 8553: 7966: 5221: 3590: 1937:mathematical formulation of quantum mechanics 1613:{\displaystyle {\hat {H}}(x)|\psi \rangle =0} 1226:For decades, each had left open at least one 891: 14385:. Stanford, California: Stanford University. 14234: 13627: 12665: 12507: 12367: 11845: 10631: 10078: 8890: 8814: 8799: 8797: 8795: 7583: 7527: 7507: 7484: 7466: 7448: 7366:. The traditional GHZ state was defined for 7327: 7269: 7245: 7227: 7151: 7130: 7109: 7088: 7058: 7013: 6992: 6971: 6950: 6920: 6345:assume the same convention when calculating 5455: 5427: 5410: 5163: 5154: 5103: 5082: 5023: 5014: 4984: 4975: 4896: 4875: 4854: 4833: 4601: 4558: 4555: 4519: 4381: 4377: 4297: 4092:are positively valued probabilities and the 3885: 3861: 3819: 3795: 3663: 3624: 3576: 3475: 3472: 3360:{\displaystyle |1\rangle _{A}|0\rangle _{B}} 3348: 3330: 3304:{\displaystyle |0\rangle _{A}|1\rangle _{B}} 3292: 3274: 3237: 3234: 3220: 3209: 3194:Now suppose Alice is an observer for system 3152:. Another way to say this is that while the 3110: 3089: 3068: 3047: 2989: 2980: 2959: 2947: 2920: 2911: 2890: 2878: 2668: 2622: 2587: 2541: 2374: 2353: 2300: 2245: 2236: 2224: 2197: 2188: 2176: 2134: 2113: 2075: 2040: 1844:can emerge from the entanglement first law. 1792: 1757: 1676: 1601: 119: 93: 13805: 12368:Kitagawa, Masahiro; Ueda, Masahito (1993). 12210: 12102: 10869:Quantum Computation and Quantum Information 10700: 8708:Introduction to Quantum Mechanics (2nd ed.) 8453: 8410: 8344: 8313: 5433:{\displaystyle |00\rangle +0.01|11\rangle } 4791:For example, the reduced density matrix of 4279:. Let the state of the composite system be 4251: 1515:Emergence of time from quantum entanglement 14674: 14660: 13322: 11783: 10586: 9550: 9533:The Cosmic Compendium: Interstellar Travel 9468: 9203: 9164: 9162: 9160: 9086:, Cambridge University Press, 2010, p. xv. 9049: 8820: 8755: 8753: 8134:Entanglement of elements of living systems 6862:Entanglement is used in some protocols of 6484:if the reduced state of each subsystem of 5245: 4632: 4380: 2518:{\displaystyle c_{ij}=c_{i}^{A}c_{j}^{B},} 1418: 1281: 1129:they considered. Following the EPR paper, 898: 884: 14540: 14493: 14446: 14430: 14324:Albert, David Z.; Galchen, Rivka (2009). 14304: 14286: 14241:O'Callaghan, Jonathan (29 October 2018). 14217: 14199: 14119: 14039: 13949: 13655: 13645: 13519: 13458: 13397: 13336: 13275: 13214: 13161: 13108: 13055: 13041: 12967: 12906: 12853: 12808: 12794: 12788: 12779: 12738: 12685: 12622: 12581: 12528: 12404: 12328: 12282: 12134: 12116: 12063: 12003: 11958: 11859: 11806: 11766: 11748: 11746: 11684: 11631: 11566: 11513: 11452: 11429: 11396: 11343: 11285: 11232: 11187: 11174:spins of the XY model in one dimension". 11122: 11093: 11055: 11030: 11012: 10957: 10909: 10783: 10720: 10651: 10553: 10524: 10471: 10457: 10410: 10357: 10342:"History states of systems and operators" 10300: 10235: 10178: 10120:"Quantum mechanics without time: A model" 10004: 9895: 9842: 9791: 9696: 9589: 9500: 9374: 9299: 9235: 9194: 9148: 9133:"On the Einstein-Poldolsky-Rosen paradox" 9114: 9089: 9059: 8954: 8836: 8792: 8567: 8392: 7547: 3243:{\displaystyle \{|0\rangle ,|1\rangle \}} 1861:that are entangled for certain values of 14367:. Cambridge: Cambridge University Press. 13694:Introduction to Computational Chemistry. 11546: 10034: 9529: 6765: 6273:{\displaystyle \lim _{p\to 0}p\log p=0,} 5663: 4784:to obtain the reduced density matrix on 3421:, such a matrix takes the general form: 3380:. This remains true even if the systems 1079: 31: 14592: 14365:An Introduction to Quantum Entanglement 13876:CMS Collaboration (17 September 2024), 13569:IEEE Transactions on Information Theory 12567: 12159: 12096: 11664: 11499: 11376: 11265: 11102: 10947: 10117: 10037:"Demonstration of quantum entanglement" 9244: 9157: 8896: 8810:(PhD thesis). University of California. 8750: 8677: 8651: 8319: 6820: 3009:, the following is an entangled state: 2094:, the state of the composite system is 1947:Consider two arbitrary quantum systems 1541:generator of infinitesimal translations 1202:, starting with the pioneering work of 1031:between the measurements, and that the 14: 15446: 14573: 14408: 14389: 14377:"Quantum Entanglement and Information" 14363:(2006). "Geometry of Quantum States". 11789: 11743: 10592: 10460:Journal of the Korean Physical Society 10164: 9990: 9635: 9556: 9486:The Royal Swedish Academy of Sciences 9346: 9285: 9279: 9250: 9100: 8803: 8760:Kocher, C. A.; Commins, E. D. (1967). 8685:. London: Jonathan Cape. p. 603. 8617: 8270:Spontaneous parametric down-conversion 7847:testing) is "StateSeparator" web-app. 7629:spontaneous parametric down-conversion 6841:Entanglement has many applications in 5544:{\displaystyle \rho _{1}\to \rho _{2}} 5504:{\displaystyle \rho _{1}\to \rho _{2}} 5387:{\displaystyle \rho _{2}\to \rho _{1}} 5347:{\displaystyle \rho _{1}\to \rho _{2}} 4780:. Colloquially, we "trace out" system 3669:{\displaystyle |\mathbf {y} -\rangle } 3630:{\displaystyle |\mathbf {z} +\rangle } 37:Spontaneous parametric down-conversion 14681: 14655: 14416:. Bristol, UK: Institute of Physics. 13566: 13200: 13006: 12892: 12628: 12165: 11989: 11898: 11731: 11607: 11435: 11061: 10895: 10889: 9530:Anderson, Rupert W. (28 March 2015). 9412: 6859:(although this is disputed by some). 6584:). For example, by properties of the 6460:}. Therefore, a bipartite pure state 3934:are positively valued probabilities, 2871:For example, given two basis vectors 2752:at least for one pair of coordinates 2690:It is inseparable if for any vectors 1086:Einstein–Podolsky–Rosen (EPR) paradox 14414:Fundamentals of Quantum Entanglement 12045:"Experimental Quantum Teleportation" 11905:Quantum Theory: Concepts and Methods 9946:. physicsworld.com. 11 December 2013 9509:Quantum Theory: Concepts and Methods 9393:, volume 175, p. 8. New York, 1984. 9130: 8652:Francis, Matthew (30 October 2012). 8647: 8645: 8613: 8611: 8549: 8547: 8501: 3990:{\textstyle \sum _{j}|c_{ij}|^{2}=1} 3643:direction, and the other with state 3582:{\displaystyle |\alpha _{i}\rangle } 3198:, and Bob is an observer for system 1920:interpretations of quantum mechanics 1021:interpretations of quantum mechanics 14382:Stanford Encyclopedia of Philosophy 14374: 12797:Quantum Information and Computation 10058:(2 ed.). Pearson. p. 68. 10035:Rozatkar, Gaurav (16 August 2018). 9739:, Vintage Books, 2008, pp. 321–324. 9150:10.1103/PhysicsPhysiqueFizika.1.195 9099:, Walker, New York, 1971. Cited in 8122:Entanglement of macroscopic objects 8099:{\displaystyle \tau \sim 10^{-21}s} 8020:{\displaystyle \tau \sim 10^{-25}s} 6879: 5722:{\displaystyle p_{1},\cdots ,p_{n}} 4260:in 1930. Consider as above systems 3701:. That is, it has the general form 3676:with spins aligned in the negative 3637:with spins aligned in the positive 2795:{\displaystyle c_{i}^{A},c_{j}^{B}} 2014:{\displaystyle H_{A}\otimes H_{B}.} 1823: 1548:treats time as a dynamical variable 1088:paper, in the May 4, 1935 issue of 1016:interval between the measurements. 27:Correlation between quantum systems 24: 14316: 13812:The European Physical Journal Plus 11732:Cerf, Nicolas J.; Cleve, Richard. 11725: 10399:General Relativity and Gravitation 10118:Rovelli, Carlo (15 October 1990). 9729: 9347:Aspect, Alain (16 December 2015). 9030: 8730: 7938:Quantum Experiments at Space Scale 7600:photons are in one mode" and "the 7589:{\displaystyle |\Psi ^{+}\rangle } 7574: 7403:Also for M>2 qubits, there are 7333:{\displaystyle |\Phi ^{+}\rangle } 7318: 7223: 7220: 7217: 7049: 6911: 4704:{\displaystyle N_{B}:=\dim(H_{B})} 4561: 4552: 4468: 4465: 4462: 4384: 4374: 4294: 3527:as representing an ensemble where 2251:{\displaystyle \{|j\rangle _{B}\}} 2203:{\displaystyle \{|i\rangle _{A}\}} 2087:{\displaystyle |\phi \rangle _{B}} 2052:{\displaystyle |\psi \rangle _{A}} 1852:In the media and popular science, 1775: 1771: 1661:is the Hamiltonian constraint and 430:Sum-over-histories (path integral) 116: 90: 46:Part of a series of articles about 25: 15465: 14613: 14344:10.1038/scientificamerican0309-32 14188:Journal of Physics Communications 13845:CMS Collaboration (6 June 2024), 9679:Zbinden, H.; et al. (2001). 9476:"The Nobel Prize in Physics 2022" 8897:Markoff, Jack (21 October 2015). 8804:Kocher, Carl Alvin (1 May 1967). 8642: 8608: 8544: 7804:{\displaystyle 0\leq p_{j}\leq 1} 7678:Testing a system for entanglement 7643:, etc. Quantum entanglement of a 6723: 1766: 1523:, between the way the concept of 1309:As an example of entanglement: a 1117:that attempted to show that "the 67: 15428: 15427: 14262: 14095: 14015: 13989: 13925: 13900: 13869: 13838: 13799: 13773: 13747: 13707: 13686: 13672: 13621: 13595: 13560: 13499: 13438: 13377: 13316: 13255: 13194: 13141: 13035: 13000: 12947: 12886: 12833: 8293: 7623:Methods of creating entanglement 7604:photons are in the other mode". 7305:which reduces to the Bell state 6798:Relative entropy of entanglement 4246:Positive Partial Transpose (PPT) 3656: 3617: 2024:If the first system is in state 1809:and other subsequent proposals. 1546:In contrast, general relativity 1543:of quantum states through time. 913:is the phenomenon of a group of 13824:10.1140/epjp/s13360-021-01902-1 12718: 12561: 12464: 12421: 12361: 12307: 12262: 12036: 11983: 11934: 11892: 11839: 11493: 11163: 10992: 10941: 10860: 10817: 10761: 10512: 10451: 10390: 10333: 10276: 10211: 10158: 10111: 10072: 10047: 10028: 9984: 9958: 9936: 9875: 9785: 9751:Letters in Mathematical Physics 9742: 9672: 9629: 9569: 9523: 9406: 9383: 9340: 9124: 9076: 9043: 8979: 8930: 8916: 8724: 8200:Faster-than-light communication 7934:Jiuquan Satellite Launch Center 6866:, but to prove the security of 6836: 6094:. The Shannon entropy is then: 1432:Violations of Bell's inequality 1111:Einstein–Podolsky–Rosen paradox 1084:Article headline regarding the 1041:faster-than-light communication 15377:Relativistic quantum mechanics 14482:Quantum Information Processing 14370:second, revised edition (2017) 13538:10.1103/PhysRevLett.103.160505 12757:10.1103/PhysRevLett.120.240403 11943:"Entanglement Rényi α entropy" 11585:10.1103/PhysRevLett.111.060502 11471:10.1103/PhysRevLett.104.020504 11176:Quantum Information Processing 10670:10.1103/PhysRevLett.115.030404 9914:10.1103/physrevlett.110.210403 9608:10.1103/PhysRevLett.107.170404 9082:Alisa Bokulich, Gregg Jaeger, 8671: 8586:10.1103/PhysRevLett.110.260407 8490: 8180:Einstein's thought experiments 8060:{\displaystyle \sim 10^{-23}s} 7947:In the 16 June 2017, issue of 7756: 7750: 7732: 7726: 7569: 7517: 7497: 7476: 7458: 7434: 7313: 7261: 7237: 7212: 7160: 7143: 7122: 7101: 7080: 7076: 7044: 7022: 7005: 6984: 6963: 6942: 6938: 6906: 6691: 6685: 6614: 6608: 6241: 6113: 6107: 6081: 6068: 6043: 6030: 5948:. If the nonnegative operator 5869: 5863: 5774: 5742: 5581: 5528: 5488: 5448: 5420: 5403: 5371: 5331: 5240:Classification of entanglement 5199:Two applications that use them 5171: 5146: 5095: 5074: 5031: 5006: 4992: 4967: 4888: 4867: 4846: 4825: 4698: 4685: 4593: 4565: 4548: 4527: 4410:onto this state. The state of 4388: 4370: 4290: 3971: 3952: 3906: 3902: 3878: 3858: 3822: 3802: 3778: 3774: 3759: 3651: 3612: 3562: 3489: 3458: 3340: 3322: 3284: 3266: 3227: 3213: 3102: 3081: 3060: 3039: 2972: 2951: 2903: 2882: 2739: 2721: 2715: 2697: 2660: 2614: 2579: 2533: 2446: 2428: 2422: 2404: 2366: 2345: 2292: 2228: 2180: 2126: 2105: 2067: 2032: 1942: 1785: 1750: 1743: 1706: 1682:{\displaystyle |\psi \rangle } 1669: 1648: 1642: 1636: 1594: 1590: 1584: 1578: 1501:Delft University of Technology 1437:Local hidden variable theories 1133:wrote a letter to Einstein in 1003:must therefore be incomplete. 580:Relativistic quantum mechanics 112: 105: 86: 13: 1: 15355:Quantum statistical mechanics 15132:Quantum differential calculus 15054:Delayed-choice quantum eraser 14837:Symmetry in quantum mechanics 14465:10.1016/j.physrep.2009.02.004 13477:10.1103/PhysRevLett.96.050503 13416:10.1103/PhysRevLett.95.230502 13074:10.1016/j.physrep.2009.02.004 12925:10.1103/PhysRevLett.92.117903 12600:10.1103/PhysRevLett.96.020501 12022:10.1088/1367-2630/16/3/033027 11703:10.1103/PhysRevLett.98.140402 11141:10.1103/PhysRevLett.93.227203 11041:10.1016/S0375-9601(96)00706-2 9050:Christandl, Matthias (2006). 8307: 8235:Quantum entanglement swapping 5659: 4222:{\displaystyle \rho _{i}^{B}} 4190:{\displaystyle \rho _{i}^{A}} 4149:{\displaystyle \rho _{i}^{B}} 4117:{\displaystyle \rho _{i}^{A}} 1848:Non-locality and entanglement 1691:wave function of the universe 1654:{\displaystyle {\hat {H}}(x)} 1440:results would always satisfy 1403:is first and others in which 620:Quantum statistical mechanics 11251:10.1103/RevModPhys.91.025001 9273:10.1016/0003-4916(71)90159-X 8706:Griffiths, David J. (2004), 8285:Ward's probability amplitude 7940:(QUESS) mission, nicknamed " 7651:, such as an electron and a 7561:This is like the Bell state 6444:As a measure of entanglement 3411:positive-semidefinite matrix 3399: 2265:. The most general state in 1926:Quantum-mechanical framework 1556:quantized general relativity 1481:this result was achieved by 7: 15454:Quantum information science 15157:Quantum stochastic calculus 15147:Quantum measurement problem 15069:Mach–Zehnder interferometer 14559:10.1103/PhysRevA.105.052441 13355:10.1103/PhysRevLett.86.3658 13294:10.1103/PhysRevLett.84.2722 13233:10.1103/PhysRevLett.84.2726 12704:10.1103/PhysRevA.106.062437 12651:10.1103/PhysRevLett.68.2981 12493:10.1103/PhysRevLett.71.1355 11878:10.1103/PhysRevLett.78.5022 11650:10.1103/PhysRevLett.80.5239 10898:American Journal of Physics 9040:, Icon Books, 2009, p. 313. 8157: 7955:Naturally entangled systems 6852:and quantum teleportation. 5551:is impossible by LOCC, but 5461:{\displaystyle |00\rangle } 5066:for the pure product state 1807:Page and Wootters mechanism 1531:, and the role it plays in 1474:spooky action at a distance 1170:spooky action at a distance 590:Quantum information science 10: 15470: 14597:. Imperial College Press. 13733:10.1038/s41586-024-07824-z 13180:10.1103/PhysRevA.74.012313 13029:10.1103/PhysRevA.82.012335 12986:10.1103/PhysRevA.70.022316 12872:10.1103/PhysRevA.68.032103 12293:10.1103/PhysRevA.72.012332 12188:10.1103/PhysRevLett.67.661 11969:10.1103/PhysRevA.93.022324 11618:entanglement in nature?". 11532:10.1103/PhysRevLett.83.436 11063:Dirac, Paul Adrien Maurice 10873:Cambridge University Press 10376:10.1103/PhysRevA.98.032108 10319:10.1103/PhysRevA.93.062127 10254:10.1103/PhysRevD.92.045033 10197:10.1103/PhysRevA.89.052122 9991:Gibney, Elizabeth (2014). 9715:10.1103/PhysRevA.63.022111 9557:Gibney, Elizabeth (2017). 9438:10.1103/PhysRevLett.67.661 9196:10.1103/PhysRevLett.28.938 8885:free online access version 8786:10.1103/PhysRevLett.18.575 8067:) and spin decorrelation ( 7967:Entanglement of top quarks 7607:Finally, there also exist 7400:rather than 2 dimensions. 6843:quantum information theory 5651: 5222:Entanglement as a resource 4268:each with a Hilbert space 1715:{\displaystyle {\hat {H}}} 1347: 1276: 1141:(translated by himself as 1137:in which he used the word 1073: 1069: 15423: 15385: 15337: 15217:Quantum complexity theory 15195:Quantum cellular automata 15170: 15102: 15036: 14949: 14913: 14900:Path integral formulation 14867: 14732: 14689: 14578:. Heildelberg: Springer. 14512:10.1007/s11128-017-1568-0 14306:10.1038/s41534-018-0110-2 13960:10.1038/s41567-020-1031-5 13888:10.48550/arXiv.2409.11067 13857:10.48550/arXiv.2406.03976 13119:10.1038/s42254-018-0003-5 12450:10.1103/PhysRevA.46.R6797 12240:10.1038/s41586-020-2401-y 11825:10.1103/RevModPhys.74.197 11795:Reviews of Modern Physics 11362:10.1103/RevModPhys.81.865 11332:Reviews of Modern Physics 11304:10.1142/S0217984922501019 11221:Reviews of Modern Physics 11198:10.1007/s11128-010-0197-7 11095:10.1017/S0305004100016108 10572:10.1103/RevModPhys.86.419 10542:Reviews of Modern Physics 10429:10.1007/s10714-010-1034-0 10006:10.1038/nature.2014.15781 9658:10.1007/s10701-004-1308-z 9237:10.1103/PhysRevLett.49.91 8628:10.1038/nature.2012.11163 8476:10.1017/S0305004100019137 8433:10.1017/S0305004100013554 8215:Pauli exclusion principle 8205:Multipartite entanglement 8185:Entanglement distillation 8112:Peres–Horodecki criterion 7922:{\displaystyle 2\oplus 2} 7896:{\displaystyle 1\oplus n} 7870:{\displaystyle 1\oplus 1} 7845:Peres-Horodecki criterion 7816:Peres–Horodecki criterion 6788:Entanglement of formation 6586:Borel functional calculus 6482:maximally entangled state 5934:Borel functional calculus 5932:In general, one uses the 5307:{\displaystyle \rho _{2}} 5280:{\displaystyle \rho _{1}} 4738:the identity operator in 4425:over the basis of system 1359:Copenhagen interpretation 1238:protocols, most famously 18:Maximally entangled state 15284:Quantum machine learning 15264:Quantum key distribution 15254:Quantum image processing 15244:Quantum error correction 15094:Wheeler's delayed choice 14633:Audio – Cain/Gay (2009) 14219:10.1088/2399-6528/aae224 13581:10.1109/TIT.2013.2257936 12547:10.1038/nphoton.2011.283 12397:10.1103/physreva.47.5138 11415:10.1103/PhysRevA.58.1833 11274:Modern Physics Letters B 10802:10.1103/PhysRevA.59.1070 10617:10.1103/PhysRevA.40.4277 10144:10.1103/PhysRevD.42.2638 10105:10.1103/PhysRevD.27.2885 10056:Modern Quantum Mechanics 8280:Stern–Gerlach experiment 8250:Quantum pseudo-telepathy 8245:Quantum phase transition 7409:squeezed coherent states 6868:quantum key distribution 6783:Distillable entanglement 5729:, in the following way: 4795:for the entangled state 4252:Reduced density matrices 3593:to represent the state. 3394:no-communication theorem 2059:and the second in state 1818:system-time entanglement 1493:separable quantum states 1490:classically communicated 1236:quantum key distribution 1214:'s experiments in 1982. 1037:faster-than-light speeds 983:, and several papers by 625:Quantum machine learning 378:Wheeler's delayed-choice 15200:Quantum finite automata 14275:npj Quantum Information 14138:10.1126/science.abf2998 14058:10.1126/science.abf5389 13657:10.1126/science.aan3211 13508:Physical Review Letters 13447:Physical Review Letters 13386:Physical Review Letters 13325:Physical Review Letters 13264:Physical Review Letters 13203:Physical Review Letters 12895:Physical Review Letters 12727:Physical Review Letters 12631:Physical Review Letters 12570:Physical Review Letters 12473:Physical Review Letters 12168:Physical Review Letters 11673:Physical Review Letters 11502:Physical Review Letters 11441:Physical Review Letters 11280:(22): 2250101–2250255. 11111:Physical Review Letters 10640:Physical Review Letters 9884:Physical Review Letters 9737:The Age of Entanglement 9578:Physical Review Letters 9400:30 January 2020 at the 9216:Physical Review Letters 9175:Physical Review Letters 9137:Physics Physique Физика 9008:10.1126/science.1211914 8965:10.1126/science.adf4272 8766:Physical Review Letters 8556:Physical Review Letters 7197:For M>2 qubits, the 7181:entropy of entanglement 6488:is the diagonal matrix 1887:{\displaystyle p_{sym}} 1560:Wheeler–DeWitt equation 1419:Hidden variables theory 1282:Meaning of entanglement 335:Leggett–Garg inequality 15304:Quantum neural network 13097:Nature Reviews Physics 12370:"Squeezed Spin States" 12145:10.1098/rspa.2002.1097 11992:New Journal of Physics 10846:10.1103/PhysRevA.51.54 9816:10.1103/PhysRev.77.136 9638:Foundations of Physics 8394:10.1103/PhysRev.47.777 8100: 8061: 8021: 7975:using techniques from 7923: 7897: 7871: 7805: 7766: 7661:Hardy's interferometer 7657:quantum wave functions 7590: 7552: 7386: 7360: 7334: 7296: 7188:basis (linear algebra) 7170: 7029: 6827:Reeh-Schlieder theorem 6808:Logarithmic negativity 6739: 6656: 6588:, we see that for any 6558: 6419: 6336: 6274: 6213: 6088: 5992: 5923: 5833: 5723: 5669: 5618: 5598: 5545: 5505: 5462: 5434: 5388: 5348: 5308: 5281: 5185: 5116: 5053: 4917: 4759: 4732: 4705: 4650: 4500: 4396: 4333: 4223: 4191: 4150: 4118: 4076: 3991: 3918: 3670: 3631: 3583: 3548: 3500: 3361: 3305: 3244: 3131: 2996: 2940:and two basis vectors 2927: 2862: 2796: 2746: 2684: 2600: 2519: 2453: 2387: 2252: 2204: 2150: 2088: 2053: 2015: 1888: 1830:AdS/CFT correspondence 1799: 1716: 1683: 1655: 1614: 1552:Hamiltonian constraint 1341:hidden variable theory 1094: 1076:Hidden-variable theory 966:wave function collapse 126: 40: 15329:Quantum teleportation 14857:Wave–particle duality 14620:Explanatory video by 14375:Bub, Jeffrey (2019). 12819:10.26421/QIC10.3-4-11 10968:10.1145/780542.780545 10709:Nature Communications 9376:10.1103/Physics.8.123 8275:Squashed entanglement 8255:Quantum teleportation 8101: 8062: 8022: 7924: 7898: 7872: 7806: 7767: 7672:entanglement swapping 7641:Hong–Ou–Mandel effect 7591: 7553: 7387: 7361: 7335: 7297: 7171: 7030: 6803:Squashed entanglement 6766:Entanglement measures 6740: 6657: 6559: 6430:statistical mechanics 6420: 6337: 6275: 6214: 6089: 5993: 5924: 5834: 5724: 5667: 5619: 5617:{\displaystyle \rho } 5599: 5546: 5506: 5463: 5435: 5389: 5349: 5309: 5282: 5246:entanglement measures 5186: 5117: 5054: 4918: 4760: 4758:{\displaystyle H_{A}} 4733: 4731:{\displaystyle I_{A}} 4706: 4651: 4479: 4397: 4334: 4224: 4192: 4151: 4119: 4077: 3992: 3919: 3671: 3632: 3584: 3549: 3547:{\displaystyle w_{i}} 3501: 3362: 3306: 3245: 3132: 2997: 2928: 2863: 2797: 2747: 2685: 2601: 2520: 2454: 2388: 2253: 2205: 2151: 2089: 2054: 2016: 1908:quantum teleportation 1889: 1800: 1717: 1684: 1656: 1615: 1483:entanglement swapping 1462:uncertainty principle 1446:number of experiments 1185:principle of locality 1177:Bohm's interpretation 1166:spukhafte Fernwirkung 1083: 1074:Further information: 1025:wavefunction collapse 320:Elitzur–Vaidman 310:Davisson–Germer 127: 35: 15360:Quantum field theory 15289:Quantum metamaterial 15234:Quantum cryptography 14964:Consistent histories 12803:(3&4): 343–360. 11912:. pp. 260–270. 10875:. pp. 112–113. 10490:10.3938/jkps.63.1094 9131:Bell, J. S. (1964). 8502:Bell, J. S. (1987). 8190:Entanglement witness 8071: 8035: 7992: 7907: 7881: 7855: 7819:not limited to) the 7776: 7686: 7565: 7430: 7405:spin squeezed states 7370: 7344: 7309: 7208: 7040: 6902: 6864:quantum cryptography 6831:quantum field theory 6821:Quantum field theory 6772:entanglement entropy 6679: 6602: 6495: 6352: 6301: 6230: 6101: 6014: 5956: 5857: 5842:Since a mixed state 5736: 5687: 5608: 5555: 5515: 5475: 5444: 5399: 5358: 5318: 5291: 5264: 5129: 5070: 4933: 4802: 4742: 4715: 4663: 4659:The sum occurs over 4436: 4353: 4286: 4201: 4169: 4128: 4096: 4007: 3938: 3708: 3647: 3608: 3558: 3531: 3428: 3318: 3262: 3206: 3016: 2944: 2875: 2806: 2756: 2694: 2610: 2529: 2463: 2401: 2288: 2221: 2173: 2101: 2063: 2028: 1982: 1865: 1854:quantum non-locality 1734: 1728:Schrödinger equation 1697: 1665: 1627: 1569: 1537:Hamiltonian operator 1200:relevant experiments 1162:theory of relativity 997:action at a distance 911:Quantum entanglement 585:Quantum field theory 497:Consistent histories 134:Schrödinger equation 61: 15345:Quantum fluctuation 15314:Quantum programming 15274:Quantum logic gates 15259:Quantum information 15239:Quantum electronics 14714:Classical mechanics 14622:Scientific American 14593:Steward EG (2008). 14551:2022PhRvA.105e2441S 14504:2017QuIP...16..118B 14457:2009PhR...474....1G 14394:. Springer Verlag. 14331:Scientific American 14297:2018npjQI...4...60K 14248:Scientific American 14210:2018JPhCo...2j1001M 14130:2021Sci...372..622K 14050:2021Sci...372..625M 13787:. 18 September 2024 13761:. 28 September 2023 13640:(6343): 1140–1144. 13530:2009PhRvL.103p0505W 13469:2006PhRvL..96e0503H 13408:2005PhRvL..95w0502S 13347:2001PhRvL..86.3658W 13286:2000PhRvL..84.2722D 13225:2000PhRvL..84.2726S 13172:2006PhRvA..74a2313L 13066:2009PhR...474....1G 13021:2010PhRvA..82a2335H 12978:2004PhRvA..70b2316G 12917:2004PhRvL..92k7903G 12864:2003PhRvA..68c2103H 12749:2018PhRvL.120x0403L 12696:2022PhRvA.106f2437G 12643:1992PhRvL..68.2981H 12592:2006PhRvL..96b0501D 12539:2012NaPho...6...45S 12485:1993PhRvL..71.1355H 12442:1992PhRvA..46.6797W 12436:(11): R6797–R6800. 12389:1993PhRvA..47.5138K 12339:2020AdOP...12.1012P 12232:2020Natur.582..501Y 12180:1991PhRvL..67..661E 12127:2003RSPSA.459.2011J 12111:(2036): 2011–2032. 12074:1997Natur.390..575B 12014:2014NJPh...16c3027H 11870:1997PhRvL..78.5022H 11817:2002RvMP...74..197V 11777:2005quant.ph..4163P 11695:2007PhRvL..98n0402W 11642:1998PhRvL..80.5239H 11626:(1998): 5239–5242. 11577:2013PhRvL.111f0502G 11524:1999PhRvL..83..436N 11463:2010PhRvL.104b0504K 11407:1998PhRvA..58.1833G 11354:2009RvMP...81..865H 11296:2022MPLB...3650101G 11243:2019RvMP...91b5001C 11133:2004PhRvL..93v7203F 11086:1930PCPS...26..376D 11023:1996PhLA..223....1H 10920:2001AmJPh..69..655L 10838:1995PhRvA..51...54J 10794:1999PhRvA..59.1070B 10731:2014NatCo...5.5297V 10662:2015PhRvL.115c0404A 10609:1989PhRvA..40.4277W 10564:2014RvMP...86..419B 10482:2013JKPS...63.1094L 10421:2010GReGr..42.2323V 10368:2018PhRvA..98c2108B 10311:2016PhRvA..93f2127B 10246:2015PhRvD..92d5033G 10189:2014PhRvA..89e2122M 10136:1990PhRvD..42.2638R 10097:1983PhRvD..27.2885P 9906:2013PhRvL.110u0403M 9853:2012NatPh...8..480M 9808:1950PhRv...77..136W 9763:1980LMaPh...4...93C 9707:2001PhRvA..63b2111Z 9650:2004FoPh...34.1643S 9600:2011PhRvL.107q0404G 9563:Scientific American 9430:1991PhRvL..67..661E 9367:2015PhyOJ...8..123A 9318:10.1038/nature15759 9310:2015Natur.526..682H 9265:1971AnPhy..65....1K 9228:1982PhRvL..49...91A 9187:1972PhRvL..28..938F 9070:2006PhDT.......289C 9000:2011Sci...334.1253L 8994:(6060): 1253–1256. 8949:(6675): 1143–1147. 8855:10.1038/nature15759 8847:2015Natur.526..682H 8778:1967PhRvL..18..575K 8578:2013PhRvL.110z0407Y 8468:1936PCPS...32..446S 8425:1935PCPS...31..555S 8385:1935PhRv...47..777E 8323:(10 October 2022). 7825:reduction criterion 7760: 7736: 7396:, i.e., systems of 7385:{\displaystyle M=3} 7359:{\displaystyle M=2} 6293:spectral resolution 5848:von Neumann entropy 5227:of which arbitrary 5122:discussed above is 4926:discussed above is 4408:projection operator 4218: 4186: 4145: 4113: 4068: 4050: 3154:von Neumann entropy 2854: 2839: 2791: 2773: 2738: 2714: 2658: 2577: 2511: 2496: 2445: 2421: 938:physical properties 373:Stern–Gerlach 170:Classical mechanics 15398:in popular culture 15180:Quantum algorithms 15028:Von Neumann–Wigner 15008:Objective collapse 14719:Old quantum theory 14390:Cramer JG (2015). 12347:10.1364/AOP.361502 10739:10.1038/ncomms6297 9771:10.1007/BF00417500 8926:. 11 October 2023. 8903:The New York Times 8330:The New York Times 8165:Bound entanglement 8096: 8057: 8017: 7977:quantum tomography 7919: 7893: 7867: 7801: 7762: 7740: 7716: 7705: 7586: 7548: 7382: 7356: 7330: 7292: 7166: 7025: 6873:device independent 6735: 6652: 6576:Boltzmann constant 6554: 6545: 6415: 6332: 6270: 6248: 6209: 6175: 6127: 6084: 5988: 5919: 5883: 5829: 5792: 5719: 5674:information theory 5670: 5614: 5594: 5541: 5501: 5458: 5430: 5384: 5344: 5304: 5277: 5229:quantum operations 5181: 5112: 5049: 4959: 4913: 4817: 4755: 4728: 4701: 4646: 4624: 4392: 4329: 4219: 4204: 4187: 4172: 4146: 4131: 4114: 4099: 4072: 4054: 4036: 4025: 3987: 3950: 3914: 3844: 3751: 3726: 3666: 3627: 3579: 3544: 3496: 3446: 3357: 3301: 3240: 3127: 3031: 2992: 2923: 2858: 2840: 2825: 2792: 2777: 2759: 2742: 2724: 2700: 2680: 2644: 2643: 2596: 2563: 2562: 2515: 2497: 2482: 2449: 2431: 2407: 2383: 2330: 2248: 2200: 2146: 2084: 2049: 2011: 1955:, with respective 1884: 1834:Mark Van Raamsdonk 1795: 1712: 1679: 1651: 1610: 1533:general relativity 1370:special relativity 1311:subatomic particle 1244:Charles H. Bennett 1119:quantum-mechanical 1115:thought experiment 1095: 1091:The New York Times 1033:mutual information 1019:According to some 1013:subtly transmitted 561:Von Neumann–Wigner 541:Objective-collapse 340:Mach–Zehnder 330:Leggett inequality 325:Franck–Hertz 175:Old quantum theory 122: 41: 15441: 15440: 15415:Quantum mysticism 15393:Schrödinger's cat 15324:Quantum simulator 15294:Quantum metrology 15222:Quantum computing 15185:Quantum amplifier 15162:Quantum spacetime 15127:Quantum cosmology 15117:Quantum chemistry 14832:Scattering theory 14780:Zero-point energy 14775:Degenerate levels 14683:Quantum mechanics 14604:978-1-86094-978-4 14585:978-3-540-92127-1 14574:Jaeger G (2009). 14423:978-0-7503-2226-3 14401:978-3-319-24642-0 14114:(6542): 622–625. 14034:(6542): 625–629. 13727:(8030): 542–547. 13702:978-0-470-01187-4 13575:(10): 6774–6778. 13331:(16): 3658–3661. 13270:(12): 2722–2725. 13209:(12): 2726–2729. 13150:Physical Review A 13009:Physical Review A 12956:Physical Review A 12842:Physical Review A 12674:Physical Review A 12637:(20): 2981–2984. 12430:Physical Review A 12377:Physical Review A 12271:Physical Review A 12226:(7813): 501–505. 12058:(6660): 575–579. 11947:Physical Review A 11854:(26): 5022–5025. 11001:Physics Letters A 10977:978-1-58113-674-6 10928:10.1119/1.1356698 10882:978-0-521-63503-5 10596:Physical Review A 10405:(10): 2323–2329. 10346:Physical Review A 10289:Physical Review A 10224:Physical Review D 10167:Physical Review A 10124:Physical Review D 10091:(12): 2885–2892. 10085:Physical Review D 10065:978-0-8053-8291-4 9861:10.1038/nphys2294 9644:(11): 1643–1673. 9484:(Press release). 9294:(7575): 682–686. 9253:Annals of Physics 8831:(7575): 682–686. 8717:978-0-13-111892-8 8710:, Prentice Hall, 8692:978-0-224-04447-9 8225:Quantum computing 8220:Quantum coherence 7832:Jon Magne Leinaas 7696: 7639:, the use of the 7542: 7541: 7445: 7287: 7286: 7074: 7073: 6936: 6935: 6857:quantum computing 6850:superdense coding 6779:Entanglement cost 6726: 6541: 6514: 6233: 6166: 6126: 5882: 5783: 4958: 4816: 4815: 4623: 4478: 4473: 4451: 4016: 3941: 3835: 3762: 3742: 3717: 3437: 3030: 3029: 2745:{\displaystyle ,} 2634: 2553: 2452:{\displaystyle ,} 2315: 1912:superdense coding 1782: 1746: 1709: 1639: 1581: 1529:quantum mechanics 1442:Bell's inequality 1355:faster than light 1318:conservation laws 1192:Bell's inequality 1181:John Stewart Bell 1131:Erwin Schrödinger 1113:(EPR paradox), a 1009:Bell's inequality 1001:quantum mechanics 985:Erwin Schrödinger 908: 907: 615:Scattering theory 595:Quantum computing 368:Schrödinger's cat 300:Bell's inequality 108: 83: 52:Quantum mechanics 16:(Redirected from 15461: 15431: 15430: 15142:Quantum geometry 15137:Quantum dynamics 14994:Superdeterminism 14890:Matrix mechanics 14745:Bra–ket notation 14676: 14669: 14662: 14653: 14652: 14608: 14589: 14570: 14544: 14523: 14497: 14476: 14450: 14427: 14405: 14386: 14368: 14355: 14311: 14310: 14308: 14290: 14266: 14260: 14259: 14257: 14255: 14238: 14232: 14231: 14221: 14203: 14179: 14173: 14172: 14170: 14168: 14123: 14099: 14093: 14092: 14090: 14088: 14043: 14019: 14013: 14012: 14010: 14008: 13993: 13987: 13986: 13984: 13982: 13953: 13929: 13923: 13922: 13920: 13918: 13904: 13898: 13897: 13896: 13894: 13873: 13867: 13866: 13865: 13863: 13842: 13836: 13835: 13803: 13797: 13796: 13794: 13792: 13777: 13771: 13770: 13768: 13766: 13751: 13745: 13744: 13711: 13705: 13690: 13684: 13683: 13676: 13670: 13669: 13659: 13649: 13625: 13619: 13618: 13616: 13614: 13609:. 16 August 2016 13607:physicsworld.com 13599: 13593: 13592: 13564: 13558: 13557: 13523: 13503: 13497: 13496: 13462: 13460:quant-ph/0507168 13442: 13436: 13435: 13401: 13399:quant-ph/0508132 13381: 13375: 13374: 13340: 13338:quant-ph/0009118 13320: 13314: 13313: 13279: 13277:quant-ph/9908056 13259: 13253: 13252: 13218: 13216:quant-ph/9909044 13198: 13192: 13191: 13165: 13163:quant-ph/0605079 13145: 13139: 13138: 13112: 13092: 13086: 13085: 13059: 13039: 13033: 13032: 13004: 12998: 12997: 12971: 12969:quant-ph/0403219 12951: 12945: 12944: 12910: 12908:quant-ph/0306194 12890: 12884: 12883: 12857: 12855:quant-ph/0212090 12837: 12831: 12830: 12812: 12792: 12786: 12783: 12777: 12776: 12742: 12722: 12716: 12715: 12689: 12669: 12663: 12662: 12626: 12620: 12619: 12585: 12583:quant-ph/0509060 12565: 12559: 12558: 12532: 12517:Nature Photonics 12511: 12505: 12504: 12479:(9): 1355–1358. 12468: 12462: 12461: 12425: 12419: 12418: 12408: 12383:(6): 5138–5143. 12374: 12365: 12359: 12358: 12332: 12323:(4): 1012–1236. 12317:Adv. Opt. Photon 12311: 12305: 12304: 12286: 12284:quant-ph/0502064 12266: 12260: 12259: 12214: 12208: 12207: 12163: 12157: 12156: 12138: 12120: 12118:quant-ph/0201143 12100: 12094: 12093: 12067: 12049: 12040: 12034: 12033: 12007: 11987: 11981: 11980: 11962: 11938: 11932: 11931: 11896: 11890: 11889: 11863: 11861:quant-ph/9703041 11843: 11837: 11836: 11810: 11808:quant-ph/0102094 11787: 11781: 11780: 11770: 11768:quant-ph/0504163 11755:Quant. Inf. Comp 11750: 11741: 11740: 11738: 11729: 11723: 11722: 11688: 11686:quant-ph/0612147 11668: 11662: 11661: 11635: 11633:quant-ph/9801069 11611: 11605: 11604: 11570: 11550: 11544: 11543: 11517: 11515:quant-ph/9811053 11497: 11491: 11490: 11456: 11433: 11427: 11426: 11400: 11398:quant-ph/9712040 11391:(3): 1833–1839. 11380: 11374: 11373: 11347: 11345:quant-ph/0702225 11327: 11316: 11315: 11289: 11269: 11263: 11262: 11236: 11216: 11210: 11209: 11191: 11167: 11161: 11160: 11126: 11124:quant-ph/0406067 11106: 11100: 11099: 11097: 11071: 11059: 11053: 11052: 11034: 11016: 11014:quant-ph/9605038 10996: 10990: 10989: 10961: 10959:quant-ph/0303055 10945: 10939: 10938: 10913: 10911:quant-ph/0209123 10893: 10887: 10886: 10864: 10858: 10857: 10821: 10815: 10813: 10787: 10785:quant-ph/9804053 10778:(2): 1070–1091. 10765: 10759: 10758: 10724: 10704: 10698: 10697: 10655: 10635: 10629: 10628: 10603:(8): 4277–4281. 10590: 10584: 10583: 10557: 10537: 10531: 10530: 10528: 10516: 10510: 10509: 10475: 10466:(5): 1094–1098. 10455: 10449: 10448: 10414: 10394: 10388: 10387: 10361: 10337: 10331: 10330: 10304: 10280: 10274: 10273: 10239: 10215: 10209: 10208: 10182: 10162: 10156: 10155: 10130:(8): 2638–2646. 10115: 10109: 10108: 10076: 10070: 10069: 10051: 10045: 10044: 10032: 10026: 10025: 10023: 10021: 10008: 9988: 9982: 9981: 9979: 9977: 9968:. Archived from 9962: 9956: 9955: 9953: 9951: 9940: 9934: 9933: 9899: 9879: 9873: 9872: 9846: 9826: 9820: 9819: 9789: 9783: 9782: 9746: 9740: 9733: 9727: 9726: 9700: 9698:quant-ph/0007009 9676: 9670: 9669: 9633: 9627: 9626: 9593: 9573: 9567: 9566: 9554: 9548: 9547: 9527: 9521: 9512:, Kluwer, 1993; 9504: 9498: 9497: 9495: 9493: 9488:. 4 October 2022 9472: 9466: 9465: 9410: 9404: 9387: 9381: 9380: 9378: 9344: 9338: 9337: 9303: 9283: 9277: 9276: 9248: 9242: 9241: 9239: 9207: 9201: 9200: 9198: 9166: 9155: 9154: 9152: 9128: 9122: 9120: 9118: 9109:(6): 1829–1841. 9093: 9087: 9080: 9074: 9073: 9063: 9061:quant-ph/0604183 9047: 9041: 9034: 9028: 9027: 8983: 8977: 8976: 8958: 8934: 8928: 8927: 8920: 8914: 8913: 8911: 8909: 8894: 8888: 8882: 8840: 8818: 8812: 8811: 8801: 8790: 8789: 8757: 8748: 8747: 8745: 8743: 8728: 8722: 8720: 8703: 8697: 8696: 8675: 8669: 8668: 8666: 8664: 8649: 8640: 8639: 8615: 8606: 8605: 8571: 8551: 8542: 8541: 8539: 8537: 8532:on 12 April 2015 8531: 8525:. Archived from 8510: 8494: 8488: 8487: 8451: 8445: 8444: 8408: 8399: 8398: 8396: 8357:Einstein, Albert 8353: 8342: 8341: 8339: 8337: 8317: 8303: 8298: 8297: 8140:living organisms 8105: 8103: 8102: 8097: 8092: 8091: 8066: 8064: 8063: 8058: 8053: 8052: 8026: 8024: 8023: 8018: 8013: 8012: 7928: 7926: 7925: 7920: 7902: 7900: 7899: 7894: 7876: 7874: 7873: 7868: 7810: 7808: 7807: 7802: 7794: 7793: 7771: 7769: 7768: 7763: 7761: 7759: 7748: 7735: 7724: 7715: 7714: 7704: 7609:twin Fock states 7595: 7593: 7592: 7587: 7582: 7581: 7572: 7557: 7555: 7554: 7549: 7543: 7537: 7536: 7535: 7534: 7525: 7520: 7515: 7514: 7505: 7500: 7492: 7491: 7479: 7474: 7473: 7461: 7455: 7447: 7446: 7443: 7437: 7391: 7389: 7388: 7383: 7365: 7363: 7362: 7357: 7339: 7337: 7336: 7331: 7326: 7325: 7316: 7301: 7299: 7298: 7293: 7288: 7282: 7281: 7280: 7279: 7264: 7256: 7255: 7240: 7234: 7226: 7215: 7175: 7173: 7172: 7167: 7159: 7158: 7146: 7138: 7137: 7125: 7117: 7116: 7104: 7096: 7095: 7083: 7075: 7069: 7065: 7057: 7056: 7047: 7034: 7032: 7031: 7026: 7021: 7020: 7008: 7000: 6999: 6987: 6979: 6978: 6966: 6958: 6957: 6945: 6937: 6931: 6927: 6919: 6918: 6909: 6880:Entangled states 6751: 6744: 6742: 6741: 6736: 6731: 6727: 6722: 6708: 6671: 6661: 6659: 6658: 6653: 6648: 6644: 6643: 6642: 6594: 6590:unitary operator 6583: 6563: 6561: 6560: 6555: 6550: 6549: 6542: 6534: 6531: 6530: 6527: 6521: 6518: 6517: 6515: 6507: 6487: 6480:is said to be a 6479: 6439: 6435: 6424: 6422: 6421: 6416: 6411: 6410: 6392: 6391: 6367: 6366: 6341: 6339: 6338: 6333: 6328: 6327: 6290: 6286: 6279: 6277: 6276: 6271: 6247: 6218: 6216: 6215: 6210: 6208: 6207: 6195: 6194: 6185: 6184: 6174: 6159: 6155: 6154: 6146: 6145: 6128: 6124: 6093: 6091: 6090: 6085: 6080: 6079: 6064: 6063: 6042: 6041: 6026: 6025: 6009: 5997: 5995: 5994: 5989: 5987: 5986: 5968: 5967: 5951: 5947: 5928: 5926: 5925: 5920: 5915: 5911: 5910: 5902: 5901: 5884: 5880: 5845: 5838: 5836: 5835: 5830: 5825: 5824: 5812: 5811: 5802: 5801: 5791: 5773: 5772: 5754: 5753: 5728: 5726: 5725: 5720: 5718: 5717: 5699: 5698: 5678: 5623: 5621: 5620: 5615: 5603: 5601: 5600: 5595: 5593: 5592: 5580: 5579: 5567: 5566: 5550: 5548: 5547: 5542: 5540: 5539: 5527: 5526: 5510: 5508: 5507: 5502: 5500: 5499: 5487: 5486: 5467: 5465: 5464: 5459: 5451: 5439: 5437: 5436: 5431: 5423: 5406: 5393: 5391: 5390: 5385: 5383: 5382: 5370: 5369: 5353: 5351: 5350: 5345: 5343: 5342: 5330: 5329: 5313: 5311: 5310: 5305: 5303: 5302: 5286: 5284: 5283: 5278: 5276: 5275: 5190: 5188: 5187: 5182: 5180: 5179: 5174: 5162: 5161: 5149: 5141: 5140: 5121: 5119: 5118: 5113: 5111: 5110: 5098: 5090: 5089: 5077: 5065: 5058: 5056: 5055: 5050: 5045: 5041: 5040: 5039: 5034: 5022: 5021: 5009: 5001: 5000: 4995: 4983: 4982: 4970: 4960: 4951: 4945: 4944: 4922: 4920: 4919: 4914: 4909: 4905: 4904: 4903: 4891: 4883: 4882: 4870: 4862: 4861: 4849: 4841: 4840: 4828: 4818: 4811: 4807: 4794: 4787: 4783: 4779: 4775: 4771: 4764: 4762: 4761: 4756: 4754: 4753: 4737: 4735: 4734: 4729: 4727: 4726: 4710: 4708: 4707: 4702: 4697: 4696: 4675: 4674: 4655: 4653: 4652: 4647: 4642: 4641: 4631: 4630: 4625: 4621: 4614: 4610: 4609: 4608: 4596: 4588: 4587: 4573: 4569: 4568: 4551: 4541: 4537: 4536: 4535: 4530: 4515: 4514: 4499: 4498: 4497: 4487: 4476: 4475: 4474: 4472: 4471: 4459: 4454: 4449: 4448: 4447: 4428: 4424: 4413: 4401: 4399: 4398: 4393: 4391: 4373: 4365: 4364: 4345: 4338: 4336: 4335: 4330: 4325: 4324: 4312: 4311: 4293: 4278: 4267: 4263: 4243: 4239: 4228: 4226: 4225: 4220: 4217: 4212: 4196: 4194: 4193: 4188: 4185: 4180: 4163: 4159: 4155: 4153: 4152: 4147: 4144: 4139: 4123: 4121: 4120: 4115: 4112: 4107: 4091: 4081: 4079: 4078: 4073: 4067: 4062: 4049: 4044: 4035: 4034: 4024: 3996: 3994: 3993: 3988: 3980: 3979: 3974: 3968: 3967: 3955: 3949: 3923: 3921: 3920: 3915: 3913: 3909: 3905: 3900: 3899: 3881: 3876: 3875: 3857: 3856: 3843: 3829: 3825: 3818: 3817: 3805: 3794: 3793: 3781: 3773: 3772: 3764: 3763: 3755: 3750: 3736: 3735: 3725: 3700: 3681: 3675: 3673: 3672: 3667: 3659: 3654: 3642: 3636: 3634: 3633: 3628: 3620: 3615: 3591:density matrices 3588: 3586: 3585: 3580: 3575: 3574: 3565: 3553: 3551: 3550: 3545: 3543: 3542: 3526: 3522: 3505: 3503: 3502: 3497: 3492: 3487: 3486: 3471: 3470: 3461: 3456: 3455: 3445: 3419:spectral theorem 3387: 3383: 3379: 3375: 3366: 3364: 3363: 3358: 3356: 3355: 3343: 3338: 3337: 3325: 3310: 3308: 3307: 3302: 3300: 3299: 3287: 3282: 3281: 3269: 3253: 3249: 3247: 3246: 3241: 3230: 3216: 3201: 3197: 3190: 3183: 3176: 3147: 3143: 3136: 3134: 3133: 3128: 3123: 3119: 3118: 3117: 3105: 3097: 3096: 3084: 3076: 3075: 3063: 3055: 3054: 3042: 3032: 3025: 3021: 3008: 3001: 2999: 2998: 2993: 2988: 2987: 2975: 2967: 2966: 2954: 2939: 2932: 2930: 2929: 2924: 2919: 2918: 2906: 2898: 2897: 2885: 2867: 2865: 2864: 2859: 2853: 2848: 2838: 2833: 2821: 2820: 2801: 2799: 2798: 2793: 2790: 2785: 2772: 2767: 2751: 2749: 2748: 2743: 2737: 2732: 2713: 2708: 2689: 2687: 2686: 2681: 2676: 2675: 2663: 2657: 2652: 2642: 2630: 2629: 2617: 2605: 2603: 2602: 2597: 2595: 2594: 2582: 2576: 2571: 2561: 2549: 2548: 2536: 2524: 2522: 2521: 2516: 2510: 2505: 2495: 2490: 2478: 2477: 2458: 2456: 2455: 2450: 2444: 2439: 2420: 2415: 2392: 2390: 2389: 2384: 2382: 2381: 2369: 2361: 2360: 2348: 2343: 2342: 2329: 2311: 2310: 2295: 2280: 2264: 2257: 2255: 2254: 2249: 2244: 2243: 2231: 2216: 2209: 2207: 2206: 2201: 2196: 2195: 2183: 2155: 2153: 2152: 2147: 2142: 2141: 2129: 2121: 2120: 2108: 2093: 2091: 2090: 2085: 2083: 2082: 2070: 2058: 2056: 2055: 2050: 2048: 2047: 2035: 2020: 2018: 2017: 2012: 2007: 2006: 1994: 1993: 1971: 1964: 1954: 1950: 1933:bra–ket notation 1893: 1891: 1890: 1885: 1883: 1882: 1824:Emergent gravity 1804: 1802: 1801: 1796: 1788: 1783: 1781: 1770: 1753: 1748: 1747: 1739: 1721: 1719: 1718: 1713: 1711: 1710: 1702: 1688: 1686: 1685: 1680: 1672: 1660: 1658: 1657: 1652: 1641: 1640: 1632: 1619: 1617: 1616: 1611: 1597: 1583: 1582: 1574: 1426:hidden variables 1411: 1402: 1389: 1380: 900: 893: 886: 527:Superdeterminism 180:Bra–ket notation 131: 129: 128: 123: 115: 110: 109: 101: 89: 84: 82: 71: 43: 42: 21: 15469: 15468: 15464: 15463: 15462: 15460: 15459: 15458: 15444: 15443: 15442: 15437: 15419: 15405:Wigner's friend 15381: 15372:Quantum gravity 15333: 15319:Quantum sensing 15299:Quantum network 15279:Quantum machine 15249:Quantum imaging 15212:Quantum circuit 15207:Quantum channel 15166: 15112:Quantum biology 15098: 15074:Elitzur–Vaidman 15049:Davisson–Germer 15032: 14984:Hidden-variable 14974:de Broglie–Bohm 14951:Interpretations 14945: 14909: 14863: 14750:Complementarity 14728: 14685: 14680: 14616: 14611: 14605: 14586: 14434:Physics Reports 14424: 14402: 14319: 14317:Further reading 14314: 14267: 14263: 14253: 14251: 14239: 14235: 14180: 14176: 14166: 14164: 14100: 14096: 14086: 14084: 14020: 14016: 14006: 14004: 13995: 13994: 13990: 13980: 13978: 13930: 13926: 13916: 13914: 13906: 13905: 13901: 13892: 13890: 13874: 13870: 13861: 13859: 13843: 13839: 13804: 13800: 13790: 13788: 13779: 13778: 13774: 13764: 13762: 13753: 13752: 13748: 13712: 13708: 13691: 13687: 13682:. 14 June 2017. 13678: 13677: 13673: 13626: 13622: 13612: 13610: 13601: 13600: 13596: 13565: 13561: 13504: 13500: 13443: 13439: 13382: 13378: 13321: 13317: 13260: 13256: 13199: 13195: 13146: 13142: 13093: 13089: 13044:Physics Reports 13040: 13036: 13005: 13001: 12952: 12948: 12891: 12887: 12838: 12834: 12793: 12789: 12784: 12780: 12723: 12719: 12670: 12666: 12627: 12623: 12566: 12562: 12512: 12508: 12469: 12465: 12426: 12422: 12372: 12366: 12362: 12312: 12308: 12267: 12263: 12215: 12211: 12164: 12160: 12136:10.1.1.251.7637 12101: 12097: 12047: 12041: 12037: 11988: 11984: 11939: 11935: 11920: 11897: 11893: 11848:Phys. Rev. Lett 11844: 11840: 11788: 11784: 11751: 11744: 11736: 11730: 11726: 11669: 11665: 11620:Phys. Rev. Lett 11612: 11608: 11555:Phys. Rev. Lett 11551: 11547: 11498: 11494: 11434: 11430: 11381: 11377: 11328: 11319: 11270: 11266: 11217: 11213: 11168: 11164: 11107: 11103: 11069: 11060: 11056: 10997: 10993: 10978: 10946: 10942: 10894: 10890: 10883: 10865: 10861: 10822: 10818: 10766: 10762: 10705: 10701: 10636: 10632: 10591: 10587: 10538: 10534: 10517: 10513: 10456: 10452: 10395: 10391: 10338: 10334: 10281: 10277: 10216: 10212: 10163: 10159: 10116: 10112: 10077: 10073: 10066: 10052: 10048: 10033: 10029: 10019: 10017: 9989: 9985: 9975: 9973: 9964: 9963: 9959: 9949: 9947: 9942: 9941: 9937: 9880: 9876: 9827: 9823: 9795:Physical Review 9790: 9786: 9747: 9743: 9734: 9730: 9677: 9673: 9634: 9630: 9574: 9570: 9555: 9551: 9544: 9528: 9524: 9505: 9501: 9491: 9489: 9474: 9473: 9469: 9418:Phys. Rev. Lett 9411: 9407: 9402:Wayback Machine 9388: 9384: 9345: 9341: 9284: 9280: 9249: 9245: 9208: 9204: 9181:(14): 938–941. 9167: 9158: 9129: 9125: 9094: 9090: 9081: 9077: 9048: 9044: 9035: 9031: 8984: 8980: 8935: 8931: 8922: 8921: 8917: 8907: 8905: 8895: 8891: 8819: 8815: 8802: 8793: 8772:(15): 575–577. 8758: 8751: 8741: 8739: 8731:Siegel, Ethan. 8729: 8725: 8718: 8704: 8700: 8693: 8676: 8672: 8662: 8660: 8650: 8643: 8616: 8609: 8552: 8545: 8535: 8533: 8529: 8523: 8508: 8495: 8491: 8452: 8448: 8409: 8402: 8379:(10): 777–780. 8361:Podolsky, Boris 8354: 8345: 8335: 8333: 8321:Overbye, Dennis 8318: 8314: 8310: 8299: 8292: 8289: 8265:Separable state 8240:Quantum network 8230:Quantum discord 8160: 8148:quantized light 8136: 8124: 8110:as well as the 8084: 8080: 8072: 8069: 8068: 8045: 8041: 8036: 8033: 8032: 8005: 8001: 7993: 7990: 7989: 7969: 7957: 7908: 7905: 7904: 7882: 7879: 7878: 7856: 7853: 7852: 7821:range criterion 7789: 7785: 7777: 7774: 7773: 7749: 7744: 7725: 7720: 7710: 7706: 7700: 7689: 7687: 7684: 7683: 7680: 7665:atomic cascades 7625: 7577: 7573: 7568: 7566: 7563: 7562: 7530: 7526: 7521: 7516: 7510: 7506: 7501: 7496: 7487: 7483: 7475: 7469: 7465: 7457: 7456: 7454: 7442: 7438: 7433: 7431: 7428: 7427: 7371: 7368: 7367: 7345: 7342: 7341: 7321: 7317: 7312: 7310: 7307: 7306: 7272: 7268: 7260: 7248: 7244: 7236: 7235: 7233: 7216: 7211: 7209: 7206: 7205: 7154: 7150: 7142: 7133: 7129: 7121: 7112: 7108: 7100: 7091: 7087: 7079: 7064: 7052: 7048: 7043: 7041: 7038: 7037: 7016: 7012: 7004: 6995: 6991: 6983: 6974: 6970: 6962: 6953: 6949: 6941: 6926: 6914: 6910: 6905: 6903: 6900: 6899: 6882: 6839: 6823: 6768: 6749: 6709: 6707: 6703: 6680: 6677: 6676: 6669: 6668:In particular, 6638: 6634: 6627: 6623: 6603: 6600: 6599: 6592: 6578: 6544: 6543: 6533: 6528: 6526: 6519: 6516: 6506: 6499: 6498: 6496: 6493: 6492: 6485: 6478: 6471: 6461: 6446: 6440:mixed states). 6437: 6433: 6406: 6402: 6387: 6383: 6362: 6358: 6353: 6350: 6349: 6323: 6319: 6302: 6299: 6298: 6288: 6284: 6283:the convention 6237: 6231: 6228: 6227: 6203: 6199: 6190: 6186: 6180: 6176: 6170: 6150: 6141: 6137: 6133: 6129: 6122: 6102: 6099: 6098: 6075: 6071: 6059: 6055: 6037: 6033: 6021: 6017: 6015: 6012: 6011: 6003: 5999: 5982: 5978: 5963: 5959: 5957: 5954: 5953: 5949: 5941: 5937: 5906: 5897: 5893: 5889: 5885: 5878: 5858: 5855: 5854: 5843: 5820: 5816: 5807: 5803: 5797: 5793: 5787: 5768: 5764: 5749: 5745: 5737: 5734: 5733: 5713: 5709: 5694: 5690: 5688: 5685: 5684: 5681:Shannon entropy 5676: 5662: 5654: 5648:are non-empty. 5627:bound entangled 5609: 5606: 5605: 5588: 5584: 5575: 5571: 5562: 5558: 5556: 5553: 5552: 5535: 5531: 5522: 5518: 5516: 5513: 5512: 5495: 5491: 5482: 5478: 5476: 5473: 5472: 5447: 5445: 5442: 5441: 5419: 5402: 5400: 5397: 5396: 5378: 5374: 5365: 5361: 5359: 5356: 5355: 5338: 5334: 5325: 5321: 5319: 5316: 5315: 5298: 5294: 5292: 5289: 5288: 5271: 5267: 5265: 5262: 5261: 5242: 5224: 5205:AKLT spin chain 5201: 5175: 5170: 5169: 5157: 5153: 5145: 5136: 5132: 5130: 5127: 5126: 5106: 5102: 5094: 5085: 5081: 5073: 5071: 5068: 5067: 5063: 5035: 5030: 5029: 5017: 5013: 5005: 4996: 4991: 4990: 4978: 4974: 4966: 4965: 4961: 4949: 4940: 4936: 4934: 4931: 4930: 4899: 4895: 4887: 4878: 4874: 4866: 4857: 4853: 4845: 4836: 4832: 4824: 4823: 4819: 4805: 4803: 4800: 4799: 4792: 4785: 4781: 4777: 4773: 4770: 4766: 4749: 4745: 4743: 4740: 4739: 4722: 4718: 4716: 4713: 4712: 4692: 4688: 4670: 4666: 4664: 4661: 4660: 4637: 4633: 4626: 4619: 4618: 4604: 4600: 4592: 4583: 4579: 4578: 4574: 4564: 4547: 4546: 4542: 4531: 4526: 4525: 4510: 4506: 4505: 4501: 4493: 4489: 4488: 4483: 4461: 4460: 4455: 4453: 4452: 4443: 4439: 4437: 4434: 4433: 4426: 4423: 4419: 4411: 4387: 4369: 4360: 4356: 4354: 4351: 4350: 4343: 4320: 4316: 4307: 4303: 4289: 4287: 4284: 4283: 4277: 4273: 4269: 4265: 4261: 4254: 4241: 4237: 4213: 4208: 4202: 4199: 4198: 4181: 4176: 4170: 4167: 4166: 4161: 4157: 4140: 4135: 4129: 4126: 4125: 4108: 4103: 4097: 4094: 4093: 4090: 4086: 4063: 4058: 4045: 4040: 4030: 4026: 4020: 4008: 4005: 4004: 3975: 3970: 3969: 3960: 3956: 3951: 3945: 3939: 3936: 3935: 3933: 3901: 3892: 3888: 3877: 3868: 3864: 3849: 3845: 3839: 3834: 3830: 3810: 3806: 3801: 3786: 3782: 3777: 3765: 3754: 3753: 3752: 3746: 3741: 3737: 3731: 3727: 3721: 3709: 3706: 3705: 3698: 3691: 3686: 3677: 3655: 3650: 3648: 3645: 3644: 3638: 3616: 3611: 3609: 3606: 3605: 3570: 3566: 3561: 3559: 3556: 3555: 3538: 3534: 3532: 3529: 3528: 3524: 3521: 3517: 3515: 3488: 3482: 3478: 3466: 3462: 3457: 3451: 3447: 3441: 3429: 3426: 3425: 3402: 3385: 3381: 3377: 3373: 3351: 3347: 3339: 3333: 3329: 3321: 3319: 3316: 3315: 3295: 3291: 3283: 3277: 3273: 3265: 3263: 3260: 3259: 3251: 3226: 3212: 3207: 3204: 3203: 3199: 3195: 3189: 3185: 3182: 3178: 3174: 3167: 3162: 3145: 3141: 3113: 3109: 3101: 3092: 3088: 3080: 3071: 3067: 3059: 3050: 3046: 3038: 3037: 3033: 3019: 3017: 3014: 3013: 3007: 3003: 2983: 2979: 2971: 2962: 2958: 2950: 2945: 2942: 2941: 2938: 2934: 2914: 2910: 2902: 2893: 2889: 2881: 2876: 2873: 2872: 2849: 2844: 2834: 2829: 2813: 2809: 2807: 2804: 2803: 2786: 2781: 2768: 2763: 2757: 2754: 2753: 2733: 2728: 2709: 2704: 2695: 2692: 2691: 2671: 2667: 2659: 2653: 2648: 2638: 2625: 2621: 2613: 2611: 2608: 2607: 2590: 2586: 2578: 2572: 2567: 2557: 2544: 2540: 2532: 2530: 2527: 2526: 2506: 2501: 2491: 2486: 2470: 2466: 2464: 2461: 2460: 2440: 2435: 2416: 2411: 2402: 2399: 2398: 2377: 2373: 2365: 2356: 2352: 2344: 2335: 2331: 2319: 2303: 2299: 2291: 2289: 2286: 2285: 2281:is of the form 2278: 2271: 2266: 2263: 2259: 2239: 2235: 2227: 2222: 2219: 2218: 2215: 2211: 2191: 2187: 2179: 2174: 2171: 2170: 2137: 2133: 2125: 2116: 2112: 2104: 2102: 2099: 2098: 2078: 2074: 2066: 2064: 2061: 2060: 2043: 2039: 2031: 2029: 2026: 2025: 2002: 1998: 1989: 1985: 1983: 1980: 1979: 1970: 1966: 1963: 1959: 1952: 1948: 1945: 1928: 1872: 1868: 1866: 1863: 1862: 1850: 1842:Induced gravity 1836:suggested that 1826: 1784: 1774: 1769: 1749: 1738: 1737: 1735: 1732: 1731: 1701: 1700: 1698: 1695: 1694: 1693:. The operator 1689:stands for the 1668: 1666: 1663: 1662: 1631: 1630: 1628: 1625: 1624: 1621: 1593: 1573: 1572: 1570: 1567: 1566: 1521:problem of time 1517: 1478:Chien-Shiung Wu 1470: 1434: 1421: 1410: 1404: 1401: 1395: 1392:inertial frames 1388: 1382: 1379: 1373: 1350: 1284: 1279: 1271:Anton Zeilinger 1248:Gilles Brassard 1204:Stuart Freedman 1099:Albert Einstein 1078: 1072: 973:Albert Einstein 928:quantum physics 904: 875: 874: 873: 638: 630: 629: 575: 574:Advanced topics 567: 566: 565: 517:Hidden-variable 507:de Broglie–Bohm 486: 484:Interpretations 476: 475: 474: 444: 436: 435: 434: 392: 384: 383: 382: 349: 305:CHSH inequality 294: 286: 285: 284: 213:Complementarity 207: 199: 198: 197: 165: 136: 111: 100: 99: 85: 75: 70: 62: 59: 58: 28: 23: 22: 15: 12: 11: 5: 15467: 15457: 15456: 15439: 15438: 15436: 15435: 15424: 15421: 15420: 15418: 15417: 15412: 15407: 15402: 15401: 15400: 15389: 15387: 15383: 15382: 15380: 15379: 15374: 15369: 15368: 15367: 15357: 15352: 15350:Casimir effect 15347: 15341: 15339: 15335: 15334: 15332: 15331: 15326: 15321: 15316: 15311: 15309:Quantum optics 15306: 15301: 15296: 15291: 15286: 15281: 15276: 15271: 15266: 15261: 15256: 15251: 15246: 15241: 15236: 15231: 15230: 15229: 15219: 15214: 15209: 15204: 15203: 15202: 15192: 15187: 15182: 15176: 15174: 15168: 15167: 15165: 15164: 15159: 15154: 15149: 15144: 15139: 15134: 15129: 15124: 15119: 15114: 15108: 15106: 15100: 15099: 15097: 15096: 15091: 15086: 15084:Quantum eraser 15081: 15076: 15071: 15066: 15061: 15056: 15051: 15046: 15040: 15038: 15034: 15033: 15031: 15030: 15025: 15020: 15015: 15010: 15005: 15000: 14999: 14998: 14997: 14996: 14981: 14976: 14971: 14966: 14961: 14955: 14953: 14947: 14946: 14944: 14943: 14938: 14933: 14928: 14923: 14917: 14915: 14911: 14910: 14908: 14907: 14902: 14897: 14892: 14887: 14882: 14877: 14871: 14869: 14865: 14864: 14862: 14861: 14860: 14859: 14854: 14844: 14839: 14834: 14829: 14824: 14819: 14814: 14809: 14804: 14799: 14794: 14789: 14784: 14783: 14782: 14777: 14772: 14767: 14757: 14755:Density matrix 14752: 14747: 14742: 14736: 14734: 14730: 14729: 14727: 14726: 14721: 14716: 14711: 14710: 14709: 14699: 14693: 14691: 14687: 14686: 14679: 14678: 14671: 14664: 14656: 14650: 14649: 14644: 14638: 14635:Astronomy Cast 14631: 14626: 14615: 14614:External links 14612: 14610: 14609: 14603: 14590: 14584: 14571: 14524: 14477: 14428: 14422: 14406: 14400: 14387: 14372: 14359:Bengtsson I.; 14356: 14320: 14318: 14315: 14313: 14312: 14261: 14233: 14194:(10): 101001. 14174: 14094: 14014: 13988: 13944:(2): 228–233. 13938:Nature Physics 13924: 13899: 13868: 13837: 13798: 13772: 13746: 13706: 13692:Frank Jensen: 13685: 13671: 13620: 13594: 13559: 13514:(16): 160505. 13498: 13437: 13392:(23): 230502. 13376: 13315: 13254: 13193: 13140: 13087: 13034: 12999: 12946: 12901:(11): 117903. 12885: 12832: 12787: 12778: 12733:(24): 240403. 12717: 12664: 12621: 12560: 12506: 12463: 12420: 12360: 12306: 12261: 12209: 12174:(6): 661–663. 12158: 12095: 12035: 11982: 11933: 11918: 11891: 11838: 11801:(1): 197–234. 11791:Vedral, Vlatko 11782: 11742: 11724: 11679:(14): 140402. 11663: 11606: 11545: 11492: 11437:Kraus, Barbara 11428: 11375: 11338:(2): 865–942. 11317: 11264: 11211: 11182:(3): 325–341. 11162: 11117:(22): 227203. 11101: 11080:(3): 376–385. 11054: 11032:10.1.1.252.496 10991: 10976: 10952:. p. 10. 10940: 10904:(6): 655–701, 10888: 10881: 10859: 10816: 10760: 10699: 10630: 10585: 10548:(2): 419–478. 10532: 10511: 10450: 10389: 10332: 10275: 10220:"Quantum time" 10210: 10157: 10110: 10071: 10064: 10046: 10027: 9983: 9972:on 4 July 2018 9957: 9935: 9890:(21): 210403. 9874: 9837:(6): 480–485. 9831:Nature Physics 9821: 9784: 9741: 9728: 9671: 9628: 9584:(17): 170404, 9568: 9549: 9542: 9522: 9499: 9467: 9424:(6): 661–663. 9405: 9382: 9339: 9278: 9243: 9202: 9156: 9143:(3): 195–200. 9123: 9116:10.1.1.20.8324 9103:SIAM J. Comput 9088: 9075: 9042: 9029: 8978: 8929: 8915: 8889: 8813: 8791: 8749: 8723: 8716: 8698: 8691: 8679:Penrose, Roger 8670: 8641: 8607: 8562:(26): 260407. 8543: 8521: 8489: 8462:(3): 446–452. 8446: 8419:(4): 555–563. 8400: 8343: 8311: 8309: 8306: 8305: 8304: 8301:Physics portal 8288: 8287: 8282: 8277: 8272: 8267: 8262: 8260:Retrocausality 8257: 8252: 8247: 8242: 8237: 8232: 8227: 8222: 8217: 8212: 8207: 8202: 8197: 8192: 8187: 8182: 8177: 8172: 8167: 8161: 8159: 8156: 8135: 8132: 8123: 8120: 8095: 8090: 8087: 8083: 8079: 8076: 8056: 8051: 8048: 8044: 8040: 8016: 8011: 8008: 8004: 8000: 7997: 7968: 7965: 7956: 7953: 7936:in China, the 7918: 7915: 7912: 7892: 7889: 7886: 7866: 7863: 7860: 7800: 7797: 7792: 7788: 7784: 7781: 7758: 7755: 7752: 7747: 7743: 7739: 7734: 7731: 7728: 7723: 7719: 7713: 7709: 7703: 7699: 7695: 7692: 7679: 7676: 7624: 7621: 7585: 7580: 7576: 7571: 7559: 7558: 7546: 7540: 7533: 7529: 7524: 7519: 7513: 7509: 7504: 7499: 7495: 7490: 7486: 7482: 7478: 7472: 7468: 7464: 7460: 7453: 7450: 7441: 7436: 7381: 7378: 7375: 7355: 7352: 7349: 7329: 7324: 7320: 7315: 7303: 7302: 7291: 7285: 7278: 7275: 7271: 7267: 7263: 7259: 7254: 7251: 7247: 7243: 7239: 7232: 7229: 7225: 7222: 7219: 7214: 7192:Bell's theorem 7183:) and form an 7177: 7176: 7165: 7162: 7157: 7153: 7149: 7145: 7141: 7136: 7132: 7128: 7124: 7120: 7115: 7111: 7107: 7103: 7099: 7094: 7090: 7086: 7082: 7078: 7072: 7068: 7063: 7060: 7055: 7051: 7046: 7035: 7024: 7019: 7015: 7011: 7007: 7003: 6998: 6994: 6990: 6986: 6982: 6977: 6973: 6969: 6965: 6961: 6956: 6952: 6948: 6944: 6940: 6934: 6930: 6925: 6922: 6917: 6913: 6908: 6881: 6878: 6838: 6835: 6822: 6819: 6811: 6810: 6805: 6800: 6795: 6790: 6785: 6780: 6767: 6764: 6746: 6745: 6734: 6730: 6725: 6721: 6718: 6715: 6712: 6706: 6702: 6699: 6696: 6693: 6690: 6687: 6684: 6663: 6662: 6651: 6647: 6641: 6637: 6633: 6630: 6626: 6622: 6619: 6616: 6613: 6610: 6607: 6565: 6564: 6553: 6548: 6540: 6537: 6532: 6529: 6525: 6522: 6520: 6513: 6510: 6505: 6504: 6502: 6476: 6469: 6445: 6442: 6426: 6425: 6414: 6409: 6405: 6401: 6398: 6395: 6390: 6386: 6382: 6379: 6376: 6373: 6370: 6365: 6361: 6357: 6343: 6342: 6331: 6326: 6322: 6318: 6315: 6312: 6309: 6306: 6281: 6280: 6269: 6266: 6263: 6260: 6257: 6254: 6251: 6246: 6243: 6240: 6236: 6221: 6220: 6206: 6202: 6198: 6193: 6189: 6183: 6179: 6173: 6169: 6165: 6162: 6158: 6153: 6149: 6144: 6140: 6136: 6132: 6121: 6118: 6115: 6112: 6109: 6106: 6083: 6078: 6074: 6070: 6067: 6062: 6058: 6054: 6051: 6048: 6045: 6040: 6036: 6032: 6029: 6024: 6020: 6001: 5985: 5981: 5977: 5974: 5971: 5966: 5962: 5939: 5930: 5929: 5918: 5914: 5909: 5905: 5900: 5896: 5892: 5888: 5877: 5874: 5871: 5868: 5865: 5862: 5840: 5839: 5828: 5823: 5819: 5815: 5810: 5806: 5800: 5796: 5790: 5786: 5782: 5779: 5776: 5771: 5767: 5763: 5760: 5757: 5752: 5748: 5744: 5741: 5716: 5712: 5708: 5705: 5702: 5697: 5693: 5661: 5658: 5653: 5650: 5631: 5630: 5613: 5591: 5587: 5583: 5578: 5574: 5570: 5565: 5561: 5538: 5534: 5530: 5525: 5521: 5498: 5494: 5490: 5485: 5481: 5469: 5457: 5454: 5450: 5429: 5426: 5422: 5418: 5415: 5412: 5409: 5405: 5381: 5377: 5373: 5368: 5364: 5341: 5337: 5333: 5328: 5324: 5301: 5297: 5274: 5270: 5257: 5241: 5238: 5223: 5220: 5216:XY spin chains 5200: 5197: 5193: 5192: 5178: 5173: 5168: 5165: 5160: 5156: 5152: 5148: 5144: 5139: 5135: 5109: 5105: 5101: 5097: 5093: 5088: 5084: 5080: 5076: 5060: 5059: 5048: 5044: 5038: 5033: 5028: 5025: 5020: 5016: 5012: 5008: 5004: 4999: 4994: 4989: 4986: 4981: 4977: 4973: 4969: 4964: 4957: 4954: 4948: 4943: 4939: 4924: 4923: 4912: 4908: 4902: 4898: 4894: 4890: 4886: 4881: 4877: 4873: 4869: 4865: 4860: 4856: 4852: 4848: 4844: 4839: 4835: 4831: 4827: 4822: 4814: 4810: 4768: 4752: 4748: 4725: 4721: 4700: 4695: 4691: 4687: 4684: 4681: 4678: 4673: 4669: 4657: 4656: 4645: 4640: 4636: 4629: 4617: 4613: 4607: 4603: 4599: 4595: 4591: 4586: 4582: 4577: 4572: 4567: 4563: 4560: 4557: 4554: 4550: 4545: 4540: 4534: 4529: 4524: 4521: 4518: 4513: 4509: 4504: 4496: 4492: 4486: 4482: 4470: 4467: 4464: 4458: 4446: 4442: 4421: 4404: 4403: 4390: 4386: 4383: 4379: 4376: 4372: 4368: 4363: 4359: 4340: 4339: 4328: 4323: 4319: 4315: 4310: 4306: 4302: 4299: 4296: 4292: 4275: 4271: 4253: 4250: 4216: 4211: 4207: 4184: 4179: 4175: 4143: 4138: 4134: 4111: 4106: 4102: 4088: 4083: 4082: 4071: 4066: 4061: 4057: 4053: 4048: 4043: 4039: 4033: 4029: 4023: 4019: 4015: 4012: 3986: 3983: 3978: 3973: 3966: 3963: 3959: 3954: 3948: 3944: 3931: 3925: 3924: 3912: 3908: 3904: 3898: 3895: 3891: 3887: 3884: 3880: 3874: 3871: 3867: 3863: 3860: 3855: 3852: 3848: 3842: 3838: 3833: 3828: 3824: 3821: 3816: 3813: 3809: 3804: 3800: 3797: 3792: 3789: 3785: 3780: 3776: 3771: 3768: 3761: 3758: 3749: 3745: 3740: 3734: 3730: 3724: 3720: 3716: 3713: 3696: 3689: 3665: 3662: 3658: 3653: 3626: 3623: 3619: 3614: 3578: 3573: 3569: 3564: 3541: 3537: 3519: 3513: 3507: 3506: 3495: 3491: 3485: 3481: 3477: 3474: 3469: 3465: 3460: 3454: 3450: 3444: 3440: 3436: 3433: 3407:density matrix 3401: 3398: 3369: 3368: 3354: 3350: 3346: 3342: 3336: 3332: 3328: 3324: 3312: 3298: 3294: 3290: 3286: 3280: 3276: 3272: 3268: 3250:eigenbasis of 3239: 3236: 3233: 3229: 3225: 3222: 3219: 3215: 3211: 3187: 3180: 3172: 3165: 3138: 3137: 3126: 3122: 3116: 3112: 3108: 3104: 3100: 3095: 3091: 3087: 3083: 3079: 3074: 3070: 3066: 3062: 3058: 3053: 3049: 3045: 3041: 3036: 3028: 3024: 3005: 2991: 2986: 2982: 2978: 2974: 2970: 2965: 2961: 2957: 2953: 2949: 2936: 2922: 2917: 2913: 2909: 2905: 2901: 2896: 2892: 2888: 2884: 2880: 2857: 2852: 2847: 2843: 2837: 2832: 2828: 2824: 2819: 2816: 2812: 2789: 2784: 2780: 2776: 2771: 2766: 2762: 2741: 2736: 2731: 2727: 2723: 2720: 2717: 2712: 2707: 2703: 2699: 2679: 2674: 2670: 2666: 2662: 2656: 2651: 2647: 2641: 2637: 2633: 2628: 2624: 2620: 2616: 2593: 2589: 2585: 2581: 2575: 2570: 2566: 2560: 2556: 2552: 2547: 2543: 2539: 2535: 2514: 2509: 2504: 2500: 2494: 2489: 2485: 2481: 2476: 2473: 2469: 2448: 2443: 2438: 2434: 2430: 2427: 2424: 2419: 2414: 2410: 2406: 2395: 2394: 2380: 2376: 2372: 2368: 2364: 2359: 2355: 2351: 2347: 2341: 2338: 2334: 2328: 2325: 2322: 2318: 2314: 2309: 2306: 2302: 2298: 2294: 2276: 2269: 2261: 2247: 2242: 2238: 2234: 2230: 2226: 2213: 2199: 2194: 2190: 2186: 2182: 2178: 2161:product states 2157: 2156: 2145: 2140: 2136: 2132: 2128: 2124: 2119: 2115: 2111: 2107: 2081: 2077: 2073: 2069: 2046: 2042: 2038: 2034: 2022: 2021: 2010: 2005: 2001: 1997: 1992: 1988: 1974:tensor product 1968: 1961: 1957:Hilbert spaces 1944: 1941: 1927: 1924: 1881: 1878: 1875: 1871: 1849: 1846: 1825: 1822: 1794: 1791: 1787: 1780: 1777: 1773: 1768: 1765: 1762: 1759: 1756: 1752: 1745: 1742: 1708: 1705: 1678: 1675: 1671: 1650: 1647: 1644: 1638: 1635: 1609: 1606: 1603: 1600: 1596: 1592: 1589: 1586: 1580: 1577: 1564: 1554:to vanish. In 1539:acting as the 1516: 1513: 1469: 1466: 1433: 1430: 1420: 1417: 1408: 1399: 1386: 1377: 1349: 1346: 1283: 1280: 1278: 1275: 1103:Boris Podolsky 1071: 1068: 977:Boris Podolsky 906: 905: 903: 902: 895: 888: 880: 877: 876: 872: 871: 866: 861: 856: 851: 846: 841: 836: 831: 826: 821: 816: 811: 806: 801: 796: 791: 786: 781: 776: 771: 766: 761: 756: 751: 746: 741: 736: 731: 726: 721: 716: 711: 706: 701: 696: 691: 686: 681: 676: 671: 666: 661: 656: 651: 646: 640: 639: 636: 635: 632: 631: 628: 627: 622: 617: 612: 610:Density matrix 607: 602: 597: 592: 587: 582: 576: 573: 572: 569: 568: 564: 563: 558: 553: 548: 543: 538: 533: 532: 531: 530: 529: 514: 509: 504: 499: 494: 488: 487: 482: 481: 478: 477: 473: 472: 467: 462: 457: 452: 446: 445: 442: 441: 438: 437: 433: 432: 427: 422: 417: 412: 407: 401: 400: 399: 393: 390: 389: 386: 385: 381: 380: 375: 370: 364: 363: 362: 361: 360: 358:Delayed-choice 353:Quantum eraser 348: 347: 342: 337: 332: 327: 322: 317: 312: 307: 302: 296: 295: 292: 291: 288: 287: 283: 282: 281: 280: 270: 265: 260: 255: 250: 245: 243:Quantum number 240: 235: 230: 225: 220: 215: 209: 208: 205: 204: 201: 200: 196: 195: 190: 184: 183: 182: 177: 172: 166: 163: 162: 159: 158: 157: 156: 151: 146: 138: 137: 132: 121: 118: 114: 107: 104: 98: 95: 92: 88: 81: 78: 74: 69: 66: 55: 54: 48: 47: 26: 9: 6: 4: 3: 2: 15466: 15455: 15452: 15451: 15449: 15434: 15426: 15425: 15422: 15416: 15413: 15411: 15408: 15406: 15403: 15399: 15396: 15395: 15394: 15391: 15390: 15388: 15384: 15378: 15375: 15373: 15370: 15366: 15363: 15362: 15361: 15358: 15356: 15353: 15351: 15348: 15346: 15343: 15342: 15340: 15336: 15330: 15327: 15325: 15322: 15320: 15317: 15315: 15312: 15310: 15307: 15305: 15302: 15300: 15297: 15295: 15292: 15290: 15287: 15285: 15282: 15280: 15277: 15275: 15272: 15270: 15269:Quantum logic 15267: 15265: 15262: 15260: 15257: 15255: 15252: 15250: 15247: 15245: 15242: 15240: 15237: 15235: 15232: 15228: 15225: 15224: 15223: 15220: 15218: 15215: 15213: 15210: 15208: 15205: 15201: 15198: 15197: 15196: 15193: 15191: 15188: 15186: 15183: 15181: 15178: 15177: 15175: 15173: 15169: 15163: 15160: 15158: 15155: 15153: 15150: 15148: 15145: 15143: 15140: 15138: 15135: 15133: 15130: 15128: 15125: 15123: 15122:Quantum chaos 15120: 15118: 15115: 15113: 15110: 15109: 15107: 15105: 15101: 15095: 15092: 15090: 15089:Stern–Gerlach 15087: 15085: 15082: 15080: 15077: 15075: 15072: 15070: 15067: 15065: 15062: 15060: 15057: 15055: 15052: 15050: 15047: 15045: 15042: 15041: 15039: 15035: 15029: 15026: 15024: 15023:Transactional 15021: 15019: 15016: 15014: 15013:Quantum logic 15011: 15009: 15006: 15004: 15001: 14995: 14992: 14991: 14990: 14987: 14986: 14985: 14982: 14980: 14977: 14975: 14972: 14970: 14967: 14965: 14962: 14960: 14957: 14956: 14954: 14952: 14948: 14942: 14939: 14937: 14934: 14932: 14929: 14927: 14924: 14922: 14919: 14918: 14916: 14912: 14906: 14903: 14901: 14898: 14896: 14893: 14891: 14888: 14886: 14883: 14881: 14878: 14876: 14873: 14872: 14870: 14866: 14858: 14855: 14853: 14850: 14849: 14848: 14847:Wave function 14845: 14843: 14840: 14838: 14835: 14833: 14830: 14828: 14825: 14823: 14822:Superposition 14820: 14818: 14817:Quantum state 14815: 14813: 14810: 14808: 14805: 14803: 14800: 14798: 14795: 14793: 14790: 14788: 14785: 14781: 14778: 14776: 14773: 14771: 14770:Excited state 14768: 14766: 14763: 14762: 14761: 14758: 14756: 14753: 14751: 14748: 14746: 14743: 14741: 14738: 14737: 14735: 14731: 14725: 14722: 14720: 14717: 14715: 14712: 14708: 14705: 14704: 14703: 14700: 14698: 14695: 14694: 14692: 14688: 14684: 14677: 14672: 14670: 14665: 14663: 14658: 14657: 14654: 14648: 14645: 14642: 14639: 14636: 14632: 14630: 14627: 14625: 14623: 14618: 14617: 14606: 14600: 14596: 14591: 14587: 14581: 14577: 14572: 14568: 14564: 14560: 14556: 14552: 14548: 14543: 14538: 14535:(5): 052441. 14534: 14530: 14525: 14521: 14517: 14513: 14509: 14505: 14501: 14496: 14491: 14487: 14483: 14478: 14474: 14470: 14466: 14462: 14458: 14454: 14449: 14444: 14441:(1–6): 1–75. 14440: 14436: 14435: 14429: 14425: 14419: 14415: 14411: 14407: 14403: 14397: 14393: 14388: 14384: 14383: 14378: 14373: 14371: 14366: 14362: 14361:Życzkowski K. 14357: 14353: 14349: 14345: 14341: 14337: 14333: 14332: 14327: 14322: 14321: 14307: 14302: 14298: 14294: 14289: 14284: 14280: 14276: 14272: 14265: 14250: 14249: 14244: 14237: 14229: 14225: 14220: 14215: 14211: 14207: 14202: 14197: 14193: 14189: 14185: 14178: 14163: 14159: 14155: 14151: 14147: 14143: 14139: 14135: 14131: 14127: 14122: 14117: 14113: 14109: 14105: 14098: 14083: 14079: 14075: 14071: 14067: 14063: 14059: 14055: 14051: 14047: 14042: 14037: 14033: 14029: 14025: 14018: 14003:. 17 May 2021 14002: 14001:Physics World 13998: 13992: 13977: 13973: 13969: 13965: 13961: 13957: 13952: 13947: 13943: 13939: 13935: 13928: 13913: 13909: 13903: 13889: 13885: 13881: 13880: 13872: 13858: 13854: 13850: 13849: 13841: 13833: 13829: 13825: 13821: 13817: 13813: 13809: 13802: 13786: 13782: 13776: 13760: 13756: 13750: 13742: 13738: 13734: 13730: 13726: 13722: 13718: 13710: 13703: 13699: 13696:Wiley, 2007, 13695: 13689: 13681: 13675: 13667: 13663: 13658: 13653: 13648: 13643: 13639: 13635: 13631: 13624: 13608: 13604: 13598: 13590: 13586: 13582: 13578: 13574: 13570: 13563: 13555: 13551: 13547: 13543: 13539: 13535: 13531: 13527: 13522: 13517: 13513: 13509: 13502: 13494: 13490: 13486: 13482: 13478: 13474: 13470: 13466: 13461: 13456: 13453:(5): 050503. 13452: 13448: 13441: 13433: 13429: 13425: 13421: 13417: 13413: 13409: 13405: 13400: 13395: 13391: 13387: 13380: 13372: 13368: 13364: 13360: 13356: 13352: 13348: 13344: 13339: 13334: 13330: 13326: 13319: 13311: 13307: 13303: 13299: 13295: 13291: 13287: 13283: 13278: 13273: 13269: 13265: 13258: 13250: 13246: 13242: 13238: 13234: 13230: 13226: 13222: 13217: 13212: 13208: 13204: 13197: 13189: 13185: 13181: 13177: 13173: 13169: 13164: 13159: 13156:(1): 012313. 13155: 13151: 13144: 13136: 13132: 13128: 13124: 13120: 13116: 13111: 13106: 13102: 13098: 13091: 13083: 13079: 13075: 13071: 13067: 13063: 13058: 13053: 13050:(1–6): 1–75. 13049: 13045: 13038: 13030: 13026: 13022: 13018: 13015:(1): 012335. 13014: 13010: 13003: 12995: 12991: 12987: 12983: 12979: 12975: 12970: 12965: 12962:(2): 022316. 12961: 12957: 12950: 12942: 12938: 12934: 12930: 12926: 12922: 12918: 12914: 12909: 12904: 12900: 12896: 12889: 12881: 12877: 12873: 12869: 12865: 12861: 12856: 12851: 12848:(3): 032103. 12847: 12843: 12836: 12828: 12824: 12820: 12816: 12811: 12806: 12802: 12798: 12791: 12782: 12774: 12770: 12766: 12762: 12758: 12754: 12750: 12746: 12741: 12736: 12732: 12728: 12721: 12713: 12709: 12705: 12701: 12697: 12693: 12688: 12683: 12680:(6): 062437. 12679: 12675: 12668: 12660: 12656: 12652: 12648: 12644: 12640: 12636: 12632: 12625: 12617: 12613: 12609: 12605: 12601: 12597: 12593: 12589: 12584: 12579: 12576:(2): 130501. 12575: 12571: 12564: 12556: 12552: 12548: 12544: 12540: 12536: 12531: 12526: 12522: 12518: 12510: 12502: 12498: 12494: 12490: 12486: 12482: 12478: 12474: 12467: 12459: 12455: 12451: 12447: 12443: 12439: 12435: 12431: 12424: 12416: 12412: 12407: 12402: 12398: 12394: 12390: 12386: 12382: 12378: 12371: 12364: 12356: 12352: 12348: 12344: 12340: 12336: 12331: 12326: 12322: 12318: 12310: 12302: 12298: 12294: 12290: 12285: 12280: 12276: 12272: 12265: 12257: 12253: 12249: 12245: 12241: 12237: 12233: 12229: 12225: 12221: 12213: 12205: 12201: 12197: 12193: 12189: 12185: 12181: 12177: 12173: 12169: 12162: 12154: 12150: 12146: 12142: 12137: 12132: 12128: 12124: 12119: 12114: 12110: 12106: 12099: 12091: 12087: 12083: 12082:10.1038/37539 12079: 12075: 12071: 12066: 12061: 12057: 12053: 12046: 12039: 12031: 12027: 12023: 12019: 12015: 12011: 12006: 12001: 11998:(3): 033027. 11997: 11993: 11986: 11978: 11974: 11970: 11966: 11961: 11956: 11952: 11948: 11944: 11937: 11929: 11925: 11921: 11919:0-7923-2549-4 11915: 11911: 11907: 11906: 11901: 11895: 11887: 11883: 11879: 11875: 11871: 11867: 11862: 11857: 11853: 11849: 11842: 11834: 11830: 11826: 11822: 11818: 11814: 11809: 11804: 11800: 11796: 11792: 11786: 11778: 11774: 11769: 11764: 11760: 11756: 11749: 11747: 11735: 11728: 11720: 11716: 11712: 11708: 11704: 11700: 11696: 11692: 11687: 11682: 11678: 11674: 11667: 11659: 11655: 11651: 11647: 11643: 11639: 11634: 11629: 11625: 11621: 11617: 11610: 11602: 11598: 11594: 11590: 11586: 11582: 11578: 11574: 11569: 11564: 11561:(6): 060502. 11560: 11556: 11549: 11541: 11537: 11533: 11529: 11525: 11521: 11516: 11511: 11507: 11503: 11496: 11488: 11484: 11480: 11476: 11472: 11468: 11464: 11460: 11455: 11450: 11447:(2): 020504. 11446: 11442: 11438: 11432: 11424: 11420: 11416: 11412: 11408: 11404: 11399: 11394: 11390: 11386: 11379: 11371: 11367: 11363: 11359: 11355: 11351: 11346: 11341: 11337: 11333: 11326: 11324: 11322: 11313: 11309: 11305: 11301: 11297: 11293: 11288: 11283: 11279: 11275: 11268: 11260: 11256: 11252: 11248: 11244: 11240: 11235: 11230: 11227:(2): 025001. 11226: 11222: 11215: 11207: 11203: 11199: 11195: 11190: 11185: 11181: 11177: 11173: 11166: 11158: 11154: 11150: 11146: 11142: 11138: 11134: 11130: 11125: 11120: 11116: 11112: 11105: 11096: 11091: 11087: 11083: 11079: 11075: 11068: 11064: 11058: 11050: 11046: 11042: 11038: 11033: 11028: 11024: 11020: 11015: 11010: 11006: 11002: 10995: 10987: 10983: 10979: 10973: 10969: 10965: 10960: 10955: 10951: 10944: 10937: 10933: 10929: 10925: 10921: 10917: 10912: 10907: 10903: 10899: 10892: 10884: 10878: 10874: 10870: 10863: 10855: 10851: 10847: 10843: 10839: 10835: 10831: 10827: 10820: 10811: 10807: 10803: 10799: 10795: 10791: 10786: 10781: 10777: 10773: 10764: 10756: 10752: 10748: 10744: 10740: 10736: 10732: 10728: 10723: 10718: 10714: 10710: 10703: 10695: 10691: 10687: 10683: 10679: 10675: 10671: 10667: 10663: 10659: 10654: 10649: 10646:(3): 030404. 10645: 10641: 10634: 10626: 10622: 10618: 10614: 10610: 10606: 10602: 10598: 10597: 10589: 10581: 10577: 10573: 10569: 10565: 10561: 10556: 10551: 10547: 10543: 10536: 10527: 10522: 10515: 10507: 10503: 10499: 10495: 10491: 10487: 10483: 10479: 10474: 10469: 10465: 10461: 10454: 10446: 10442: 10438: 10434: 10430: 10426: 10422: 10418: 10413: 10408: 10404: 10400: 10393: 10385: 10381: 10377: 10373: 10369: 10365: 10360: 10355: 10352:(3): 032108. 10351: 10347: 10343: 10336: 10328: 10324: 10320: 10316: 10312: 10308: 10303: 10298: 10295:(6): 062127. 10294: 10290: 10286: 10279: 10271: 10267: 10263: 10259: 10255: 10251: 10247: 10243: 10238: 10233: 10230:(4): 045033. 10229: 10225: 10221: 10214: 10206: 10202: 10198: 10194: 10190: 10186: 10181: 10176: 10173:(5): 052122. 10172: 10168: 10161: 10153: 10149: 10145: 10141: 10137: 10133: 10129: 10125: 10121: 10114: 10106: 10102: 10098: 10094: 10090: 10086: 10082: 10075: 10067: 10061: 10057: 10050: 10042: 10038: 10031: 10016: 10012: 10007: 10002: 9998: 9994: 9987: 9971: 9967: 9961: 9945: 9939: 9931: 9927: 9923: 9919: 9915: 9911: 9907: 9903: 9898: 9893: 9889: 9885: 9878: 9870: 9866: 9862: 9858: 9854: 9850: 9845: 9840: 9836: 9832: 9825: 9817: 9813: 9809: 9805: 9801: 9797: 9796: 9788: 9780: 9776: 9772: 9768: 9764: 9760: 9757:(2): 93–100. 9756: 9752: 9745: 9738: 9732: 9724: 9720: 9716: 9712: 9708: 9704: 9699: 9694: 9690: 9686: 9682: 9675: 9667: 9663: 9659: 9655: 9651: 9647: 9643: 9639: 9632: 9625: 9621: 9617: 9613: 9609: 9605: 9601: 9597: 9592: 9587: 9583: 9579: 9572: 9564: 9560: 9553: 9545: 9543:9781329022027 9539: 9535: 9534: 9526: 9519: 9518:0-7923-2549-4 9515: 9511: 9510: 9506:Asher Peres, 9503: 9487: 9483: 9482: 9477: 9471: 9463: 9459: 9455: 9451: 9447: 9443: 9439: 9435: 9431: 9427: 9423: 9419: 9415: 9409: 9403: 9399: 9396: 9392: 9386: 9377: 9372: 9368: 9364: 9360: 9356: 9355: 9350: 9343: 9335: 9331: 9327: 9323: 9319: 9315: 9311: 9307: 9302: 9297: 9293: 9289: 9282: 9274: 9270: 9266: 9262: 9258: 9254: 9247: 9238: 9233: 9229: 9225: 9221: 9217: 9213: 9206: 9197: 9192: 9188: 9184: 9180: 9176: 9172: 9165: 9163: 9161: 9151: 9146: 9142: 9138: 9134: 9127: 9117: 9112: 9108: 9104: 9098: 9092: 9085: 9079: 9071: 9067: 9062: 9057: 9053: 9046: 9039: 9033: 9025: 9021: 9017: 9013: 9009: 9005: 9001: 8997: 8993: 8989: 8982: 8974: 8970: 8966: 8962: 8957: 8952: 8948: 8944: 8940: 8933: 8925: 8919: 8904: 8900: 8893: 8886: 8880: 8876: 8872: 8868: 8864: 8860: 8856: 8852: 8848: 8844: 8839: 8834: 8830: 8826: 8825: 8817: 8809: 8808: 8800: 8798: 8796: 8787: 8783: 8779: 8775: 8771: 8767: 8763: 8756: 8754: 8738: 8734: 8727: 8719: 8713: 8709: 8702: 8694: 8688: 8684: 8680: 8674: 8659: 8655: 8648: 8646: 8637: 8633: 8629: 8625: 8621: 8614: 8612: 8603: 8599: 8595: 8591: 8587: 8583: 8579: 8575: 8570: 8565: 8561: 8557: 8550: 8548: 8528: 8524: 8518: 8514: 8507: 8506: 8499: 8493: 8485: 8481: 8477: 8473: 8469: 8465: 8461: 8457: 8450: 8442: 8438: 8434: 8430: 8426: 8422: 8418: 8414: 8407: 8405: 8395: 8390: 8386: 8382: 8378: 8374: 8370: 8366: 8365:Rosen, Nathan 8362: 8358: 8352: 8350: 8348: 8332: 8331: 8326: 8322: 8316: 8312: 8302: 8296: 8291: 8286: 8283: 8281: 8278: 8276: 8273: 8271: 8268: 8266: 8263: 8261: 8258: 8256: 8253: 8251: 8248: 8246: 8243: 8241: 8238: 8236: 8233: 8231: 8228: 8226: 8223: 8221: 8218: 8216: 8213: 8211: 8208: 8206: 8203: 8201: 8198: 8196: 8193: 8191: 8188: 8186: 8183: 8181: 8178: 8176: 8173: 8171: 8168: 8166: 8163: 8162: 8155: 8151: 8149: 8145: 8141: 8131: 8129: 8119: 8117: 8113: 8109: 8093: 8088: 8085: 8081: 8077: 8074: 8054: 8049: 8046: 8042: 8038: 8030: 8029:hadronization 8014: 8009: 8006: 8002: 7998: 7995: 7987: 7983: 7978: 7974: 7964: 7962: 7952: 7950: 7945: 7943: 7939: 7935: 7930: 7916: 7913: 7910: 7890: 7887: 7884: 7864: 7861: 7858: 7848: 7846: 7841: 7837: 7833: 7828: 7826: 7822: 7817: 7812: 7798: 7795: 7790: 7786: 7782: 7779: 7753: 7745: 7741: 7737: 7729: 7721: 7717: 7711: 7707: 7701: 7697: 7693: 7690: 7675: 7673: 7668: 7666: 7662: 7658: 7654: 7650: 7646: 7642: 7638: 7634: 7633:fiber coupler 7630: 7620: 7616: 7614: 7610: 7605: 7603: 7599: 7578: 7544: 7538: 7531: 7522: 7511: 7502: 7493: 7488: 7480: 7470: 7462: 7451: 7439: 7426: 7425: 7424: 7422: 7418: 7413: 7410: 7407:, a class of 7406: 7401: 7399: 7395: 7379: 7376: 7373: 7353: 7350: 7347: 7322: 7289: 7283: 7276: 7273: 7265: 7257: 7252: 7249: 7241: 7230: 7204: 7203: 7202: 7200: 7195: 7193: 7189: 7186: 7182: 7163: 7155: 7147: 7139: 7134: 7126: 7118: 7113: 7105: 7097: 7092: 7084: 7070: 7066: 7061: 7053: 7036: 7017: 7009: 7001: 6996: 6988: 6980: 6975: 6967: 6959: 6954: 6946: 6932: 6928: 6923: 6915: 6898: 6897: 6896: 6894: 6890: 6885: 6877: 6875: 6874: 6869: 6865: 6860: 6858: 6853: 6851: 6846: 6844: 6834: 6832: 6828: 6818: 6816: 6809: 6806: 6804: 6801: 6799: 6796: 6794: 6791: 6789: 6786: 6784: 6781: 6778: 6777: 6776: 6773: 6763: 6761: 6760:Rényi entropy 6757: 6755: 6732: 6728: 6719: 6716: 6713: 6710: 6704: 6700: 6697: 6694: 6688: 6682: 6675: 6674: 6673: 6666: 6649: 6645: 6639: 6635: 6631: 6628: 6624: 6620: 6617: 6611: 6605: 6598: 6597: 6596: 6591: 6587: 6581: 6577: 6573: 6568: 6551: 6546: 6538: 6535: 6523: 6511: 6508: 6500: 6491: 6490: 6489: 6483: 6475: 6468: 6464: 6459: 6455: 6450: 6441: 6431: 6412: 6407: 6403: 6399: 6396: 6393: 6388: 6384: 6380: 6377: 6374: 6371: 6368: 6363: 6359: 6355: 6348: 6347: 6346: 6329: 6324: 6320: 6316: 6313: 6310: 6307: 6304: 6297: 6296: 6295: 6294: 6267: 6264: 6261: 6258: 6255: 6252: 6249: 6244: 6238: 6226: 6225: 6224: 6204: 6200: 6196: 6191: 6187: 6181: 6177: 6171: 6167: 6163: 6160: 6156: 6151: 6147: 6142: 6138: 6134: 6130: 6119: 6116: 6110: 6104: 6097: 6096: 6095: 6076: 6072: 6065: 6060: 6056: 6052: 6049: 6046: 6038: 6034: 6027: 6022: 6018: 6007: 5983: 5979: 5975: 5972: 5969: 5964: 5960: 5945: 5935: 5916: 5912: 5907: 5903: 5898: 5894: 5890: 5886: 5875: 5872: 5866: 5860: 5853: 5852: 5851: 5849: 5826: 5821: 5817: 5813: 5808: 5804: 5798: 5794: 5788: 5784: 5780: 5777: 5769: 5765: 5761: 5758: 5755: 5750: 5746: 5739: 5732: 5731: 5730: 5714: 5710: 5706: 5703: 5700: 5695: 5691: 5682: 5675: 5672:In classical 5666: 5657: 5649: 5646: 5644: 5639: 5637: 5628: 5611: 5589: 5585: 5576: 5572: 5568: 5563: 5559: 5536: 5532: 5523: 5519: 5496: 5492: 5483: 5479: 5470: 5452: 5424: 5416: 5413: 5407: 5379: 5375: 5366: 5362: 5339: 5335: 5326: 5322: 5299: 5295: 5272: 5268: 5258: 5255: 5251: 5250: 5249: 5247: 5237: 5235: 5230: 5219: 5217: 5212: 5210: 5206: 5196: 5176: 5166: 5158: 5150: 5142: 5137: 5133: 5125: 5124: 5123: 5107: 5099: 5091: 5086: 5078: 5046: 5042: 5036: 5026: 5018: 5010: 5002: 4997: 4987: 4979: 4971: 4962: 4955: 4952: 4946: 4941: 4937: 4929: 4928: 4927: 4910: 4906: 4900: 4892: 4884: 4879: 4871: 4863: 4858: 4850: 4842: 4837: 4829: 4820: 4812: 4808: 4798: 4797: 4796: 4789: 4776:on subsystem 4750: 4746: 4723: 4719: 4693: 4689: 4682: 4679: 4676: 4671: 4667: 4643: 4638: 4634: 4627: 4615: 4611: 4605: 4597: 4589: 4584: 4580: 4575: 4570: 4543: 4538: 4532: 4522: 4516: 4511: 4507: 4502: 4494: 4490: 4484: 4480: 4456: 4444: 4440: 4432: 4431: 4430: 4417: 4416:partial trace 4409: 4406:which is the 4366: 4361: 4357: 4349: 4348: 4347: 4326: 4321: 4317: 4313: 4308: 4304: 4300: 4282: 4281: 4280: 4259: 4249: 4247: 4235: 4230: 4214: 4209: 4205: 4182: 4177: 4173: 4141: 4136: 4132: 4109: 4104: 4100: 4069: 4064: 4059: 4055: 4051: 4046: 4041: 4037: 4031: 4027: 4021: 4017: 4013: 4010: 4003: 4002: 4001: 3998: 3984: 3981: 3976: 3964: 3961: 3957: 3946: 3942: 3930: 3910: 3896: 3893: 3889: 3882: 3872: 3869: 3865: 3853: 3850: 3846: 3840: 3836: 3831: 3826: 3814: 3811: 3807: 3798: 3790: 3787: 3783: 3769: 3766: 3756: 3747: 3743: 3738: 3732: 3728: 3722: 3718: 3714: 3711: 3704: 3703: 3702: 3699: 3692: 3683: 3680: 3660: 3641: 3621: 3603: 3599: 3594: 3592: 3571: 3567: 3539: 3535: 3512: 3493: 3483: 3479: 3467: 3463: 3452: 3448: 3442: 3438: 3434: 3431: 3424: 3423: 3422: 3420: 3416: 3412: 3409:, which is a 3408: 3397: 3395: 3389: 3352: 3344: 3334: 3326: 3313: 3296: 3288: 3278: 3270: 3257: 3256: 3255: 3231: 3223: 3217: 3192: 3175: 3168: 3160: 3155: 3151: 3124: 3120: 3114: 3106: 3098: 3093: 3085: 3077: 3072: 3064: 3056: 3051: 3043: 3034: 3026: 3022: 3012: 3011: 3010: 2984: 2976: 2968: 2963: 2955: 2915: 2907: 2899: 2894: 2886: 2869: 2855: 2850: 2845: 2841: 2835: 2830: 2826: 2822: 2817: 2814: 2810: 2787: 2782: 2778: 2774: 2769: 2764: 2760: 2734: 2729: 2725: 2718: 2710: 2705: 2701: 2677: 2672: 2664: 2654: 2649: 2645: 2639: 2635: 2631: 2626: 2618: 2591: 2583: 2573: 2568: 2564: 2558: 2554: 2550: 2545: 2537: 2512: 2507: 2502: 2498: 2492: 2487: 2483: 2479: 2474: 2471: 2467: 2441: 2436: 2432: 2425: 2417: 2412: 2408: 2378: 2370: 2362: 2357: 2349: 2339: 2336: 2332: 2326: 2323: 2320: 2316: 2312: 2307: 2304: 2296: 2284: 2283: 2282: 2279: 2272: 2240: 2232: 2192: 2184: 2169: 2164: 2162: 2143: 2138: 2130: 2122: 2117: 2109: 2097: 2096: 2095: 2079: 2071: 2044: 2036: 2008: 2003: 1999: 1995: 1990: 1986: 1978: 1977: 1976: 1975: 1958: 1940: 1938: 1934: 1923: 1921: 1917: 1913: 1909: 1903: 1901: 1895: 1879: 1876: 1873: 1869: 1860: 1859:Werner states 1855: 1845: 1843: 1839: 1835: 1831: 1821: 1819: 1815: 1810: 1808: 1789: 1778: 1763: 1760: 1754: 1740: 1729: 1725: 1724:Hilbert space 1703: 1692: 1673: 1645: 1633: 1620: 1607: 1604: 1598: 1587: 1575: 1563: 1561: 1557: 1553: 1549: 1544: 1542: 1538: 1534: 1530: 1526: 1522: 1512: 1508: 1504: 1502: 1498: 1497:Ronald Hanson 1494: 1491: 1486: 1484: 1479: 1475: 1465: 1463: 1459: 1454: 1452: 1447: 1443: 1438: 1429: 1427: 1416: 1413: 1407: 1398: 1393: 1385: 1376: 1371: 1367: 1362: 1360: 1356: 1345: 1342: 1337: 1335: 1334:singlet state 1331: 1327: 1323: 1319: 1315: 1312: 1307: 1305: 1301: 1297: 1292: 1290: 1289:superposition 1274: 1272: 1268: 1264: 1259: 1257: 1253: 1249: 1245: 1241: 1237: 1232: 1229: 1224: 1221: 1215: 1213: 1209: 1205: 1201: 1197: 1196:local realism 1193: 1188: 1186: 1182: 1178: 1173: 1171: 1167: 1163: 1159: 1155: 1151: 1146: 1144: 1140: 1139:Verschränkung 1136: 1132: 1128: 1127:quantum state 1124: 1120: 1116: 1112: 1108: 1104: 1100: 1093: 1092: 1087: 1082: 1077: 1067: 1065: 1064:quantum radar 1061: 1057: 1056:communication 1053: 1049: 1044: 1042: 1038: 1034: 1030: 1026: 1022: 1017: 1014: 1010: 1004: 1002: 998: 994: 993:local realism 990: 986: 982: 978: 974: 969: 967: 963: 959: 955: 951: 947: 943: 939: 935: 931: 929: 925: 920: 919:quantum state 916: 912: 901: 896: 894: 889: 887: 882: 881: 879: 878: 870: 867: 865: 862: 860: 857: 855: 852: 850: 847: 845: 842: 840: 837: 835: 832: 830: 827: 825: 822: 820: 817: 815: 812: 810: 807: 805: 802: 800: 797: 795: 792: 790: 787: 785: 782: 780: 777: 775: 772: 770: 767: 765: 762: 760: 757: 755: 752: 750: 747: 745: 742: 740: 737: 735: 732: 730: 727: 725: 722: 720: 717: 715: 712: 710: 707: 705: 702: 700: 697: 695: 692: 690: 687: 685: 682: 680: 677: 675: 672: 670: 667: 665: 662: 660: 657: 655: 652: 650: 647: 645: 642: 641: 634: 633: 626: 623: 621: 618: 616: 613: 611: 608: 606: 603: 601: 600:Quantum chaos 598: 596: 593: 591: 588: 586: 583: 581: 578: 577: 571: 570: 562: 559: 557: 556:Transactional 554: 552: 549: 547: 546:Quantum logic 544: 542: 539: 537: 534: 528: 525: 524: 523: 520: 519: 518: 515: 513: 510: 508: 505: 503: 500: 498: 495: 493: 490: 489: 485: 480: 479: 471: 468: 466: 463: 461: 458: 456: 453: 451: 448: 447: 440: 439: 431: 428: 426: 423: 421: 418: 416: 413: 411: 408: 406: 403: 402: 398: 395: 394: 388: 387: 379: 376: 374: 371: 369: 366: 365: 359: 356: 355: 354: 351: 350: 346: 343: 341: 338: 336: 333: 331: 328: 326: 323: 321: 318: 316: 313: 311: 308: 306: 303: 301: 298: 297: 290: 289: 279: 276: 275: 274: 273:Wave function 271: 269: 266: 264: 261: 259: 256: 254: 253:Superposition 251: 249: 246: 244: 241: 239: 236: 234: 231: 229: 226: 224: 221: 219: 216: 214: 211: 210: 203: 202: 194: 191: 189: 186: 185: 181: 178: 176: 173: 171: 168: 167: 161: 160: 155: 152: 150: 147: 145: 142: 141: 140: 139: 135: 102: 96: 79: 76: 72: 64: 57: 56: 53: 50: 49: 45: 44: 38: 34: 30: 19: 15152:Quantum mind 15064:Franck–Hertz 14926:Klein–Gordon 14875:Formulations 14868:Formulations 14797:Interference 14787:Entanglement 14786: 14765:Ground state 14760:Energy level 14733:Fundamentals 14697:Introduction 14637:Entanglement 14621: 14594: 14575: 14532: 14529:Phys. Rev. A 14528: 14485: 14481: 14438: 14432: 14413: 14391: 14381: 14364: 14338:(3): 32–39. 14335: 14329: 14278: 14274: 14264: 14252:. Retrieved 14246: 14236: 14191: 14187: 14177: 14165:. Retrieved 14111: 14107: 14097: 14085:. Retrieved 14031: 14027: 14017: 14005:. Retrieved 14000: 13991: 13979:. Retrieved 13941: 13937: 13927: 13915:. Retrieved 13911: 13902: 13893:21 September 13891:, retrieved 13878: 13871: 13862:21 September 13860:, retrieved 13847: 13840: 13815: 13811: 13801: 13791:21 September 13789:. Retrieved 13784: 13775: 13765:21 September 13763:. Retrieved 13758: 13749: 13724: 13720: 13709: 13693: 13688: 13674: 13637: 13633: 13623: 13611:. Retrieved 13606: 13597: 13572: 13568: 13562: 13511: 13507: 13501: 13450: 13446: 13440: 13389: 13385: 13379: 13328: 13324: 13318: 13267: 13263: 13257: 13206: 13202: 13196: 13153: 13149: 13143: 13100: 13096: 13090: 13047: 13043: 13037: 13012: 13008: 13002: 12959: 12955: 12949: 12898: 12894: 12888: 12845: 12841: 12835: 12800: 12796: 12790: 12781: 12730: 12726: 12720: 12677: 12673: 12667: 12634: 12630: 12624: 12573: 12569: 12563: 12523:(1): 45–59. 12520: 12516: 12509: 12476: 12472: 12466: 12433: 12429: 12423: 12380: 12376: 12363: 12320: 12316: 12309: 12274: 12270: 12264: 12223: 12219: 12212: 12171: 12167: 12161: 12108: 12104: 12098: 12055: 12051: 12038: 11995: 11991: 11985: 11950: 11946: 11936: 11904: 11900:Peres, Asher 11894: 11851: 11847: 11841: 11798: 11794: 11785: 11758: 11754: 11727: 11676: 11672: 11666: 11623: 11619: 11615: 11609: 11558: 11554: 11548: 11505: 11501: 11495: 11444: 11440: 11431: 11388: 11385:Phys. Rev. A 11384: 11378: 11335: 11331: 11277: 11273: 11267: 11224: 11220: 11214: 11179: 11175: 11171: 11165: 11114: 11110: 11104: 11077: 11073: 11057: 11004: 11000: 10994: 10949: 10943: 10901: 10897: 10891: 10868: 10862: 10832:(1): 54–67. 10829: 10825: 10819: 10775: 10772:Phys. Rev. A 10771: 10763: 10712: 10708: 10702: 10643: 10639: 10633: 10600: 10594: 10588: 10545: 10541: 10535: 10514: 10463: 10459: 10453: 10402: 10398: 10392: 10349: 10345: 10335: 10292: 10288: 10278: 10262:1721.1/98287 10227: 10223: 10213: 10170: 10166: 10160: 10127: 10123: 10113: 10088: 10084: 10074: 10055: 10049: 10040: 10030: 10018:. Retrieved 9996: 9986: 9974:. Retrieved 9970:the original 9960: 9948:. Retrieved 9938: 9887: 9883: 9877: 9834: 9830: 9824: 9799: 9793: 9787: 9754: 9750: 9744: 9736: 9731: 9691:(2): 22111. 9688: 9685:Phys. Rev. A 9684: 9674: 9641: 9637: 9631: 9581: 9577: 9571: 9562: 9552: 9532: 9525: 9507: 9502: 9490:. Retrieved 9479: 9470: 9421: 9417: 9408: 9390: 9385: 9358: 9352: 9342: 9291: 9287: 9281: 9256: 9252: 9246: 9222:(2): 91–94. 9219: 9215: 9205: 9178: 9174: 9140: 9136: 9126: 9106: 9102: 9096: 9091: 9083: 9078: 9051: 9045: 9037: 9032: 8991: 8987: 8981: 8946: 8942: 8932: 8918: 8906:. Retrieved 8902: 8892: 8828: 8822: 8816: 8806: 8769: 8765: 8740:. Retrieved 8736: 8726: 8707: 8701: 8682: 8673: 8661:. Retrieved 8658:Ars Technica 8657: 8619: 8559: 8555: 8534:. Retrieved 8527:the original 8504: 8497: 8492: 8459: 8455: 8449: 8416: 8412: 8376: 8372: 8334:. Retrieved 8328: 8315: 8152: 8137: 8125: 7971:In 2023 the 7970: 7958: 7948: 7946: 7931: 7849: 7829: 7813: 7681: 7669: 7649:antiparticle 7626: 7617: 7606: 7601: 7597: 7560: 7414: 7402: 7397: 7304: 7196: 7178: 6886: 6883: 6871: 6861: 6854: 6847: 6840: 6837:Applications 6824: 6812: 6769: 6758: 6747: 6667: 6664: 6579: 6569: 6566: 6481: 6473: 6466: 6462: 6457: 6453: 6451: 6447: 6427: 6344: 6285:0 log(0) = 0 6282: 6222: 6005: 5943: 5931: 5841: 5671: 5655: 5641: 5634: 5632: 5253: 5243: 5225: 5213: 5209:proportional 5202: 5194: 5061: 4925: 4790: 4658: 4405: 4341: 4255: 4231: 4084: 3999: 3928: 3926: 3694: 3687: 3684: 3678: 3639: 3595: 3510: 3508: 3403: 3390: 3370: 3193: 3170: 3163: 3139: 2870: 2396: 2274: 2267: 2217:and a basis 2165: 2158: 2023: 1946: 1929: 1904: 1896: 1851: 1827: 1817: 1811: 1806: 1722:acts on the 1622: 1565: 1545: 1524: 1520: 1518: 1509: 1505: 1487: 1473: 1471: 1458:incompatible 1455: 1451:relativistic 1435: 1422: 1414: 1405: 1396: 1383: 1374: 1363: 1351: 1338: 1308: 1293: 1285: 1267:John Clauser 1263:Alain Aspect 1260: 1233: 1225: 1216: 1212:Alain Aspect 1210:in 1972 and 1208:John Clauser 1189: 1174: 1165: 1153: 1149: 1147: 1143:entanglement 1142: 1138: 1123:entanglement 1122: 1107:Nathan Rosen 1096: 1089: 1045: 1018: 1005: 981:Nathan Rosen 970: 954:polarization 934:Measurements 932: 910: 909: 455:Klein–Gordon 391:Formulations 228:Energy level 223:Entanglement 222: 206:Fundamentals 193:Interference 144:Introduction 29: 15410:EPR paradox 15190:Quantum bus 15059:Double-slit 15037:Experiments 15003:Many-worlds 14941:Schrödinger 14905:Phase space 14895:Schrödinger 14885:Interaction 14842:Uncertainty 14812:Nonlocality 14807:Measurement 14802:Decoherence 14792:Hamiltonian 12406:11094/77656 10715:(1): 5297. 9481:Nobel Prize 9414:Ekert, A.K. 9259:(1): 1–18. 9036:Kumar, M., 8620:Nature News 8170:Concurrence 8108:concurrence 7840:Eirik Ovrum 7836:Jan Myrheim 7637:quantum dot 7185:orthonormal 6893:Bell states 6793:Concurrence 6754:Hamiltonian 4248:condition. 3415:trace class 3159:Bell states 3148:a definite 1943:Pure states 1916:foundations 1900:distillable 1527:is used in 1256:Artur Ekert 1152:but rather 1060:computation 1029:correlation 989:EPR paradox 962:paradoxical 844:von Neumann 829:Schrödinger 605:EPR paradox 536:Many-worlds 470:Schrödinger 425:Schrödinger 420:Phase-space 410:Interaction 315:Double-slit 293:Experiments 268:Uncertainty 238:Nonlocality 233:Measurement 218:Decoherence 188:Hamiltonian 15338:Extensions 15172:Technology 15018:Relational 14969:Copenhagen 14880:Heisenberg 14827:Tunnelling 14690:Background 14542:1706.01448 14495:1607.00164 14488:(5): 118. 14288:1711.06485 14254:29 October 14201:1702.08075 14121:2004.05515 14041:2009.12902 13951:2003.11310 13647:1707.01339 13613:7 December 13110:1906.10929 12740:1712.00706 12687:2212.07502 12330:1906.01645 12277:: 012332. 12065:1901.11004 11960:1504.03909 11508:(2): 436. 11287:2206.13180 11234:1806.06107 11007:(1): 210. 10678:2117/78836 10359:1806.00956 10302:1512.07313 10237:1504.04215 10020:13 October 9976:24 October 9802:(1): 136. 9301:1508.05949 8956:2210.06309 8908:21 October 8863:2117/79298 8838:1508.05949 8522:0521334950 8336:10 October 8308:References 8118:detector. 7961:calculated 7613:Fock state 7421:NOON state 5660:Definition 4258:Paul Dirac 4236:. For the 4085:where the 3927:where the 3509:where the 3150:pure state 3144:or system 1390:there are 958:correlated 839:Sommerfeld 754:Heisenberg 749:Gutzwiller 689:de Broglie 637:Scientists 551:Relational 502:Copenhagen 405:Heisenberg 263:Tunnelling 164:Background 15044:Bell test 14914:Equations 14740:Born rule 14567:239885759 14473:119288569 14448:0811.2803 14410:Duarte FJ 14228:119236759 14162:233872863 14146:0036-8075 14082:221971015 14066:0036-8075 13981:9 October 13976:214641162 13968:1745-2481 13917:9 October 13832:2190-5444 13741:1476-4687 13521:0909.0147 13188:119443360 13135:125658647 13127:2522-5820 13103:: 72–87. 13082:119288569 13057:0811.2803 12994:118952931 12810:0810.4507 12712:254685902 12530:1108.3309 12355:174799187 12301:119052621 12256:219692094 12131:CiteSeerX 12030:118556793 12005:1305.5941 11977:2469-9926 11658:111379972 11568:1304.7259 11454:0909.5152 11312:250072286 11259:119194947 11189:1002.2931 11027:CiteSeerX 10936:123349369 10826:Phys. Rev 10722:1405.4502 10653:1407.3114 10580:119194006 10555:1303.2849 10526:1405.2933 10506:118494859 10498:0374-4884 10473:1001.5445 10445:189843725 10437:0001-7701 10412:1005.3035 10327:119245348 10205:118638346 10180:1310.4691 10015:124976589 9897:1209.4191 9869:119208488 9844:1203.4834 9779:120680226 9666:123642560 9591:1106.3224 9520:, p. 115. 9492:5 October 9446:0031-9007 9334:205246446 9111:CiteSeerX 9024:206536690 8973:0036-8075 8883:See also 8879:205246446 8742:6 January 8663:22 August 8636:124852641 8602:119293698 8569:1303.0614 8484:122822435 8441:121278681 8373:Phys. Rev 8175:CNOT gate 8086:− 8078:∼ 8075:τ 8047:− 8039:∼ 8007:− 7999:∼ 7996:τ 7914:⊕ 7888:⊕ 7862:⊕ 7796:≤ 7783:≤ 7742:ρ 7738:⊗ 7718:ρ 7698:∑ 7691:ρ 7584:⟩ 7575:Ψ 7528:⟩ 7508:⟩ 7485:⟩ 7467:⟩ 7449:⟩ 7440:ψ 7419:modes, a 7328:⟩ 7319:Φ 7274:⊗ 7270:⟩ 7250:⊗ 7246:⟩ 7228:⟩ 7199:GHZ state 7152:⟩ 7140:⊗ 7131:⟩ 7119:± 7110:⟩ 7098:⊗ 7089:⟩ 7059:⟩ 7054:± 7050:Ψ 7014:⟩ 7002:⊗ 6993:⟩ 6981:± 6972:⟩ 6960:⊗ 6951:⟩ 6921:⟩ 6916:± 6912:Φ 6724:ℏ 6711:− 6701:⁡ 6640:∗ 6632:ρ 6612:ρ 6524:⋱ 6408:λ 6397:λ 6394:⁡ 6381:λ 6378:∫ 6372:ρ 6369:⁡ 6356:ρ 6325:λ 6314:λ 6311:∫ 6305:ρ 6256:⁡ 6242:→ 6201:λ 6197:⁡ 6178:λ 6168:∑ 6164:− 6152:ρ 6148:⁡ 6135:ρ 6120:− 6111:ρ 6073:λ 6066:⁡ 6050:⋯ 6035:λ 6028:⁡ 5980:λ 5973:⋯ 5961:λ 5908:ρ 5904:⁡ 5891:ρ 5876:− 5867:ρ 5814:⁡ 5785:∑ 5781:− 5759:⋯ 5704:⋯ 5643:steerable 5636:non-local 5612:ρ 5586:ρ 5582:→ 5573:ρ 5569:⊗ 5560:ρ 5533:ρ 5529:→ 5520:ρ 5493:ρ 5489:→ 5480:ρ 5456:⟩ 5428:⟩ 5411:⟩ 5376:ρ 5372:→ 5363:ρ 5336:ρ 5332:→ 5323:ρ 5296:ρ 5269:ρ 5167:ψ 5164:⟨ 5155:⟩ 5151:ψ 5134:ρ 5104:⟩ 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Quantum entanglement

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