85:, more specifically in secure quantum key exchange. MUBs are used in many protocols since the outcome is random when a measurement is made in a basis unbiased to that in which the state was prepared. When two remote parties share two non-orthogonal quantum states, attempts by an eavesdropper to distinguish between these by measurements will affect the system and this can be detected. While many quantum cryptography protocols have relied on 1-
19:
5088:
4864:
5899:
bases it is known that large sets of mutually unbiased bases exist which exhibit very little uncertainty. This means merely being mutually unbiased does not lead to high uncertainty, except when considering measurements in only two bases. Yet there do exist other measurements that are very uncertain.
1958:
1742:
1526:
1310:
4886:
4669:
810:
688:
3631:
3073:
Other bases which are unbiased to both the standard basis and the basis generated by the
Fourier matrix can be generated using Weyl groups. The dimension of the Hilbert space is important when generating sets of mutually unbiased bases using Weyl groups. When
6124:
3818:
22:
An example of three bases in a two-dimensional space, where bases B1 and B2 are mutually unbiased, whereas the vectors of basis B3 do not have equal overlap with the vectors of basis B1 (as well as B2) and so B3 is not mutually unbiased with B1 (and
2637:, the analogy can be taken further: a complete set of mutually unbiased bases can be directly constructed from a SIC-POVM. The 9 vectors of the SIC-POVM, together with the 12 vectors of the mutually unbiased bases, form a set that can be used in a
481:
5552:, we aim for measurement bases such that full knowledge of a state with respect to one basis implies minimal knowledge of the state with respect to the other bases. This implies a high entropy of measurement outcomes, and thus we call these
4223:
2608:, a type of structure whose definition is identical to that of a finite projective plane with the roles of points and lines exchanged. In this sense, the problems of SIC-POVMs and of mutually unbiased bases are dual to one another.
5854:
6810:
6973:
4121:
1094:
3350:
2271:
6475:
4603:
6290:
3258:
1748:
1532:
1316:
1100:
2795:
2162:
69:
Mutually unbiased bases (MUBs) and their existence problem is now known to have several closely related problems and equivalent avatars in several other branches of mathematics and quantum sciences, such as
2353:
6421:, respectively. Weigert and Wilkinson were first to notice that also a linear combination of these operators have eigenbases, which have some features typical for the mutually unbiased bases. An operator
5908:
While there has been investigation into mutually unbiased bases in infinite dimension
Hilbert space, their existence remains an open question. It is conjectured that in a continuous Hilbert space, two
572:
4516:
253:
185:
5083:{\displaystyle H_{4}(\phi )={\frac {1}{2}}{\begin{bmatrix}1&1&1&1\\1&e^{i\phi }&-1&-e^{i\phi }\\1&-1&1&-1\\1&-e^{i\phi }&-1&e^{i\phi }\end{bmatrix}}}
4859:{\displaystyle U={\frac {1}{\sqrt {d}}}{\begin{bmatrix}1&1&1\\e^{i\phi _{10}}&e^{i\phi _{11}}&e^{i\phi _{12}}\\e^{i\phi _{20}}&e^{i\phi _{21}}&e^{i\phi _{22}}\end{bmatrix}}}
3068:
5406:
3951:
2891:
694:
5999:
5258:
5186:
6419:
5336:
578:
5640:
2463:
3482:
3413:
3459:
6211:
6169:
2641:. However, in 6-dimensional Hilbert space, a SIC-POVM is known, but no complete set of mutually unbiased bases has yet been discovered, and it is widely believed that no such set exists.
2483: = 6. This is also the smallest dimension for which the number of mutually unbiased bases is not known. The methods used to determine the number of mutually unbiased bases when
6295:
The existence of mutually unbiased bases in a continuous
Hilbert space remains open for debate, as further research in their existence is required before any conclusions can be reached.
4376:
7421:
3690:
864:
6691:, the overlap between any eigenstate of the linear combination and any eigenstate of the position operator (both states normalized to the Dirac delta) is constant, but dependent on
5949:
901:
4638:
6876:
6536:
2532:
3698:
6574:
2056:
2015:
4437:
3886:
4026:
3990:
346:
311:
4408:
8043:
Damgaard, Ivan B.; Fehr, Serge; Renner, Renato; Salvail, Louis; Schaffner, Christian (2006). "A Tight High-Order
Entropic Quantum Uncertainty Relation with Applications".
6689:
6663:
2491: = 6, both by using Hadamard matrices and numerical methods have been unsuccessful. The general belief is that the maximum number of mutually unbiased bases for
365:
7448:
6905:
6841:
6381:
6352:
6324:
3168:
3139:
2985:
2953:
2924:
2713:
2684:
5474:
5440:
6617:
6597:
5733:
3854:
2841:
2821:
2401:
6709:
6637:
6007:
5697:
5670:
5532:
5505:
4655:
Given that one basis in a
Hilbert space is the standard basis, then all bases which are unbiased with respect to this basis can be represented by the columns of a
5897:
2635:
2594:
4127:
5859:
which is stronger than the relation we would get from pairing up the sets and then using the
Maassen and Uffink equation. Thus we have a characterization of
5748:
5559:
For two bases, the lower bound of the uncertainty relation is maximized when the measurement bases are mutually unbiased, since mutually unbiased bases are
6717:
6910:
4034:
7946:
Bandyopadhyay, Somshubhro; Oscar Boykin, P.; Roychowdhury, Vwani; Vatan, Farrokh (2002). "A new proof for the existence of mutually unbiased bases".
66:
in 1960, and the first person to consider applications of mutually unbiased bases was I. D. Ivanovic in the problem of quantum state determination.
942:
3264:
2181:
5113:
6424:
4521:
3086:
is not a prime number, then it is possible that the maximal number of mutually unbiased bases which can be generated using this method is 3.
5870: + 1 bases is well studied, very little is known about uncertainty relations for mutually unbiased bases in other circumstances.
1953:{\displaystyle M_{4}=\left\{{\frac {1}{2}}(1,-i,-1,-i),{\frac {1}{2}}(1,-i,1,i),{\frac {1}{2}}(1,i,-1,i),{\frac {1}{2}}(1,i,1,-i)\right\}}
1737:{\displaystyle M_{3}=\left\{{\frac {1}{2}}(1,-i,-i,-1),{\frac {1}{2}}(1,-i,i,1),{\frac {1}{2}}(1,i,i,-1),{\frac {1}{2}}(1,i,-i,1)\right\}}
1521:{\displaystyle M_{2}=\left\{{\frac {1}{2}}(1,-1,-i,-i),{\frac {1}{2}}(1,-1,i,i),{\frac {1}{2}}(1,1,i,-i),{\frac {1}{2}}(1,1,-i,i)\right\}}
1305:{\displaystyle M_{1}=\left\{{\frac {1}{2}}(1,1,1,1),{\frac {1}{2}}(1,1,-1,-1),{\frac {1}{2}}(1,-1,-1,1),{\frac {1}{2}}(1,-1,1,-1)\right\}}
6223:
3176:
5563:: the outcome of a measurement made in a basis unbiased to that in which the state is prepared in is completely random. In fact, for a
2729:
2075:
93:, allows for better security against eavesdropping. This motivates the study of mutually unbiased bases in higher-dimensional spaces.
8426:
2282:
5105:
2556:
1978:
512:
4456:
190:
122:
8421:
805:{\displaystyle M_{2}=\left\{{\frac {|0\rangle +i|1\rangle }{\sqrt {2}}},{\frac {|0\rangle -i|1\rangle }{\sqrt {2}}}\right\}}
2997:
2487:
is an integer power of a prime number cannot be used in this case. Searches for a set of four mutually unbiased bases when
5344:
3899:
683:{\displaystyle M_{1}=\left\{{\frac {|0\rangle +|1\rangle }{\sqrt {2}}},{\frac {|0\rangle -|1\rangle }{\sqrt {2}}}\right\}}
2850:
7115:
M. Planat et al, A Survey of Finite
Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements,
5191:
5119:
2366: + 1 mutually unbiased bases. This can be seen in the previous equation, as the prime number decomposition of
6386:
5266:
3626:{\displaystyle {\hat {X}},{\hat {Z}},{\hat {X}}{\hat {Z}},{\hat {X}}{\hat {Z}}^{2},\ldots ,{\hat {X}}{\hat {Z}}^{d-1}.}
2991:, we can generate another basis unbiased to it using a Fourier matrix. The elements of the Fourier matrix are given by
5477:
8461:
8446:
7919:
5573:
2605:
2409:
7695:
P. Butterley, W. Hall "Numerical evidence for the maximum number of mutually unbiased bases in dimension six, 2007,
3358:
6984:
3418:
491:
490:
in the following sense: if a system is prepared in a state belonging to one of the bases, then all outcomes of the
47:
6174:
6132:
5549:
4234:
7383:
5954:
4643:
The joint eigenbases of the operators in one set are mutually unbiased to that of any other set. We thus have
3647:
8033:
H. Maassen, J.B.M. Uffink: Generalized entropic uncertainty relations: Phys. Rev. Lett. 60, 1103–1106 (1988).
5703:: If we measure a state from one of the bases then the outcome has entropy 0 in that basis and an entropy of
829:
8242:
5914:
3813:{\displaystyle |\psi _{t}^{k}\rangle ={\frac {1}{\sqrt {d}}}\sum _{j=0}^{d-1}\omega ^{tj+kj^{2}}|j\rangle .}
869:
8416:
4608:
43:
6846:
6480:
2498:
8451:
8441:
7129:
Wootters, W. K.; Fields, B. D. (1989). "Optimal State-Determination by
Mutually Unbiased Measurements".
6541:
2028:
1987:
7911:
7826:
Bengtsson, Ingemar; Blanchfield, Kate; Cabello, Adán (2012). "A Kochen–Specker inequality from a SIC".
7709:
Brierley, S.; Weigert, S. (2008). "Maximal sets of mutually unbiased quantum states in dimension six".
2638:
50:
with respect to the other basis are predicted to occur with an equal probability inexorably equal to 1/
8301:
Wehner, S.; A. Winter (2008). "Higher entropic uncertainty relations for anti-commuting observables".
7174:
Gottesman, D. (1996). "Class of quantum error-correcting codes saturating the quantum
Hamming bound".
4413:
3862:
476:{\displaystyle |\langle e_{j}|f_{k}\rangle |^{2}={\frac {1}{d}},\quad \forall j,k\in \{1,\dots ,d\}.}
8156:
8079:
7634:
5098:
There is an alternative characterization of mutually unbiased bases that considers them in terms of
3995:
3959:
316:
281:
8436:
5863: + 1 mutually unbiased bases as those for which the uncertainty relations are strongest.
4384:
101:
82:
7288:
Huang, Yichen (29 July 2010). "Entanglement criteria via concave-function uncertainty relations".
6668:
6642:
8431:
7426:
4656:
2844:
2597:
271:
75:
8357:
Weigert, Stefan; Wilkinson, Michael (2008). "Mutually unbiased bases for continuous variables".
8188:
Wu, S.; Yu, S.; Mølmer, K. (2009). "Entropic uncertainty relation for mutually unbiased bases".
6881:
6818:
6357:
6329:
6301:
3144:
3115:
2961:
2929:
2900:
2689:
2660:
6119:{\displaystyle |\langle \psi _{s}^{b}|\psi _{s'}^{b'}\rangle |^{2}=k>0,s,s'\in \mathbb {R} }
5445:
5411:
349:
7651:
Durt, T.; Englert, B.-G.; Bengtsson, I.; Życzkowski, K. (2010). "On mutually unbiased bases".
4880:
An example of a one parameter family of
Hadamard matrices in a 4-dimensional Hilbert space is
8143:
8066:
7621:
7598:
Klappenecker, Andreas; Roetteler, Martin (2003). "Constructions of Mutually Unbiased Bases".
6602:
6579:
5706:
5538:
5109:
5099:
3826:
2826:
2806:
2373:
2168:
6694:
6622:
4381:
is an additive character over the field and the addition and multiplication in the kets and
3892: + 1 mutually unbiased bases. We label the elements of the computational basis of
8456:
8376:
8320:
8267:
8207:
8102:
8054:
7965:
7847:
7728:
7609:
7512:
7459:
7346:
7297:
7254:
7235:
Calderbank, A. R.; et al. (1997). "Quantum Error Correction and Orthogonal Geometry".
7193:
7138:
7087:
7032:
5675:
5648:
5510:
5483:
105:
7493:
Englert, B.-G.; Aharonov, Y. (2001). "The mean king's problem: prime degrees of freedom".
7116:
8:
8128:
Ambainis, Andris (2009). "Limits on entropic uncertainty relations for 3 and more MUBs".
6999:
5876:
4218:{\displaystyle {\hat {Z_{b}}}=\sum _{c\in \mathbb {F} _{d}}\chi (bc)|c\rangle \langle c|}
2614:
2573:
824:
28:
8380:
8324:
8271:
8211:
8106:
8058:
7969:
7903:
7851:
7732:
7613:
7516:
7463:
7350:
7301:
7258:
7197:
7142:
7091:
7036:
8392:
8366:
8336:
8310:
8283:
8257:
8223:
8197:
8129:
8044:
8016:
7981:
7955:
7882:
7863:
7837:
7796:
7763:
7744:
7718:
7678:
7660:
7599:
7580:
7562:
7528:
7502:
7362:
7336:
7270:
7244:
7217:
7183:
97:
8243:"Entropic uncertainty relations and locking: tight bounds for mutually unbiased bases"
7524:
7099:
7055:
7020:
5849:{\displaystyle \sum _{k=1}^{d+1}H_{B_{k}}\geq (d+1)\log \left({\frac {d+1}{2}}\right)}
2276:
then the maximum number of mutually unbiased bases which can be constructed satisfies
8340:
8227:
7925:
7915:
7828:
7682:
7475:
7209:
7150:
7060:
6805:{\displaystyle |\langle x_{\theta }|x\rangle |^{2}={\frac {1}{2\pi |\sin \theta |}},}
5909:
2716:
117:
90:
8396:
8287:
7867:
7584:
7532:
7366:
7274:
7221:
923: + 1 = 5 mutually unbiased bases where each basis is denoted by
8384:
8328:
8275:
8215:
8175:
8110:
8008:
7985:
7973:
7859:
7855:
7806:
7748:
7736:
7670:
7572:
7520:
7467:
7354:
7324:
7305:
7262:
7201:
7154:
7146:
7095:
7050:
7040:
6968:{\displaystyle \cos \theta {\hat {x}}-i\sin \theta {\frac {\partial }{\partial x}}}
3110:
2567:
267:
63:
7078:
Ivanovic, I. D. (1981). "Geometrical description of quantal state determination".
4116:{\displaystyle {\hat {X_{a}}}=\sum _{c\in \mathbb {F} _{d}}|c+a\rangle \langle c|}
7471:
5541:, as they are not phrased in terms of the state to be measured, but in terms of
3082: + 1 mutually unbiased bases can be generated using Weyl groups. When
8388:
8279:
8219:
8114:
7740:
7358:
7309:
7266:
2988:
494:
with respect to the other basis are predicted to occur with equal probability.
7977:
7674:
7159:
8410:
7929:
7205:
4447:
3462:
275:
256:
32:
7557:
Bengtsson, Ingemar (2007). "Three Ways to Look at Mutually Unbiased Bases".
1975:
What is the maximum number of MUBs in any given non-prime-power dimension d?
1089:{\displaystyle M_{0}=\left\{(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)\right\}}
7479:
7064:
3857:
3345:{\displaystyle {\hat {Z}}=\sum _{k=0}^{d-1}\omega ^{k}|k\rangle \langle k|}
3106:
3091:
2801:
820:
7945:
7213:
7045:
2266:{\displaystyle p_{1}^{n_{1}}<p_{2}^{n_{2}}<\cdots <p_{k}^{n_{k}}}
8262:
8049:
7960:
7908:
Geometry of quantum states : an introduction to quantum entanglement
7887:
7768:
7762:
Wootters, William K. (2004). "Quantum measurements and finite geometry".
7696:
7604:
7567:
7507:
7249:
7188:
2601:
2472:
is an integer power of a prime number, but it is not known for arbitrary
7811:
7784:
6470:{\displaystyle \alpha {\hat {x}}-i\beta {\frac {\partial }{\partial x}}}
8020:
4598:{\displaystyle \{{\hat {X_{s}}}{\hat {Z_{sr}}}|s\in \mathbb {F} _{d}\}}
3469:
2894:
2650:
2541:
1967:
8332:
7576:
2562:
The MUBs problem seems similar in nature to the symmetric property of
353:
8012:
6994:
2563:
2550:
2362:
is an integer power of a prime number, then it is possible to find
71:
8371:
8315:
8202:
8134:
7842:
7801:
7723:
7665:
7341:
6285:{\displaystyle |\langle q|p\rangle |^{2}={\frac {1}{2\pi \hbar }}}
3253:{\displaystyle {\hat {X}}=\sum _{k=0}^{d-1}|k+1\rangle \langle k|}
2479:
The smallest dimension that is not an integer power of a prime is
2468:
Thus, the maximum number of mutually unbiased bases is known when
7380:
Vaidman, L.; et al. (1987). "How to ascertain the values of
5738:
If the dimension of the space is a prime power, we can construct
18:
2790:{\displaystyle {\hat {Z}}{\hat {X}}=\omega {\hat {X}}{\hat {Z}}}
2157:{\displaystyle d=p_{1}^{n_{1}}p_{2}^{n_{2}}\cdots p_{k}^{n_{k}}}
7881:
Grassl, Markus (2004). "On SIC-POVMs and MUBs in Dimension 6".
3089:
2348:{\displaystyle p_{1}^{n_{1}}+1\leq {\mathfrak {M}}(d)\leq d+1.}
7650:
7327:(2012). "Entanglement detection via mutually unbiased bases".
7322:
62:
The notion of mutually unbiased bases was first introduced by
86:
4873:+1 mutually unbiased bases therefore corresponds to finding
2017:
denote the maximum number of mutually unbiased bases in the
8093:
Deutsch, D. (1982). "Uncertainty in Quantum Measurements".
7825:
6989:
5537:
Entropic uncertainty relations are often preferable to the
2025:. It is an open question how many mutually unbiased bases,
815:
provide the simplest example of mutually unbiased bases in
89:
technologies, employing higher-dimensional states, such as
567:{\displaystyle M_{0}=\left\{|0\rangle ,|1\rangle \right\}}
8042:
4511:{\displaystyle \{{\hat {Z_{s}}}|s\in \mathbb {F} _{d}\}}
248:{\displaystyle \{|f_{1}\rangle ,\dots ,|f_{d}\rangle \}}
180:{\displaystyle \{|e_{1}\rangle ,\dots ,|e_{d}\rangle \}}
7117:
http://hal.ccsd.cnrs.fr/docs/00/07/99/18/PDF/MUB_FP.pdf
2596:
unbiased bases yields a geometric structure known as a
8174:
S. Wehner and A. Winter, 2010 New J. Phys. 12 025009:
7323:
Spengler, C.; Huber, M.; Brierley, S.; Adaktylos, T.;
6129:
For the generalized position and momentum eigenstates
4927:
4696:
7597:
7429:
7386:
6913:
6884:
6849:
6821:
6720:
6697:
6671:
6645:
6625:
6605:
6582:
6544:
6483:
6427:
6389:
6360:
6332:
6304:
6226:
6177:
6135:
6010:
5957:
5917:
5879:
5751:
5742: + 1 MUBs, and then it has been found that
5709:
5678:
5651:
5576:
5513:
5486:
5448:
5414:
5347:
5269:
5194:
5122:
4889:
4672:
4611:
4524:
4459:
4416:
4387:
4237:
4130:
4037:
3998:
3962:
3902:
3865:
3829:
3701:
3650:
3485:
3421:
3361:
3267:
3179:
3147:
3118:
3063:{\displaystyle F_{ab}=\omega ^{ab},0\leq a,b\leq N-1}
3000:
2964:
2932:
2903:
2853:
2829:
2809:
2732:
2692:
2663:
2617:
2576:
2501:
2412:
2376:
2285:
2184:
2078:
2031:
1990:
1751:
1535:
1319:
1103:
945:
872:
832:
697:
581:
515:
368:
319:
284:
193:
125:
5903:
5401:{\displaystyle c=\max |\langle a_{j}|b_{k}\rangle |}
3946:{\displaystyle \{|a\rangle |a\in \mathbb {F} _{d}\}}
2358:
It follows that if the dimension of a Hilbert space
7901:
2886:{\displaystyle \omega \equiv e^{\frac {2\pi i}{d}}}
8240:
7999:Hirschman, I. I. Jr. (1957). "A note on entropy".
7785:"SIC-POVMs and Compatibility among Quantum States"
7442:
7415:
6967:
6899:
6870:
6835:
6804:
6703:
6683:
6657:
6631:
6611:
6591:
6568:
6530:
6469:
6413:
6375:
6346:
6318:
6284:
6205:
6163:
6118:
5993:
5943:
5891:
5848:
5727:
5691:
5664:
5634:
5526:
5499:
5468:
5434:
5400:
5330:
5253:{\displaystyle B_{2}=\{|b_{j}\rangle _{j=1}^{d}\}}
5252:
5181:{\displaystyle B_{1}=\{|a_{i}\rangle _{i=1}^{d}\}}
5180:
5082:
4858:
4663: = 3 these matrices would have the form
4632:
4597:
4510:
4431:
4402:
4370:
4217:
4115:
4020:
3984:
3945:
3880:
3848:
3812:
3684:
3625:
3453:
3407:
3344:
3252:
3162:
3133:
3062:
2979:
2947:
2918:
2885:
2835:
2815:
2789:
2707:
2678:
2629:
2588:
2526:
2457:
2395:
2347:
2265:
2156:
2050:
2009:
1952:
1736:
1520:
1304:
1088:
895:
858:
804:
682:
566:
475:
340:
305:
247:
179:
6414:{\displaystyle -i{\frac {\partial }{\partial x}}}
5331:{\displaystyle H_{B_{1}}+H_{B_{2}}\geq -2\log c.}
5116:and J. B. M. Uffink found that for any two bases
5093:
8408:
8356:
8176:http://iopscience.iop.org/1367-2630/12/2/025009/
5354:
3476: + 1 operators are mutually unbiased:
8300:
7708:
7492:
5635:{\displaystyle H_{B_{1}}+H_{B_{2}}\geq \log(d)}
2600:, while a SIC-POVM (in any dimension that is a
2543:
2458:{\displaystyle {\mathfrak {M}}(p^{n})=p^{n}+1.}
1969:
7910:(Second ed.). Cambridge, United Kingdom:
7128:
5873:When considering more than two, and less than
3408:{\displaystyle \{|k\rangle |0\leq k\leq d-1\}}
96:Other uses of mutually unbiased bases include
46:of one of the bases, then all outcomes of the
7552:
7550:
7548:
7546:
7544:
7542:
4877:mutually unbiased complex Hadamard matrices.
3454:{\displaystyle \omega =e^{\frac {2\pi i}{d}}}
7653:International Journal of Quantum Information
7646:
7644:
7021:"Unitary Operator Bases, Harvard University"
6865:
6830:
6747:
6726:
6341:
6313:
6246:
6232:
6206:{\displaystyle |p\rangle ,p\in \mathbb {R} }
6186:
6164:{\displaystyle |q\rangle ,q\in \mathbb {R} }
6144:
6064:
6016:
5988:
5938:
5390:
5362:
5247:
5227:
5208:
5175:
5155:
5136:
4592:
4525:
4505:
4460:
4204:
4201:
4102:
4099:
3940:
3914:
3903:
3804:
3722:
3402:
3373:
3362:
3331:
3328:
3239:
3236:
786:
769:
744:
727:
664:
650:
625:
611:
556:
542:
467:
449:
402:
374:
335:
300:
242:
239:
212:
194:
174:
171:
144:
126:
8352:
8350:
108:, and the so-called "mean king's problem".
8187:
8170:
8168:
8166:
7539:
7234:
4659:multiplied by a normalization factor. For
4371:{\displaystyle \chi (\theta )=\exp \left,}
3856:is a power of a prime, we make use of the
111:
8370:
8314:
8261:
8234:
8201:
8133:
8048:
7998:
7959:
7941:
7939:
7886:
7841:
7810:
7800:
7767:
7722:
7664:
7641:
7603:
7566:
7556:
7506:
7340:
7248:
7187:
7173:
7158:
7111:
7109:
7054:
7044:
7018:
6527:
6199:
6157:
6112:
5866:Although the case for two bases, and for
4650:
4620:
4582:
4495:
4419:
4166:
4073:
3930:
3868:
8347:
8127:
7761:
7416:{\displaystyle \sigma _{x},\sigma _{y},}
7077:
6354:are eigenvectors of Hermitian operators
5994:{\displaystyle |\psi _{s'}^{b'}\rangle }
5645:for any pair of mutually unbiased bases
4647: + 1 mutually unbiased bases.
3685:{\displaystyle {\hat {X}}{\hat {Z}}^{k}}
17:
8163:
8092:
7379:
2557:(more unsolved problems in mathematics)
1979:(more unsolved problems in mathematics)
859:{\displaystyle \sigma _{z},\sigma _{x}}
8409:
7936:
7880:
7782:
7697:https://arxiv.org/abs/quant-ph/0701122
7106:
5944:{\displaystyle |\psi _{s}^{b}\rangle }
896:{\displaystyle \sigma _{x}\sigma _{z}}
819:. The above bases are composed of the
7287:
4633:{\displaystyle r\in \mathbb {F} _{d}}
6871:{\displaystyle |x_{\theta }\rangle }
6531:{\displaystyle \exp(i(ax^{2}+bx))\,}
6001:are said to be mutually unbiased if
2649:
2527:{\displaystyle {\mathfrak {M}}(6)=3}
1963:
906:
497:
8181:
6477:has eigenfunctions proportional to
2537:
2504:
2415:
2319:
2034:
1993:
74:, finite projective/affine planes,
42:if when a system is prepared in an
13:
6956:
6952:
6576:and the corresponding eigenvalues
6569:{\displaystyle \alpha +2\beta a=0}
6458:
6454:
6402:
6398:
3078:is a prime number, then the usual
2570:points out that a complete set of
2051:{\displaystyle {\mathfrak {M}}(d)}
2010:{\displaystyle {\mathfrak {M}}(d)}
434:
14:
8473:
8241:Ballester, M.; S. Wehner (2007).
7561:. Vol. 889. pp. 40–51.
6276:
5904:Infinite dimension Hilbert spaces
2644:
8427:Unsolved problems in mathematics
6985:Measurement in quantum mechanics
5556:entropic uncertainty relations.
5539:Heisenberg uncertainty principle
5534:, when measuring a given state.
5110:Heisenberg uncertainty principle
4869:The problem of finding a set of
4432:{\displaystyle \mathbb {F} _{d}}
3881:{\displaystyle \mathbb {F} _{d}}
3644:-th eigenvector of the operator
8303:Journal of Mathematical Physics
8294:
8121:
8086:
8036:
8027:
8001:American Journal of Mathematics
7992:
7895:
7874:
7819:
7776:
7755:
7702:
7689:
7591:
7486:
3090:Unitary operators method using
2544:Unsolved problem in mathematics
1970:Unsolved problem in mathematics
433:
7860:10.1016/j.physleta.2011.12.011
7373:
7316:
7281:
7228:
7167:
7122:
7071:
7012:
6929:
6891:
6851:
6823:
6792:
6778:
6752:
6740:
6722:
6524:
6521:
6496:
6490:
6437:
6367:
6334:
6306:
6251:
6239:
6228:
6179:
6137:
6069:
6035:
6012:
5959:
5919:
5811:
5799:
5722:
5716:
5629:
5623:
5394:
5376:
5358:
5212:
5140:
5106:Entropic uncertainty relations
5094:Entropic uncertainty relations
4906:
4900:
4570:
4563:
4541:
4483:
4476:
4397:
4391:
4247:
4241:
4211:
4194:
4190:
4181:
4144:
4109:
4086:
4051:
4021:{\displaystyle {\hat {Z_{b}}}}
4012:
3985:{\displaystyle {\hat {X_{a}}}}
3976:
3918:
3907:
3888:to construct a maximal set of
3797:
3703:
3670:
3657:
3602:
3589:
3562:
3549:
3534:
3522:
3507:
3492:
3377:
3366:
3338:
3321:
3274:
3246:
3223:
3186:
3154:
3125:
2971:
2958:By choosing the eigenbasis of
2939:
2910:
2781:
2769:
2751:
2739:
2699:
2670:
2515:
2509:
2433:
2420:
2330:
2324:
2045:
2039:
2004:
1998:
1942:
1915:
1899:
1872:
1856:
1829:
1813:
1780:
1726:
1699:
1683:
1656:
1640:
1613:
1597:
1564:
1510:
1483:
1467:
1440:
1424:
1397:
1381:
1348:
1294:
1264:
1248:
1218:
1202:
1172:
1156:
1132:
1078:
1054:
1048:
1024:
1018:
994:
988:
964:
919: = 4, an example of
779:
762:
737:
720:
657:
643:
618:
604:
549:
535:
407:
388:
370:
341:{\displaystyle |f_{k}\rangle }
321:
306:{\displaystyle |e_{j}\rangle }
286:
225:
198:
157:
130:
102:quantum error correction codes
1:
7525:10.1016/s0375-9601(01)00271-7
7005:
5567:-dimensional space, we have:
4403:{\displaystyle \chi (\cdot )}
8422:Unsolved problems in physics
7151:10.1016/0003-4916(89)90322-9
7025:Proc. Natl. Acad. Sci. U.S.A
6878:stand for eigenfunctions of
6684:{\displaystyle \sin \theta }
6658:{\displaystyle \cos \theta }
98:quantum state reconstruction
7:
7472:10.1103/PhysRevLett.58.1385
7443:{\displaystyle \sigma _{z}}
7100:10.1088/0305-4470/14/12/019
6978:
2021:-dimensional Hilbert space
57:
10:
8478:
8389:10.1103/PhysRevA.78.020303
8280:10.1103/PhysRevA.75.012319
8220:10.1103/physreva.79.022104
8115:10.1103/physrevlett.50.631
7912:Cambridge University Press
7741:10.1103/physreva.78.042312
7559:AIP Conference Proceedings
7359:10.1103/physreva.86.022311
7310:10.1103/PhysRevA.82.012335
7267:10.1103/physrevlett.78.405
6900:{\displaystyle {\hat {x}}}
6836:{\displaystyle |x\rangle }
6376:{\displaystyle {\hat {x}}}
6347:{\displaystyle |p\rangle }
6319:{\displaystyle |q\rangle }
3415:is the standard basis and
3163:{\displaystyle {\hat {Z}}}
3134:{\displaystyle {\hat {X}}}
2980:{\displaystyle {\hat {Z}}}
2948:{\displaystyle {\hat {Z}}}
2919:{\displaystyle {\hat {X}}}
2708:{\displaystyle {\hat {Z}}}
2679:{\displaystyle {\hat {X}}}
936:≤ 4, is given as follows:
31:theory, a set of bases in
7978:10.1007/s00453-002-0980-7
7783:Stacey, Blake C. (2016).
7675:10.1142/s0219749910006502
7450:of a spin-1/2 particle".
5469:{\displaystyle H_{B_{2}}}
5435:{\displaystyle H_{B_{1}}}
2169:prime-power factorization
278:between any basis states
76:complex Hadamard matrices
8462:Computer-assisted proofs
8447:Euclidean plane geometry
7206:10.1103/physreva.54.1862
5550:quantum key distribution
3956:We define the operators
3896:using the finite field:
2553:exist in all dimensions?
83:quantum key distribution
8095:Physical Review Letters
6612:{\displaystyle \alpha }
6592:{\displaystyle b\beta }
5728:{\displaystyle \log(d)}
4657:complex Hadamard matrix
4446: + 1 sets of
3849:{\displaystyle d=p^{r}}
3692:is given explicitly by
2955:are mutually unbiased.
2845:primitive root of unity
2836:{\displaystyle \omega }
2816:{\displaystyle \omega }
2598:finite projective plane
2396:{\displaystyle d=p^{n}}
112:Definition and examples
81:MUBs are important for
8151:Cite journal requires
8074:Cite journal requires
7629:Cite journal requires
7444:
7417:
7019:Schwinger, J. (1960).
6969:
6901:
6872:
6837:
6806:
6705:
6704:{\displaystyle \beta }
6685:
6659:
6633:
6632:{\displaystyle \beta }
6613:
6593:
6570:
6532:
6471:
6415:
6377:
6348:
6320:
6286:
6207:
6165:
6120:
5995:
5945:
5893:
5850:
5778:
5729:
5693:
5666:
5636:
5561:maximally incompatible
5528:
5501:
5470:
5436:
5402:
5332:
5254:
5182:
5084:
4860:
4651:Hadamard matrix method
4634:
4599:
4512:
4433:
4404:
4372:
4219:
4117:
4022:
3986:
3947:
3882:
3850:
3814:
3766:
3686:
3627:
3455:
3409:
3346:
3309:
3254:
3221:
3164:
3135:
3064:
2981:
2949:
2920:
2887:
2837:
2817:
2791:
2709:
2680:
2631:
2590:
2528:
2459:
2397:
2349:
2267:
2158:
2052:
2011:
1954:
1738:
1522:
1306:
1090:
897:
860:
806:
684:
568:
477:
342:
307:
249:
181:
24:
7445:
7418:
7046:10.1073/pnas.46.4.570
6970:
6902:
6873:
6838:
6807:
6706:
6686:
6660:
6634:
6614:
6594:
6571:
6533:
6472:
6416:
6378:
6349:
6321:
6287:
6208:
6166:
6121:
5996:
5946:
5894:
5851:
5752:
5730:
5694:
5692:{\displaystyle B_{2}}
5667:
5665:{\displaystyle B_{1}}
5637:
5548:In scenarios such as
5529:
5527:{\displaystyle B_{2}}
5502:
5500:{\displaystyle B_{1}}
5471:
5437:
5403:
5333:
5255:
5183:
5108:are analogous to the
5100:uncertainty relations
5085:
4861:
4635:
4600:
4513:
4434:
4405:
4373:
4220:
4118:
4028:in the following way
4023:
3987:
3948:
3883:
3851:
3815:
3740:
3687:
3628:
3456:
3410:
3347:
3283:
3255:
3195:
3165:
3136:
3065:
2982:
2950:
2921:
2888:
2838:
2818:
2792:
2719:in the Hilbert space
2710:
2681:
2632:
2591:
2529:
2460:
2398:
2350:
2268:
2159:
2053:
2012:
1955:
1739:
1523:
1307:
1091:
898:
861:
807:
685:
569:
478:
343:
308:
266:, if and only if the
250:
182:
21:
7914:. pp. 313–354.
7902:Bengtsson, Ingemar;
7427:
7384:
6911:
6882:
6847:
6819:
6718:
6695:
6669:
6643:
6623:
6603:
6599:. If we parametrize
6580:
6542:
6481:
6425:
6387:
6358:
6330:
6326:and momentum states
6302:
6224:
6175:
6133:
6008:
5955:
5915:
5877:
5749:
5707:
5676:
5649:
5574:
5511:
5484:
5446:
5412:
5345:
5267:
5192:
5120:
4887:
4670:
4609:
4522:
4457:
4414:
4385:
4235:
4128:
4035:
3996:
3960:
3900:
3863:
3827:
3699:
3648:
3483:
3419:
3359:
3265:
3177:
3145:
3116:
2998:
2962:
2930:
2901:
2851:
2827:
2807:
2730:
2690:
2661:
2639:Kochen–Specker proof
2615:
2574:
2499:
2410:
2374:
2283:
2182:
2076:
2029:
1988:
1749:
1533:
1317:
1101:
943:
870:
830:
695:
579:
513:
366:
317:
282:
191:
123:
106:quantum entanglement
8417:Quantum measurement
8381:2008PhRvA..78b0303W
8325:2008JMP....49f2105W
8272:2007PhRvA..75a2319C
8212:2009PhRvA..79b2104W
8107:1983PhRvL..50..631D
8059:2006quant.ph.12014D
7970:2001quant.ph..3162B
7852:2012PhLA..376..374B
7812:10.3390/math4020036
7733:2008PhRvA..78d2312B
7614:2003quant.ph..9120K
7517:2001PhLA..284....1E
7464:1987PhRvL..58.1385V
7351:2012PhRvA..86b2311S
7302:2010PhRvA..82a2335H
7259:1997PhRvL..78..405C
7198:1996PhRvA..54.1862G
7143:1989AnPhy.191..363W
7092:1981JPhA...14.3241I
7037:1960PNAS...46..570S
6063:
6033:
5987:
5937:
5892:{\displaystyle d+1}
5246:
5174:
4450:unitary operators:
3721:
2630:{\displaystyle d=3}
2606:finite affine plane
2589:{\displaystyle d+1}
2307:
2262:
2231:
2206:
2153:
2128:
2106:
825:Pauli spin matrices
29:quantum information
8452:Algebraic geometry
8442:Incidence geometry
7440:
7413:
7160:10338.dmlcz/141471
6965:
6897:
6868:
6833:
6802:
6701:
6681:
6655:
6629:
6609:
6589:
6566:
6528:
6467:
6411:
6373:
6344:
6316:
6282:
6203:
6161:
6116:
6039:
6019:
5991:
5963:
5941:
5923:
5889:
5846:
5725:
5689:
5662:
5632:
5524:
5497:
5476:is the respective
5466:
5432:
5398:
5328:
5250:
5226:
5178:
5154:
5080:
5074:
4856:
4850:
4630:
4595:
4508:
4429:
4400:
4368:
4215:
4177:
4113:
4084:
4018:
3982:
3943:
3878:
3846:
3810:
3707:
3682:
3623:
3451:
3405:
3342:
3250:
3160:
3131:
3060:
2977:
2945:
2916:
2883:
2833:
2813:
2787:
2705:
2676:
2627:
2586:
2524:
2495: = 6 is
2455:
2393:
2345:
2286:
2263:
2241:
2210:
2185:
2154:
2132:
2107:
2085:
2058:, one can find in
2048:
2007:
1950:
1734:
1518:
1302:
1086:
893:
866:and their product
856:
802:
680:
564:
473:
338:
303:
245:
177:
25:
8359:Physical Review A
8333:10.1063/1.2943685
8250:Physical Review A
7904:Życzkowski, Karol
7829:Physics Letters A
7577:10.1063/1.2713445
7458:(14): 1385–1387.
7290:Physical Review A
7086:(12): 3241–3245.
6963:
6932:
6894:
6797:
6465:
6440:
6409:
6370:
6280:
5910:orthonormal bases
5840:
4920:
4689:
4688:
4566:
4544:
4479:
4280:
4153:
4147:
4060:
4054:
4015:
3979:
3738:
3737:
3673:
3660:
3605:
3592:
3565:
3552:
3537:
3525:
3510:
3495:
3472:of the following
3448:
3277:
3189:
3157:
3128:
3111:unitary operators
2974:
2942:
2913:
2880:
2784:
2772:
2754:
2742:
2717:unitary operators
2702:
2673:
1964:Existence problem
1913:
1870:
1827:
1778:
1697:
1654:
1611:
1562:
1481:
1438:
1395:
1346:
1262:
1216:
1170:
1130:
795:
794:
753:
752:
673:
672:
634:
633:
428:
264:mutually unbiased
118:orthonormal bases
40:mutually unbiased
8469:
8401:
8400:
8374:
8354:
8345:
8344:
8318:
8298:
8292:
8291:
8265:
8263:quant-ph/0606244
8247:
8238:
8232:
8231:
8205:
8185:
8179:
8172:
8161:
8160:
8154:
8149:
8147:
8139:
8137:
8125:
8119:
8118:
8090:
8084:
8083:
8077:
8072:
8070:
8062:
8052:
8050:quant-ph/0612014
8040:
8034:
8031:
8025:
8024:
7996:
7990:
7989:
7963:
7961:quant-ph/0103162
7943:
7934:
7933:
7899:
7893:
7892:
7890:
7888:quant-ph/0406175
7878:
7872:
7871:
7845:
7823:
7817:
7816:
7814:
7804:
7780:
7774:
7773:
7771:
7769:quant-ph/0406032
7759:
7753:
7752:
7726:
7706:
7700:
7693:
7687:
7686:
7668:
7648:
7639:
7638:
7632:
7627:
7625:
7617:
7607:
7605:quant-ph/0309120
7595:
7589:
7588:
7570:
7568:quant-ph/0610216
7554:
7537:
7536:
7510:
7508:quant-ph/0101134
7490:
7484:
7483:
7449:
7447:
7446:
7441:
7439:
7438:
7422:
7420:
7419:
7414:
7409:
7408:
7396:
7395:
7377:
7371:
7370:
7344:
7320:
7314:
7313:
7285:
7279:
7278:
7252:
7250:quant-ph/9605005
7232:
7226:
7225:
7191:
7189:quant-ph/9604038
7182:(3): 1862–1868.
7171:
7165:
7164:
7162:
7126:
7120:
7113:
7104:
7103:
7075:
7069:
7068:
7058:
7048:
7016:
6974:
6972:
6971:
6966:
6964:
6962:
6951:
6934:
6933:
6925:
6906:
6904:
6903:
6898:
6896:
6895:
6887:
6877:
6875:
6874:
6869:
6864:
6863:
6854:
6842:
6840:
6839:
6834:
6826:
6811:
6809:
6808:
6803:
6798:
6796:
6795:
6781:
6766:
6761:
6760:
6755:
6743:
6738:
6737:
6725:
6710:
6708:
6707:
6702:
6690:
6688:
6687:
6682:
6664:
6662:
6661:
6656:
6638:
6636:
6635:
6630:
6618:
6616:
6615:
6610:
6598:
6596:
6595:
6590:
6575:
6573:
6572:
6567:
6537:
6535:
6534:
6529:
6511:
6510:
6476:
6474:
6473:
6468:
6466:
6464:
6453:
6442:
6441:
6433:
6420:
6418:
6417:
6412:
6410:
6408:
6397:
6382:
6380:
6379:
6374:
6372:
6371:
6363:
6353:
6351:
6350:
6345:
6337:
6325:
6323:
6322:
6317:
6309:
6298:Position states
6291:
6289:
6288:
6283:
6281:
6279:
6265:
6260:
6259:
6254:
6242:
6231:
6212:
6210:
6209:
6204:
6202:
6182:
6170:
6168:
6167:
6162:
6160:
6140:
6125:
6123:
6122:
6117:
6115:
6107:
6078:
6077:
6072:
6062:
6061:
6052:
6051:
6038:
6032:
6027:
6015:
6000:
5998:
5997:
5992:
5986:
5985:
5976:
5975:
5962:
5950:
5948:
5947:
5942:
5936:
5931:
5922:
5898:
5896:
5895:
5890:
5855:
5853:
5852:
5847:
5845:
5841:
5836:
5825:
5795:
5794:
5793:
5792:
5777:
5766:
5734:
5732:
5731:
5726:
5699:. This bound is
5698:
5696:
5695:
5690:
5688:
5687:
5671:
5669:
5668:
5663:
5661:
5660:
5641:
5639:
5638:
5633:
5613:
5612:
5611:
5610:
5593:
5592:
5591:
5590:
5533:
5531:
5530:
5525:
5523:
5522:
5506:
5504:
5503:
5498:
5496:
5495:
5475:
5473:
5472:
5467:
5465:
5464:
5463:
5462:
5441:
5439:
5438:
5433:
5431:
5430:
5429:
5428:
5407:
5405:
5404:
5399:
5397:
5389:
5388:
5379:
5374:
5373:
5361:
5337:
5335:
5334:
5329:
5306:
5305:
5304:
5303:
5286:
5285:
5284:
5283:
5259:
5257:
5256:
5251:
5245:
5240:
5225:
5224:
5215:
5204:
5203:
5187:
5185:
5184:
5179:
5173:
5168:
5153:
5152:
5143:
5132:
5131:
5089:
5087:
5086:
5081:
5079:
5078:
5071:
5070:
5048:
5047:
4995:
4994:
4969:
4968:
4921:
4913:
4899:
4898:
4865:
4863:
4862:
4857:
4855:
4854:
4847:
4846:
4845:
4844:
4825:
4824:
4823:
4822:
4803:
4802:
4801:
4800:
4779:
4778:
4777:
4776:
4757:
4756:
4755:
4754:
4735:
4734:
4733:
4732:
4690:
4684:
4680:
4639:
4637:
4636:
4631:
4629:
4628:
4623:
4604:
4602:
4601:
4596:
4591:
4590:
4585:
4573:
4568:
4567:
4562:
4561:
4549:
4546:
4545:
4540:
4539:
4530:
4517:
4515:
4514:
4509:
4504:
4503:
4498:
4486:
4481:
4480:
4475:
4474:
4465:
4438:
4436:
4435:
4430:
4428:
4427:
4422:
4409:
4407:
4406:
4401:
4377:
4375:
4374:
4369:
4364:
4360:
4359:
4355:
4354:
4353:
4352:
4351:
4322:
4321:
4320:
4319:
4302:
4301:
4281:
4276:
4265:
4224:
4222:
4221:
4216:
4214:
4197:
4176:
4175:
4174:
4169:
4149:
4148:
4143:
4142:
4133:
4122:
4120:
4119:
4114:
4112:
4089:
4083:
4082:
4081:
4076:
4056:
4055:
4050:
4049:
4040:
4027:
4025:
4024:
4019:
4017:
4016:
4011:
4010:
4001:
3991:
3989:
3988:
3983:
3981:
3980:
3975:
3974:
3965:
3952:
3950:
3949:
3944:
3939:
3938:
3933:
3921:
3910:
3887:
3885:
3884:
3879:
3877:
3876:
3871:
3855:
3853:
3852:
3847:
3845:
3844:
3819:
3817:
3816:
3811:
3800:
3795:
3794:
3793:
3792:
3765:
3754:
3739:
3733:
3729:
3720:
3715:
3706:
3691:
3689:
3688:
3683:
3681:
3680:
3675:
3674:
3666:
3662:
3661:
3653:
3632:
3630:
3629:
3624:
3619:
3618:
3607:
3606:
3598:
3594:
3593:
3585:
3573:
3572:
3567:
3566:
3558:
3554:
3553:
3545:
3539:
3538:
3530:
3527:
3526:
3518:
3512:
3511:
3503:
3497:
3496:
3488:
3460:
3458:
3457:
3452:
3450:
3449:
3444:
3433:
3414:
3412:
3411:
3406:
3380:
3369:
3351:
3349:
3348:
3343:
3341:
3324:
3319:
3318:
3308:
3297:
3279:
3278:
3270:
3259:
3257:
3256:
3251:
3249:
3226:
3220:
3209:
3191:
3190:
3182:
3169:
3167:
3166:
3161:
3159:
3158:
3150:
3140:
3138:
3137:
3132:
3130:
3129:
3121:
3109:, we define the
3069:
3067:
3066:
3061:
3029:
3028:
3013:
3012:
2986:
2984:
2983:
2978:
2976:
2975:
2967:
2954:
2952:
2951:
2946:
2944:
2943:
2935:
2925:
2923:
2922:
2917:
2915:
2914:
2906:
2892:
2890:
2889:
2884:
2882:
2881:
2876:
2865:
2842:
2840:
2839:
2834:
2822:
2820:
2819:
2814:
2796:
2794:
2793:
2788:
2786:
2785:
2777:
2774:
2773:
2765:
2756:
2755:
2747:
2744:
2743:
2735:
2714:
2712:
2711:
2706:
2704:
2703:
2695:
2685:
2683:
2682:
2677:
2675:
2674:
2666:
2636:
2634:
2633:
2628:
2595:
2593:
2592:
2587:
2568:William Wootters
2545:
2538:Related problems
2533:
2531:
2530:
2525:
2508:
2507:
2464:
2462:
2461:
2456:
2448:
2447:
2432:
2431:
2419:
2418:
2402:
2400:
2399:
2394:
2392:
2391:
2354:
2352:
2351:
2346:
2323:
2322:
2306:
2305:
2304:
2294:
2272:
2270:
2269:
2264:
2261:
2260:
2259:
2249:
2230:
2229:
2228:
2218:
2205:
2204:
2203:
2193:
2163:
2161:
2160:
2155:
2152:
2151:
2150:
2140:
2127:
2126:
2125:
2115:
2105:
2104:
2103:
2093:
2062:, for arbitrary
2057:
2055:
2054:
2049:
2038:
2037:
2016:
2014:
2013:
2008:
1997:
1996:
1971:
1959:
1957:
1956:
1951:
1949:
1945:
1914:
1906:
1871:
1863:
1828:
1820:
1779:
1771:
1761:
1760:
1743:
1741:
1740:
1735:
1733:
1729:
1698:
1690:
1655:
1647:
1612:
1604:
1563:
1555:
1545:
1544:
1527:
1525:
1524:
1519:
1517:
1513:
1482:
1474:
1439:
1431:
1396:
1388:
1347:
1339:
1329:
1328:
1311:
1309:
1308:
1303:
1301:
1297:
1263:
1255:
1217:
1209:
1171:
1163:
1131:
1123:
1113:
1112:
1095:
1093:
1092:
1087:
1085:
1081:
955:
954:
903:, respectively.
902:
900:
899:
894:
892:
891:
882:
881:
865:
863:
862:
857:
855:
854:
842:
841:
811:
809:
808:
803:
801:
797:
796:
790:
789:
782:
765:
759:
754:
748:
747:
740:
723:
717:
707:
706:
689:
687:
686:
681:
679:
675:
674:
668:
667:
660:
646:
640:
635:
629:
628:
621:
607:
601:
591:
590:
573:
571:
570:
565:
563:
559:
552:
538:
525:
524:
506:The three bases
486:These bases are
482:
480:
479:
474:
429:
421:
416:
415:
410:
401:
400:
391:
386:
385:
373:
347:
345:
344:
339:
334:
333:
324:
312:
310:
309:
304:
299:
298:
289:
254:
252:
251:
246:
238:
237:
228:
211:
210:
201:
186:
184:
183:
178:
170:
169:
160:
143:
142:
133:
64:Julian Schwinger
8477:
8476:
8472:
8471:
8470:
8468:
8467:
8466:
8437:Operator theory
8407:
8406:
8405:
8404:
8355:
8348:
8299:
8295:
8245:
8239:
8235:
8186:
8182:
8173:
8164:
8152:
8150:
8141:
8140:
8126:
8122:
8091:
8087:
8075:
8073:
8064:
8063:
8041:
8037:
8032:
8028:
8013:10.2307/2372390
7997:
7993:
7944:
7937:
7922:
7900:
7896:
7879:
7875:
7824:
7820:
7781:
7777:
7760:
7756:
7707:
7703:
7694:
7690:
7649:
7642:
7630:
7628:
7619:
7618:
7596:
7592:
7555:
7540:
7491:
7487:
7452:Phys. Rev. Lett
7434:
7430:
7428:
7425:
7424:
7404:
7400:
7391:
7387:
7385:
7382:
7381:
7378:
7374:
7325:Hiesmayr, B. C.
7321:
7317:
7286:
7282:
7237:Phys. Rev. Lett
7233:
7229:
7172:
7168:
7127:
7123:
7114:
7107:
7076:
7072:
7017:
7013:
7008:
6981:
6955:
6950:
6924:
6923:
6912:
6909:
6908:
6886:
6885:
6883:
6880:
6879:
6859:
6855:
6850:
6848:
6845:
6844:
6822:
6820:
6817:
6816:
6791:
6777:
6770:
6765:
6756:
6751:
6750:
6739:
6733:
6729:
6721:
6719:
6716:
6715:
6696:
6693:
6692:
6670:
6667:
6666:
6644:
6641:
6640:
6624:
6621:
6620:
6604:
6601:
6600:
6581:
6578:
6577:
6543:
6540:
6539:
6506:
6502:
6482:
6479:
6478:
6457:
6452:
6432:
6431:
6426:
6423:
6422:
6401:
6396:
6388:
6385:
6384:
6362:
6361:
6359:
6356:
6355:
6333:
6331:
6328:
6327:
6305:
6303:
6300:
6299:
6269:
6264:
6255:
6250:
6249:
6238:
6227:
6225:
6222:
6221:
6213:, the value of
6198:
6178:
6176:
6173:
6172:
6156:
6136:
6134:
6131:
6130:
6111:
6100:
6073:
6068:
6067:
6054:
6053:
6044:
6043:
6034:
6028:
6023:
6011:
6009:
6006:
6005:
5978:
5977:
5968:
5967:
5958:
5956:
5953:
5952:
5932:
5927:
5918:
5916:
5913:
5912:
5906:
5878:
5875:
5874:
5826:
5824:
5820:
5788:
5784:
5783:
5779:
5767:
5756:
5750:
5747:
5746:
5708:
5705:
5704:
5683:
5679:
5677:
5674:
5673:
5656:
5652:
5650:
5647:
5646:
5606:
5602:
5601:
5597:
5586:
5582:
5581:
5577:
5575:
5572:
5571:
5518:
5514:
5512:
5509:
5508:
5491:
5487:
5485:
5482:
5481:
5458:
5454:
5453:
5449:
5447:
5444:
5443:
5424:
5420:
5419:
5415:
5413:
5410:
5409:
5393:
5384:
5380:
5375:
5369:
5365:
5357:
5346:
5343:
5342:
5299:
5295:
5294:
5290:
5279:
5275:
5274:
5270:
5268:
5265:
5264:
5241:
5230:
5220:
5216:
5211:
5199:
5195:
5193:
5190:
5189:
5169:
5158:
5148:
5144:
5139:
5127:
5123:
5121:
5118:
5117:
5096:
5073:
5072:
5063:
5059:
5057:
5049:
5040:
5036:
5031:
5025:
5024:
5016:
5011:
5003:
4997:
4996:
4987:
4983:
4978:
4970:
4961:
4957:
4955:
4949:
4948:
4943:
4938:
4933:
4923:
4922:
4912:
4894:
4890:
4888:
4885:
4884:
4849:
4848:
4840:
4836:
4832:
4828:
4826:
4818:
4814:
4810:
4806:
4804:
4796:
4792:
4788:
4784:
4781:
4780:
4772:
4768:
4764:
4760:
4758:
4750:
4746:
4742:
4738:
4736:
4728:
4724:
4720:
4716:
4713:
4712:
4707:
4702:
4692:
4691:
4679:
4671:
4668:
4667:
4653:
4624:
4619:
4618:
4610:
4607:
4606:
4586:
4581:
4580:
4569:
4554:
4550:
4548:
4547:
4535:
4531:
4529:
4528:
4523:
4520:
4519:
4499:
4494:
4493:
4482:
4470:
4466:
4464:
4463:
4458:
4455:
4454:
4423:
4418:
4417:
4415:
4412:
4411:
4386:
4383:
4382:
4341:
4337:
4336:
4332:
4315:
4311:
4310:
4306:
4297:
4293:
4286:
4282:
4266:
4264:
4263:
4259:
4236:
4233:
4232:
4210:
4193:
4170:
4165:
4164:
4157:
4138:
4134:
4132:
4131:
4129:
4126:
4125:
4108:
4085:
4077:
4072:
4071:
4064:
4045:
4041:
4039:
4038:
4036:
4033:
4032:
4006:
4002:
4000:
3999:
3997:
3994:
3993:
3970:
3966:
3964:
3963:
3961:
3958:
3957:
3934:
3929:
3928:
3917:
3906:
3901:
3898:
3897:
3872:
3867:
3866:
3864:
3861:
3860:
3840:
3836:
3828:
3825:
3824:
3796:
3788:
3784:
3771:
3767:
3755:
3744:
3728:
3716:
3711:
3702:
3700:
3697:
3696:
3676:
3665:
3664:
3663:
3652:
3651:
3649:
3646:
3645:
3608:
3597:
3596:
3595:
3584:
3583:
3568:
3557:
3556:
3555:
3544:
3543:
3529:
3528:
3517:
3516:
3502:
3501:
3487:
3486:
3484:
3481:
3480:
3434:
3432:
3428:
3420:
3417:
3416:
3376:
3365:
3360:
3357:
3356:
3337:
3320:
3314:
3310:
3298:
3287:
3269:
3268:
3266:
3263:
3262:
3245:
3222:
3210:
3199:
3181:
3180:
3178:
3175:
3174:
3149:
3148:
3146:
3143:
3142:
3120:
3119:
3117:
3114:
3113:
3095:
3021:
3017:
3005:
3001:
2999:
2996:
2995:
2966:
2965:
2963:
2960:
2959:
2934:
2933:
2931:
2928:
2927:
2905:
2904:
2902:
2899:
2898:
2866:
2864:
2860:
2852:
2849:
2848:
2828:
2825:
2824:
2808:
2805:
2804:
2776:
2775:
2764:
2763:
2746:
2745:
2734:
2733:
2731:
2728:
2727:
2694:
2693:
2691:
2688:
2687:
2665:
2664:
2662:
2659:
2658:
2655:
2647:
2616:
2613:
2612:
2575:
2572:
2571:
2560:
2559:
2554:
2547:
2540:
2503:
2502:
2500:
2497:
2496:
2443:
2439:
2427:
2423:
2414:
2413:
2411:
2408:
2407:
2387:
2383:
2375:
2372:
2371:
2318:
2317:
2300:
2296:
2295:
2290:
2284:
2281:
2280:
2255:
2251:
2250:
2245:
2224:
2220:
2219:
2214:
2199:
2195:
2194:
2189:
2183:
2180:
2179:
2146:
2142:
2141:
2136:
2121:
2117:
2116:
2111:
2099:
2095:
2094:
2089:
2077:
2074:
2073:
2069:In general, if
2033:
2032:
2030:
2027:
2026:
1992:
1991:
1989:
1986:
1985:
1982:
1981:
1976:
1973:
1966:
1905:
1862:
1819:
1770:
1769:
1765:
1756:
1752:
1750:
1747:
1746:
1689:
1646:
1603:
1554:
1553:
1549:
1540:
1536:
1534:
1531:
1530:
1473:
1430:
1387:
1338:
1337:
1333:
1324:
1320:
1318:
1315:
1314:
1254:
1208:
1162:
1122:
1121:
1117:
1108:
1104:
1102:
1099:
1098:
963:
959:
950:
946:
944:
941:
940:
931:
913:
887:
883:
877:
873:
871:
868:
867:
850:
846:
837:
833:
831:
828:
827:
778:
761:
760:
758:
736:
719:
718:
716:
715:
711:
702:
698:
696:
693:
692:
656:
642:
641:
639:
617:
603:
602:
600:
599:
595:
586:
582:
580:
577:
576:
548:
534:
533:
529:
520:
516:
514:
511:
510:
504:
420:
411:
406:
405:
396:
392:
387:
381:
377:
369:
367:
364:
363:
329:
325:
320:
318:
315:
314:
294:
290:
285:
283:
280:
279:
262:are said to be
233:
229:
224:
206:
202:
197:
192:
189:
188:
165:
161:
156:
138:
134:
129:
124:
121:
120:
114:
104:, detection of
60:
38:are said to be
12:
11:
5:
8475:
8465:
8464:
8459:
8454:
8449:
8444:
8439:
8434:
8432:Hilbert spaces
8429:
8424:
8419:
8403:
8402:
8346:
8293:
8233:
8180:
8162:
8153:|journal=
8120:
8101:(9): 631–633.
8085:
8076:|journal=
8035:
8026:
8007:(1): 152–156.
7991:
7954:(4): 512–528.
7935:
7920:
7894:
7873:
7836:(4): 374–376.
7818:
7775:
7754:
7701:
7688:
7659:(4): 535–640.
7640:
7631:|journal=
7590:
7538:
7485:
7437:
7433:
7412:
7407:
7403:
7399:
7394:
7390:
7372:
7315:
7280:
7243:(3): 405–408.
7227:
7166:
7137:(2): 363–381.
7121:
7105:
7070:
7010:
7009:
7007:
7004:
7003:
7002:
6997:
6992:
6987:
6980:
6977:
6961:
6958:
6954:
6949:
6946:
6943:
6940:
6937:
6931:
6928:
6922:
6919:
6916:
6893:
6890:
6867:
6862:
6858:
6853:
6832:
6829:
6825:
6813:
6812:
6801:
6794:
6790:
6787:
6784:
6780:
6776:
6773:
6769:
6764:
6759:
6754:
6749:
6746:
6742:
6736:
6732:
6728:
6724:
6700:
6680:
6677:
6674:
6654:
6651:
6648:
6628:
6608:
6588:
6585:
6565:
6562:
6559:
6556:
6553:
6550:
6547:
6526:
6523:
6520:
6517:
6514:
6509:
6505:
6501:
6498:
6495:
6492:
6489:
6486:
6463:
6460:
6456:
6451:
6448:
6445:
6439:
6436:
6430:
6407:
6404:
6400:
6395:
6392:
6369:
6366:
6343:
6340:
6336:
6315:
6312:
6308:
6293:
6292:
6278:
6275:
6272:
6268:
6263:
6258:
6253:
6248:
6245:
6241:
6237:
6234:
6230:
6201:
6197:
6194:
6191:
6188:
6185:
6181:
6159:
6155:
6152:
6149:
6146:
6143:
6139:
6127:
6126:
6114:
6110:
6106:
6103:
6099:
6096:
6093:
6090:
6087:
6084:
6081:
6076:
6071:
6066:
6060:
6057:
6050:
6047:
6042:
6037:
6031:
6026:
6022:
6018:
6014:
5990:
5984:
5981:
5974:
5971:
5966:
5961:
5940:
5935:
5930:
5926:
5921:
5905:
5902:
5888:
5885:
5882:
5857:
5856:
5844:
5839:
5835:
5832:
5829:
5823:
5819:
5816:
5813:
5810:
5807:
5804:
5801:
5798:
5791:
5787:
5782:
5776:
5773:
5770:
5765:
5762:
5759:
5755:
5735:in the other.
5724:
5721:
5718:
5715:
5712:
5686:
5682:
5659:
5655:
5643:
5642:
5631:
5628:
5625:
5622:
5619:
5616:
5609:
5605:
5600:
5596:
5589:
5585:
5580:
5521:
5517:
5494:
5490:
5461:
5457:
5452:
5427:
5423:
5418:
5396:
5392:
5387:
5383:
5378:
5372:
5368:
5364:
5360:
5356:
5353:
5350:
5339:
5338:
5327:
5324:
5321:
5318:
5315:
5312:
5309:
5302:
5298:
5293:
5289:
5282:
5278:
5273:
5249:
5244:
5239:
5236:
5233:
5229:
5223:
5219:
5214:
5210:
5207:
5202:
5198:
5177:
5172:
5167:
5164:
5161:
5157:
5151:
5147:
5142:
5138:
5135:
5130:
5126:
5095:
5092:
5091:
5090:
5077:
5069:
5066:
5062:
5058:
5056:
5053:
5050:
5046:
5043:
5039:
5035:
5032:
5030:
5027:
5026:
5023:
5020:
5017:
5015:
5012:
5010:
5007:
5004:
5002:
4999:
4998:
4993:
4990:
4986:
4982:
4979:
4977:
4974:
4971:
4967:
4964:
4960:
4956:
4954:
4951:
4950:
4947:
4944:
4942:
4939:
4937:
4934:
4932:
4929:
4928:
4926:
4919:
4916:
4911:
4908:
4905:
4902:
4897:
4893:
4867:
4866:
4853:
4843:
4839:
4835:
4831:
4827:
4821:
4817:
4813:
4809:
4805:
4799:
4795:
4791:
4787:
4783:
4782:
4775:
4771:
4767:
4763:
4759:
4753:
4749:
4745:
4741:
4737:
4731:
4727:
4723:
4719:
4715:
4714:
4711:
4708:
4706:
4703:
4701:
4698:
4697:
4695:
4687:
4683:
4678:
4675:
4652:
4649:
4641:
4640:
4627:
4622:
4617:
4614:
4594:
4589:
4584:
4579:
4576:
4572:
4565:
4560:
4557:
4553:
4543:
4538:
4534:
4527:
4507:
4502:
4497:
4492:
4489:
4485:
4478:
4473:
4469:
4462:
4426:
4421:
4399:
4396:
4393:
4390:
4379:
4378:
4367:
4363:
4358:
4350:
4347:
4344:
4340:
4335:
4331:
4328:
4325:
4318:
4314:
4309:
4305:
4300:
4296:
4292:
4289:
4285:
4279:
4275:
4272:
4269:
4262:
4258:
4255:
4252:
4249:
4246:
4243:
4240:
4226:
4225:
4213:
4209:
4206:
4203:
4200:
4196:
4192:
4189:
4186:
4183:
4180:
4173:
4168:
4163:
4160:
4156:
4152:
4146:
4141:
4137:
4123:
4111:
4107:
4104:
4101:
4098:
4095:
4092:
4088:
4080:
4075:
4070:
4067:
4063:
4059:
4053:
4048:
4044:
4014:
4009:
4005:
3978:
3973:
3969:
3942:
3937:
3932:
3927:
3924:
3920:
3916:
3913:
3909:
3905:
3875:
3870:
3843:
3839:
3835:
3832:
3821:
3820:
3809:
3806:
3803:
3799:
3791:
3787:
3783:
3780:
3777:
3774:
3770:
3764:
3761:
3758:
3753:
3750:
3747:
3743:
3736:
3732:
3727:
3724:
3719:
3714:
3710:
3705:
3679:
3672:
3669:
3659:
3656:
3634:
3633:
3622:
3617:
3614:
3611:
3604:
3601:
3591:
3588:
3582:
3579:
3576:
3571:
3564:
3561:
3551:
3548:
3542:
3536:
3533:
3524:
3521:
3515:
3509:
3506:
3500:
3494:
3491:
3447:
3443:
3440:
3437:
3431:
3427:
3424:
3404:
3401:
3398:
3395:
3392:
3389:
3386:
3383:
3379:
3375:
3372:
3368:
3364:
3353:
3352:
3340:
3336:
3333:
3330:
3327:
3323:
3317:
3313:
3307:
3304:
3301:
3296:
3293:
3290:
3286:
3282:
3276:
3273:
3260:
3248:
3244:
3241:
3238:
3235:
3232:
3229:
3225:
3219:
3216:
3213:
3208:
3205:
3202:
3198:
3194:
3188:
3185:
3156:
3153:
3127:
3124:
3094:
3088:
3071:
3070:
3059:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3027:
3024:
3020:
3016:
3011:
3008:
3004:
2989:standard basis
2973:
2970:
2941:
2938:
2912:
2909:
2879:
2875:
2872:
2869:
2863:
2859:
2856:
2847:, for example
2832:
2812:
2798:
2797:
2783:
2780:
2771:
2768:
2762:
2759:
2753:
2750:
2741:
2738:
2701:
2698:
2672:
2669:
2654:
2648:
2646:
2645:Search methods
2643:
2626:
2623:
2620:
2585:
2582:
2579:
2555:
2548:
2542:
2539:
2536:
2523:
2520:
2517:
2514:
2511:
2506:
2466:
2465:
2454:
2451:
2446:
2442:
2438:
2435:
2430:
2426:
2422:
2417:
2390:
2386:
2382:
2379:
2356:
2355:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2321:
2316:
2313:
2310:
2303:
2299:
2293:
2289:
2274:
2273:
2258:
2254:
2248:
2244:
2240:
2237:
2234:
2227:
2223:
2217:
2213:
2209:
2202:
2198:
2192:
2188:
2165:
2164:
2149:
2145:
2139:
2135:
2131:
2124:
2120:
2114:
2110:
2102:
2098:
2092:
2088:
2084:
2081:
2047:
2044:
2041:
2036:
2006:
2003:
2000:
1995:
1977:
1974:
1968:
1965:
1962:
1961:
1960:
1948:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1912:
1909:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1869:
1866:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1837:
1834:
1831:
1826:
1823:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1777:
1774:
1768:
1764:
1759:
1755:
1744:
1732:
1728:
1725:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1696:
1693:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1653:
1650:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1610:
1607:
1602:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1561:
1558:
1552:
1548:
1543:
1539:
1528:
1516:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1485:
1480:
1477:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1437:
1434:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1394:
1391:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1350:
1345:
1342:
1336:
1332:
1327:
1323:
1312:
1300:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1269:
1266:
1261:
1258:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1215:
1212:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1169:
1166:
1161:
1158:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1129:
1126:
1120:
1116:
1111:
1107:
1096:
1084:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
962:
958:
953:
949:
927:
912:
905:
890:
886:
880:
876:
853:
849:
845:
840:
836:
813:
812:
800:
793:
788:
785:
781:
777:
774:
771:
768:
764:
757:
751:
746:
743:
739:
735:
732:
729:
726:
722:
714:
710:
705:
701:
690:
678:
671:
666:
663:
659:
655:
652:
649:
645:
638:
632:
627:
624:
620:
616:
613:
610:
606:
598:
594:
589:
585:
574:
562:
558:
555:
551:
547:
544:
541:
537:
532:
528:
523:
519:
503:
496:
484:
483:
472:
469:
466:
463:
460:
457:
454:
451:
448:
445:
442:
439:
436:
432:
427:
424:
419:
414:
409:
404:
399:
395:
390:
384:
380:
376:
372:
337:
332:
328:
323:
302:
297:
293:
288:
244:
241:
236:
232:
227:
223:
220:
217:
214:
209:
205:
200:
196:
176:
173:
168:
164:
159:
155:
152:
149:
146:
141:
137:
132:
128:
113:
110:
59:
56:
9:
6:
4:
3:
2:
8474:
8463:
8460:
8458:
8455:
8453:
8450:
8448:
8445:
8443:
8440:
8438:
8435:
8433:
8430:
8428:
8425:
8423:
8420:
8418:
8415:
8414:
8412:
8398:
8394:
8390:
8386:
8382:
8378:
8373:
8368:
8365:(2): 020303.
8364:
8360:
8353:
8351:
8342:
8338:
8334:
8330:
8326:
8322:
8317:
8312:
8309:(6): 062105.
8308:
8304:
8297:
8289:
8285:
8281:
8277:
8273:
8269:
8264:
8259:
8256:(1): 022319.
8255:
8251:
8244:
8237:
8229:
8225:
8221:
8217:
8213:
8209:
8204:
8199:
8196:(2): 022104.
8195:
8191:
8184:
8177:
8171:
8169:
8167:
8158:
8145:
8136:
8131:
8124:
8116:
8112:
8108:
8104:
8100:
8096:
8089:
8081:
8068:
8060:
8056:
8051:
8046:
8039:
8030:
8022:
8018:
8014:
8010:
8006:
8002:
7995:
7987:
7983:
7979:
7975:
7971:
7967:
7962:
7957:
7953:
7949:
7942:
7940:
7931:
7927:
7923:
7921:9781107026254
7917:
7913:
7909:
7905:
7898:
7889:
7884:
7877:
7869:
7865:
7861:
7857:
7853:
7849:
7844:
7839:
7835:
7831:
7830:
7822:
7813:
7808:
7803:
7798:
7794:
7790:
7786:
7779:
7770:
7765:
7758:
7750:
7746:
7742:
7738:
7734:
7730:
7725:
7720:
7717:(4): 042312.
7716:
7712:
7705:
7698:
7692:
7684:
7680:
7676:
7672:
7667:
7662:
7658:
7654:
7647:
7645:
7636:
7623:
7615:
7611:
7606:
7601:
7594:
7586:
7582:
7578:
7574:
7569:
7564:
7560:
7553:
7551:
7549:
7547:
7545:
7543:
7534:
7530:
7526:
7522:
7518:
7514:
7509:
7504:
7500:
7496:
7495:Phys. Lett. A
7489:
7481:
7477:
7473:
7469:
7465:
7461:
7457:
7453:
7435:
7431:
7410:
7405:
7401:
7397:
7392:
7388:
7376:
7368:
7364:
7360:
7356:
7352:
7348:
7343:
7338:
7335:(2): 022311.
7334:
7330:
7326:
7319:
7311:
7307:
7303:
7299:
7296:(1): 012335.
7295:
7291:
7284:
7276:
7272:
7268:
7264:
7260:
7256:
7251:
7246:
7242:
7238:
7231:
7223:
7219:
7215:
7211:
7207:
7203:
7199:
7195:
7190:
7185:
7181:
7177:
7170:
7161:
7156:
7152:
7148:
7144:
7140:
7136:
7132:
7125:
7118:
7112:
7110:
7101:
7097:
7093:
7089:
7085:
7081:
7074:
7066:
7062:
7057:
7052:
7047:
7042:
7038:
7034:
7030:
7026:
7022:
7015:
7011:
7001:
6998:
6996:
6993:
6991:
6988:
6986:
6983:
6982:
6976:
6959:
6947:
6944:
6941:
6938:
6935:
6926:
6920:
6917:
6914:
6888:
6860:
6856:
6827:
6799:
6788:
6785:
6782:
6774:
6771:
6767:
6762:
6757:
6744:
6734:
6730:
6714:
6713:
6712:
6698:
6678:
6675:
6672:
6652:
6649:
6646:
6626:
6606:
6586:
6583:
6563:
6560:
6557:
6554:
6551:
6548:
6545:
6518:
6515:
6512:
6507:
6503:
6499:
6493:
6487:
6484:
6461:
6449:
6446:
6443:
6434:
6428:
6405:
6393:
6390:
6364:
6338:
6310:
6296:
6273:
6270:
6266:
6261:
6256:
6243:
6235:
6220:
6219:
6218:
6216:
6195:
6192:
6189:
6183:
6153:
6150:
6147:
6141:
6108:
6104:
6101:
6097:
6094:
6091:
6088:
6085:
6082:
6079:
6074:
6058:
6055:
6048:
6045:
6040:
6029:
6024:
6020:
6004:
6003:
6002:
5982:
5979:
5972:
5969:
5964:
5933:
5928:
5924:
5911:
5901:
5886:
5883:
5880:
5871:
5869:
5864:
5862:
5842:
5837:
5833:
5830:
5827:
5821:
5817:
5814:
5808:
5805:
5802:
5796:
5789:
5785:
5780:
5774:
5771:
5768:
5763:
5760:
5757:
5753:
5745:
5744:
5743:
5741:
5736:
5719:
5713:
5710:
5702:
5684:
5680:
5657:
5653:
5626:
5620:
5617:
5614:
5607:
5603:
5598:
5594:
5587:
5583:
5578:
5570:
5569:
5568:
5566:
5562:
5557:
5555:
5551:
5546:
5544:
5540:
5535:
5519:
5515:
5492:
5488:
5480:of the bases
5479:
5459:
5455:
5450:
5425:
5421:
5416:
5385:
5381:
5370:
5366:
5351:
5348:
5325:
5322:
5319:
5316:
5313:
5310:
5307:
5300:
5296:
5291:
5287:
5280:
5276:
5271:
5263:
5262:
5261:
5242:
5237:
5234:
5231:
5221:
5217:
5205:
5200:
5196:
5170:
5165:
5162:
5159:
5149:
5145:
5133:
5128:
5124:
5115:
5111:
5107:
5103:
5101:
5075:
5067:
5064:
5060:
5054:
5051:
5044:
5041:
5037:
5033:
5028:
5021:
5018:
5013:
5008:
5005:
5000:
4991:
4988:
4984:
4980:
4975:
4972:
4965:
4962:
4958:
4952:
4945:
4940:
4935:
4930:
4924:
4917:
4914:
4909:
4903:
4895:
4891:
4883:
4882:
4881:
4878:
4876:
4872:
4851:
4841:
4837:
4833:
4829:
4819:
4815:
4811:
4807:
4797:
4793:
4789:
4785:
4773:
4769:
4765:
4761:
4751:
4747:
4743:
4739:
4729:
4725:
4721:
4717:
4709:
4704:
4699:
4693:
4685:
4681:
4676:
4673:
4666:
4665:
4664:
4662:
4658:
4648:
4646:
4625:
4615:
4612:
4587:
4577:
4574:
4558:
4555:
4551:
4536:
4532:
4500:
4490:
4487:
4471:
4467:
4453:
4452:
4451:
4449:
4445:
4442:Then we form
4440:
4424:
4394:
4388:
4365:
4361:
4356:
4348:
4345:
4342:
4338:
4333:
4329:
4326:
4323:
4316:
4312:
4307:
4303:
4298:
4294:
4290:
4287:
4283:
4277:
4273:
4270:
4267:
4260:
4256:
4253:
4250:
4244:
4238:
4231:
4230:
4229:
4207:
4198:
4187:
4184:
4178:
4171:
4161:
4158:
4154:
4150:
4139:
4135:
4124:
4105:
4096:
4093:
4090:
4078:
4068:
4065:
4061:
4057:
4046:
4042:
4031:
4030:
4029:
4007:
4003:
3971:
3967:
3954:
3935:
3925:
3922:
3911:
3895:
3891:
3873:
3859:
3841:
3837:
3833:
3830:
3807:
3801:
3789:
3785:
3781:
3778:
3775:
3772:
3768:
3762:
3759:
3756:
3751:
3748:
3745:
3741:
3734:
3730:
3725:
3717:
3712:
3708:
3695:
3694:
3693:
3677:
3667:
3654:
3643:
3639:
3620:
3615:
3612:
3609:
3599:
3586:
3580:
3577:
3574:
3569:
3559:
3546:
3540:
3531:
3519:
3513:
3504:
3498:
3489:
3479:
3478:
3477:
3475:
3471:
3466:
3464:
3463:root of unity
3445:
3441:
3438:
3435:
3429:
3425:
3422:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3370:
3334:
3325:
3315:
3311:
3305:
3302:
3299:
3294:
3291:
3288:
3284:
3280:
3271:
3261:
3242:
3233:
3230:
3227:
3217:
3214:
3211:
3206:
3203:
3200:
3196:
3192:
3183:
3173:
3172:
3171:
3151:
3122:
3112:
3108:
3104:
3101: =
3100:
3093:
3092:finite fields
3087:
3085:
3081:
3077:
3057:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
3025:
3022:
3018:
3014:
3009:
3006:
3002:
2994:
2993:
2992:
2990:
2968:
2956:
2936:
2907:
2896:
2877:
2873:
2870:
2867:
2861:
2857:
2854:
2846:
2830:
2810:
2803:
2778:
2766:
2760:
2757:
2748:
2736:
2726:
2725:
2724:
2722:
2718:
2696:
2667:
2652:
2642:
2640:
2624:
2621:
2618:
2611:In dimension
2609:
2607:
2603:
2599:
2583:
2580:
2577:
2569:
2565:
2558:
2552:
2535:
2521:
2518:
2512:
2494:
2490:
2486:
2482:
2477:
2475:
2471:
2452:
2449:
2444:
2440:
2436:
2428:
2424:
2406:
2405:
2404:
2403:. Therefore,
2388:
2384:
2380:
2377:
2369:
2365:
2361:
2342:
2339:
2336:
2333:
2327:
2314:
2311:
2308:
2301:
2297:
2291:
2287:
2279:
2278:
2277:
2256:
2252:
2246:
2242:
2238:
2235:
2232:
2225:
2221:
2215:
2211:
2207:
2200:
2196:
2190:
2186:
2178:
2177:
2176:
2174:
2170:
2147:
2143:
2137:
2133:
2129:
2122:
2118:
2112:
2108:
2100:
2096:
2090:
2086:
2082:
2079:
2072:
2071:
2070:
2067:
2065:
2061:
2042:
2024:
2020:
2001:
1980:
1946:
1939:
1936:
1933:
1930:
1927:
1924:
1921:
1918:
1910:
1907:
1902:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1867:
1864:
1859:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1824:
1821:
1816:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1775:
1772:
1766:
1762:
1757:
1753:
1745:
1730:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1694:
1691:
1686:
1680:
1677:
1674:
1671:
1668:
1665:
1662:
1659:
1651:
1648:
1643:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1608:
1605:
1600:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1559:
1556:
1550:
1546:
1541:
1537:
1529:
1514:
1507:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1478:
1475:
1470:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1435:
1432:
1427:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1392:
1389:
1384:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1343:
1340:
1334:
1330:
1325:
1321:
1313:
1298:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1259:
1256:
1251:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1213:
1210:
1205:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1167:
1164:
1159:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1127:
1124:
1118:
1114:
1109:
1105:
1097:
1082:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1051:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1021:
1015:
1012:
1009:
1006:
1003:
1000:
997:
991:
985:
982:
979:
976:
973:
970:
967:
960:
956:
951:
947:
939:
938:
937:
935:
930:
926:
922:
918:
910:
904:
888:
884:
878:
874:
851:
847:
843:
838:
834:
826:
822:
818:
798:
791:
783:
775:
772:
766:
755:
749:
741:
733:
730:
724:
712:
708:
703:
699:
691:
676:
669:
661:
653:
647:
636:
630:
622:
614:
608:
596:
592:
587:
583:
575:
560:
553:
545:
539:
530:
526:
521:
517:
509:
508:
507:
501:
495:
493:
489:
470:
464:
461:
458:
455:
452:
446:
443:
440:
437:
430:
425:
422:
417:
412:
397:
393:
382:
378:
362:
361:
360:
358:
355:
351:
330:
326:
295:
291:
277:
276:inner product
273:
269:
265:
261:
258:
257:Hilbert space
234:
230:
221:
218:
215:
207:
203:
166:
162:
153:
150:
147:
139:
135:
119:
109:
107:
103:
99:
94:
92:
88:
84:
79:
77:
73:
67:
65:
55:
53:
49:
45:
41:
37:
34:
33:Hilbert space
30:
20:
16:
8362:
8358:
8306:
8302:
8296:
8253:
8249:
8236:
8193:
8190:Phys. Rev. A
8189:
8183:
8144:cite journal
8123:
8098:
8094:
8088:
8067:cite journal
8038:
8029:
8004:
8000:
7994:
7951:
7948:Algorithmica
7947:
7907:
7897:
7876:
7833:
7827:
7821:
7792:
7788:
7778:
7757:
7714:
7711:Phys. Rev. A
7710:
7704:
7691:
7656:
7652:
7622:cite journal
7593:
7558:
7498:
7494:
7488:
7455:
7451:
7375:
7332:
7329:Phys. Rev. A
7328:
7318:
7293:
7289:
7283:
7240:
7236:
7230:
7179:
7176:Phys. Rev. A
7175:
7169:
7134:
7130:
7124:
7083:
7079:
7073:
7031:(4): 570–9.
7028:
7024:
7014:
6814:
6297:
6294:
6214:
6128:
5907:
5872:
5867:
5865:
5860:
5858:
5739:
5737:
5700:
5644:
5564:
5560:
5558:
5553:
5547:
5542:
5536:
5340:
5114:Hans Maassen
5104:
5097:
4879:
4874:
4870:
4868:
4660:
4654:
4644:
4642:
4443:
4441:
4380:
4227:
3955:
3893:
3889:
3858:finite field
3822:
3641:
3637:
3635:
3473:
3467:
3354:
3102:
3098:
3096:
3083:
3079:
3075:
3072:
2957:
2802:phase factor
2799:
2720:
2656:
2610:
2561:
2492:
2488:
2484:
2480:
2478:
2473:
2469:
2467:
2367:
2363:
2359:
2357:
2275:
2172:
2166:
2068:
2063:
2059:
2022:
2018:
1983:
933:
928:
924:
920:
916:
914:
908:
907:Example for
821:eigenvectors
816:
814:
505:
499:
498:Example for
487:
485:
356:
263:
259:
115:
95:
80:
68:
61:
51:
39:
35:
26:
15:
8457:Hypergraphs
7789:Mathematics
4410:is that of
2604:) yields a
2602:prime power
492:measurement
348:equals the
78:and more .
48:measurement
8411:Categories
7501:(1): 1–5.
7080:J. Phys. A
7006:References
3470:eigenbases
2987:to be the
2895:eigenbases
2723:such that
2651:Weyl group
2370:simply is
116:A pair of
44:eigenstate
8372:0802.0394
8341:118268095
8316:0710.1185
8228:119222014
8203:0811.2298
8135:0909.3720
7930:967938939
7843:1109.6514
7802:1404.3774
7795:(2): 36.
7724:0808.1614
7683:118551747
7666:1004.3348
7432:σ
7402:σ
7389:σ
7342:1202.5058
7131:Ann. Phys
6957:∂
6953:∂
6948:θ
6945:
6936:−
6930:^
6921:θ
6918:
6892:^
6866:⟩
6861:θ
6831:⟩
6789:θ
6786:
6775:π
6748:⟩
6735:θ
6727:⟨
6699:β
6679:θ
6676:
6653:θ
6650:
6627:β
6607:α
6587:β
6555:β
6546:α
6488:
6459:∂
6455:∂
6450:β
6444:−
6438:^
6429:α
6403:∂
6399:∂
6391:−
6368:^
6342:⟩
6314:⟩
6277:ℏ
6274:π
6247:⟩
6233:⟨
6196:∈
6187:⟩
6154:∈
6145:⟩
6109:∈
6065:⟩
6041:ψ
6021:ψ
6017:⟨
5989:⟩
5965:ψ
5939:⟩
5925:ψ
5818:
5797:≥
5754:∑
5714:
5621:
5615:≥
5391:⟩
5363:⟨
5320:
5311:−
5308:≥
5228:⟩
5156:⟩
5068:ϕ
5052:−
5045:ϕ
5034:−
5019:−
5006:−
4992:ϕ
4981:−
4973:−
4966:ϕ
4904:ϕ
4838:ϕ
4816:ϕ
4794:ϕ
4770:ϕ
4748:ϕ
4726:ϕ
4616:∈
4605:for each
4578:∈
4564:^
4542:^
4491:∈
4477:^
4448:commuting
4395:⋅
4389:χ
4346:−
4334:θ
4327:⋯
4308:θ
4295:θ
4288:θ
4271:π
4257:
4245:θ
4239:χ
4205:⟨
4202:⟩
4179:χ
4162:∈
4155:∑
4145:^
4103:⟨
4100:⟩
4069:∈
4062:∑
4052:^
4013:^
3977:^
3926:∈
3915:⟩
3805:⟩
3769:ω
3760:−
3742:∑
3723:⟩
3709:ψ
3671:^
3658:^
3613:−
3603:^
3590:^
3578:…
3563:^
3550:^
3535:^
3523:^
3508:^
3493:^
3468:Then the
3439:π
3423:ω
3397:−
3391:≤
3385:≤
3374:⟩
3332:⟨
3329:⟩
3312:ω
3303:−
3285:∑
3275:^
3240:⟨
3237:⟩
3215:−
3197:∑
3187:^
3155:^
3126:^
3055:−
3049:≤
3037:≤
3019:ω
2972:^
2940:^
2911:^
2893:then the
2871:π
2858:≡
2855:ω
2831:ω
2811:ω
2800:for some
2782:^
2770:^
2761:ω
2752:^
2740:^
2700:^
2671:^
2564:SIC-POVMs
2551:SIC-POVMs
2334:≤
2315:≤
2236:⋯
2130:⋯
1937:−
1888:−
1839:−
1808:−
1799:−
1790:−
1715:−
1678:−
1623:−
1592:−
1583:−
1574:−
1499:−
1462:−
1407:−
1376:−
1367:−
1358:−
1289:−
1274:−
1237:−
1228:−
1197:−
1188:−
885:σ
875:σ
848:σ
835:σ
787:⟩
773:−
770:⟩
745:⟩
728:⟩
665:⟩
654:−
651:⟩
626:⟩
612:⟩
557:⟩
543:⟩
459:…
447:∈
435:∀
403:⟩
375:⟨
354:dimension
336:⟩
301:⟩
272:magnitude
240:⟩
219:…
213:⟩
172:⟩
151:…
145:⟩
72:SIC-POVMs
8397:67784632
8288:41654752
7906:(2017).
7868:55755390
7585:12395501
7533:14848100
7480:10034422
7367:34502667
7275:15326700
7222:16407184
7065:16590645
6995:SIC-POVM
6979:See also
6105:′
6059:′
6049:′
5983:′
5973:′
3636:For odd
2175:, where
488:unbiased
58:Overview
8377:Bibcode
8321:Bibcode
8268:Bibcode
8208:Bibcode
8103:Bibcode
8055:Bibcode
8021:2372390
7986:1280557
7966:Bibcode
7848:Bibcode
7749:9295036
7729:Bibcode
7610:Bibcode
7513:Bibcode
7460:Bibcode
7347:Bibcode
7298:Bibcode
7255:Bibcode
7214:9913672
7194:Bibcode
7139:Bibcode
7088:Bibcode
7033:Bibcode
5701:optimal
5478:entropy
2715:be two
2167:is the
823:of the
352:of the
350:inverse
274:of the
270:of the
91:qutrits
8395:
8339:
8286:
8226:
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7866:
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7056:222876
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6815:where
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5341:where
5112:, and
4228:where
3640:, the
3355:where
2653:method
932:, 0 ≤
268:square
8393:S2CID
8367:arXiv
8337:S2CID
8311:arXiv
8284:S2CID
8258:arXiv
8246:(PDF)
8224:S2CID
8198:arXiv
8130:arXiv
8045:arXiv
8017:JSTOR
7982:S2CID
7956:arXiv
7883:arXiv
7864:S2CID
7838:arXiv
7797:arXiv
7764:arXiv
7745:S2CID
7719:arXiv
7679:S2CID
7661:arXiv
7600:arXiv
7581:S2CID
7563:arXiv
7529:S2CID
7503:arXiv
7363:S2CID
7337:arXiv
7271:S2CID
7245:arXiv
7218:S2CID
7184:arXiv
7000:QBism
6538:with
3823:When
3461:is a
3107:prime
3097:When
2843:is a
2823:. If
87:qubit
8157:help
8080:help
8005:1957
7926:OCLC
7916:ISBN
7635:help
7476:PMID
7423:and
7210:PMID
7061:PMID
6990:POVM
6907:and
6843:and
6665:and
6619:and
6383:and
6171:and
6086:>
5951:and
5672:and
5507:and
5442:and
5408:and
5188:and
4518:and
3992:and
3141:and
2926:and
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2657:Let
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1984:Let
915:For
313:and
187:and
23:B2).
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7834:376
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7737:doi
7671:doi
7573:doi
7521:doi
7499:284
7468:doi
7355:doi
7306:doi
7263:doi
7202:doi
7155:hdl
7147:doi
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7096:doi
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7041:doi
6942:sin
6915:cos
6783:sin
6673:sin
6647:cos
6639:as
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6217:is
5815:log
5711:log
5618:log
5355:max
5317:log
4254:exp
3170:by
3105:is
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2171:of
911:= 4
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255:in
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