819:
3198:
can be performed. Computational machinery is constructed out of physical matter that follows quantum mechanics, and each operation, if it is to be unambiguous, must be a transition of the system from one state to an orthogonal state. Suppose the computing machinery is a physical system evolving under
70:
and complexity. In 2017, QSLs were studied in a quantum oscillator at high temperature. In 2018, it was shown that QSL are not restricted to the quantum domain and that similar bounds hold in classical systems. In 2021, both the
Mandelstam-Tamm and the Margolus–Levitin QSL bounds were concurrently
2366:
2202:
This provides a result for time varying states. Although it also provides a bound for mixed states, the bound (for mixed states) can be so loose as to be uninformative. The
Margolus–Levitin theorem has not yet been established in time-dependent quantum systems, whose Hamiltonians
2798:
3312:
requires only modest resources. Brute-forcing a 256-bit key requires planetary-scale computers, while a brute-force search of 512-bit keys is effectively unattainable within the lifetime of the universe, even if galactic-sized computers were applied to the problem.
585:
3054:
2194:
2055:
1083:
71:
tested in a single experiment which indicated there are "two different regimes: one where the
Mandelstam-Tamm limit constrains the evolution at all times, and a second where a crossover to the Margolus-Levitin limit occurs at longer times."
3284:
2902:
1527:
1434:
1191:
3320:
limits the amount of information that can be stored within a volume of space. The maximal rate of change of information within that volume of space is given by the quantum speed limit. This product of limits is sometimes called the
573:
1788:
1610:
363:
2695:
2259:
62:
and their evolution is also subject to QSL. Quite remarkably it was shown that environmental effects, such as non-Markovian dynamics can speed up quantum processes, which was verified in a cavity QED experiment.
1906:
1682:
160:
42:) is a limitation on the minimum time for a quantum system to evolve between two distinguishable (orthogonal) states. QSL theorems are closely related to time-energy uncertainty relations. In 1945,
2703:
423:
3716:
Cimmarusti, A. D.; Yan, Z.; Patterson, B. D.; Corcos, L. P.; Orozco, L. A.; Deffner, S. (11 June 2015). "Environment-Assisted Speed-up of the Field
Evolution in Cavity Quantum Electrodynamics".
3103:
1940:
3138:
2452:
814:{\displaystyle (\delta E)^{2}=\left\langle \psi |H^{2}|\psi \right\rangle -(\left\langle \psi |H|\psi \right\rangle )^{2}={\frac {1}{2}}\sum _{n,m}|c_{n}|^{2}|c_{m}|^{2}(E_{n}-E_{m})^{2}}
2963:
2832:
58:
showed that the speed of evolution cannot exceed the mean energy, a result known as the
Margolus–Levitin theorem. Realistic physical systems in contact with an environment are known as
2522:
2487:
2615:
1841:
3178:
2979:
1935:
2075:
2932:
1266:
246:
218:
947:
3420:
Mandelstam, L.; Tamm, Ig. (1991). "The
Uncertainty Relation Between Energy and Time in Non-relativistic Quantum Mechanics". In Bolotovskii, Boris M.; Frenkel, Victor Ya.;
459:
4467:
1354:
939:
494:
190:
3199:
Hamiltonian that does not change with time. Then, according to the
Margolus–Levitin theorem, the number of operations per unit time per unit energy is bounded above by
2579:
2402:
3205:
2840:
2553:
2228:
1550:
1329:
1309:
1222:
906:
1445:
1289:
1094:
847:
502:
1687:
1362:
1558:
258:
3408:
2620:
2361:{\displaystyle \left|\psi _{q}\right\rangle ={\frac {1}{\sqrt {2}}}\left(\left|E_{0}\right\rangle +e^{i\varphi }\left|E_{1}\right\rangle \right)}
3297:
of energy. This is including "classical" computers, since even classical computers are still made of matter that follows quantum mechanics.
50:
derived a time-energy uncertainty relation that bounds the speed of evolution in terms of the energy dispersion. Over half a century later,
17:
2065:
The
Margolus–Levitin theorem can also be generalized to the case where the Hamiltonian varies with time, and the system is described by a
3349:
Deffner, S.; Campbell, S. (10 October 2017). "Quantum speed limits: from
Heisenberg's uncertainty principle to optimal quantum control".
1857:
3523:
Taddei, M. M.; Escher, B. M.; Davidovich, L.; de Matos Filho, R. L. (30 January 2013). "Quantum Speed Limit for
Physical Processes".
3289:
This establishes a strict upper limit on the number of calculations that can be performed by physical matter. The processing rate of
1618:
853:. The quantum evolution is independent of the particular Hamiltonian used to transport the quantum system along a given curve in the
93:
87:; they take a simpler form for pure states. An arbitrary pure state can be written as a linear combination of energy eigenstates:
3406:
Mandelshtam, L. I.; Tamm, I. E. (1945). "The uncertainty relation between energy and time in nonrelativistic quantum mechanics".
2793:{\displaystyle t_{\perp }\geq \max \left({\frac {\pi \hbar }{2\,\delta E}}\;,\;{\frac {\pi \hbar }{2\,E_{\text{avg}}}}\right)}
3443:
877:
must be employed in place of the Fubini–Study metric. A mixed state can be understood as a sum over pure states, weighted by
374:
3062:
4485:. 5th Berkeley Symposium on Mathematical Statistics and Probability; Univ. of California Press, Berkeley, California.
3108:
2407:
873:
The Mandelstam–Tamm limit can also be stated for mixed states and for time-varying Hamiltonians. In this case, the
2057:
This form applies when the Hamiltonian is not time-dependent, and the ground-state energy is defined to be zero.
1532:
is the time-averaged uncertainty in energy. For a pure state evolving under a time-varying Hamiltonian, the time
2937:
2806:
3470:
3301:
2966:
2492:
2457:
4611:
3049:{\displaystyle {\frac {\pi \hbar }{E_{\text{max}}}}\leq t_{\perp }\leq {\frac {2\pi \hbar }{E_{\text{max}}}}}
2584:
4098:
Ness, Gal; Lam, Manolo R.; Alt, Wolfgang; Meschede, Dieter; Sagi, Yoav; Alberti, Andrea (22 December 2021).
2803:
Levitin and Toffoli also provide a bound for the average energy in terms of the maximum. For any pure state
1793:
3143:
881:; likewise, the Bures metric is a weighted sum of the Fubini–Study metric. For a time-varying Hamiltonian
2189:{\displaystyle \int _{0}^{\tau }|{\text{tr}}(\rho _{t}H_{t})|dt\geq \hbar D_{B}(\rho _{0},\rho _{\tau })}
2050:{\displaystyle \langle E\rangle =E_{\text{avg}}=\langle \psi |H|\psi \rangle =\sum _{n}|c_{n}|^{2}E_{n}.}
4472:
In: Self-Organizing systems 1962, edited M.C. Yovits et al., Spartan Books, Washington, D.C. pp. 93–106.
4606:
1914:
461:
required to achieve this condition is called the orthogonalization interval or orthogonalization time.
1078:{\displaystyle \sigma _{H}^{2}(t)=|{\text{tr}}(\rho _{t}H_{t}^{2})|-|{\text{tr}}(\rho _{t}H_{t})|^{2}}
4275:
2910:
1235:
223:
195:
3351:
2970:
2525:
854:
431:
4228:
4158:
4037:
3980:
3850:
3718:
3655:
3592:
3525:
858:
3468:
Margolus, Norman; Levitin, Lev B. (September 1998). "The maximum speed of dynamical evolution".
914:
472:
249:
168:
3917:
3279:{\displaystyle {\frac {2}{\hbar \pi }}=6\times 10^{33}\mathrm {s} ^{-1}\cdot \mathrm {J} ^{-1}}
2561:
4411:
2897:{\displaystyle {\frac {E_{\text{max}}}{4}}\leq E_{\text{avg}}\leq {\frac {E_{\text{max}}}{2}}}
865:, as it can be understood as the arccos of the inner product of the initial and final states.
3189:
2374:
2251:
1522:{\displaystyle {\overline {\sigma }}_{H}={\frac {1}{\tau }}\int _{0}^{\tau }\sigma _{H}(t)dt}
67:
4035:
Okuyama, Manaka; Ohzeki, Masayuki (12 February 2018). "Quantum Speed Limit is Not Quantum".
2581:
This result is sharp, in that this state also satisfies the Margolus–Levitin bound, in that
1334:
1232:, giving the shortest possible distance of any continuous curve connecting two points, with
4569:
4520:
4433:
4376:
4297:
4237:
4177:
4056:
3999:
3936:
3869:
3798:
3737:
3674:
3611:
3544:
3489:
3369:
2531:
2206:
1535:
1314:
1294:
1200:
884:
850:
4153:
1186:{\displaystyle \int _{0}^{\tau }\sigma _{H}(t)dt\geq \hbar D_{B}(\rho _{0},\rho _{\tau })}
8:
4309:
3329:. That is, Hawking radiation is emitted at the maximal allowed rate set by these bounds.
2066:
1552:
taken to evolve from one pure state to another pure state orthogonal to it is bounded as
878:
568:{\displaystyle t_{\perp }\geq {\frac {\pi \hbar }{2\,\delta E}}={\frac {h}{4\,\delta E}}}
84:
59:
4573:
4524:
4437:
4380:
4301:
4241:
4181:
4060:
4003:
3940:
3873:
3802:
3741:
3678:
3615:
3548:
3493:
3373:
1783:{\displaystyle D_{B}(\rho _{0},\rho _{t})=\arccos |\langle \psi _{0}|\psi _{t}\rangle |}
1429:{\displaystyle \tau \geq {\frac {\hbar }{{\overline {\sigma }}_{H}}}D_{B}(\rho ,\rho ')}
1271:
4585:
4559:
4536:
4510:
4423:
4366:
4329:
4287:
4209:
4167:
4126:
4099:
4080:
4046:
3989:
3926:
3893:
3859:
3830:
3788:
3761:
3727:
3698:
3664:
3653:
Deffner, S.; Lutz, E. (3 July 2013). "Quantum speed limit for non-Markovian dynamics".
3635:
3601:
3568:
3534:
3505:
3479:
3425:
3385:
3359:
3305:
832:
4352:
3501:
1605:{\displaystyle \tau \geq {\frac {\hbar }{{\overline {\sigma }}_{H}}}{\frac {\pi }{2}}}
4550:
4540:
4501:
4449:
4392:
4357:
4333:
4321:
4253:
4201:
4193:
4131:
4072:
4017:
3885:
3822:
3814:
3753:
3690:
3627:
3560:
3439:
3326:
2246:
states that the precise bound for the Mandelstam–Tamm theorem is attained only for a
43:
31:
4589:
4213:
3765:
3702:
3572:
4581:
4577:
4528:
4482:
4441:
4384:
4313:
4305:
4245:
4189:
4185:
4121:
4111:
4084:
4068:
4064:
4012:
4007:
3975:
3954:
3944:
3897:
3877:
3806:
3749:
3745:
3686:
3682:
3639:
3623:
3619:
3587:
3556:
3552:
3497:
3431:
3389:
3377:
3317:
4548:
Sinitsyn, Nikolai A. (2018). "Is there a quantum limit on speed of computation?".
3881:
3509:
3293:
forms of computation cannot be higher than about 6 Ă— 10 operations per second per
2230:
are driven by arbitrary time-dependent parameters, except for the adiabatic case.
1851:
For the case of a pure state, Margolus and Levitin obtain a different limit, that
358:{\displaystyle |\psi _{t}\rangle =\sum _{n}c_{n}e^{itE_{n}/\hbar }|E_{n}\rangle .}
4499:
Jordan, Stephen P. (2017). "Fast quantum computation at arbitrarily low energy".
4348:
3834:
2243:
51:
4249:
3435:
27:
Limitation on the minimum time for a quantum system to evolve between two states
4532:
4445:
4388:
4154:"Fundamental Limit on the Rate of Quantum Dynamics: The Unified Bound Is Tight"
3949:
3912:
3421:
3381:
3309:
2239:
1225:
909:
3300:
This bound is not merely a fanciful limit: it has practical ramifications for
4600:
4453:
4396:
4325:
4197:
3818:
862:
4257:
4205:
4135:
4116:
4076:
4021:
3889:
3826:
3757:
3694:
3631:
3564:
2690:{\displaystyle t_{\perp }=\hbar \pi /2E_{\text{avg}}=\hbar \pi /2\delta E.}
2556:
874:
4412:"Comment on 'Energy-time uncertainty relation for driven quantum systems'"
1268:
understood as an infinitessimal path length along a curve parametrized by
496:
required for a state to evolve into an orthogonal state is bounded below:
469:
For pure states, the Mandelstam–Tamm theorem states that the minimum time
4226:
Aharonov, Yakir; Anandan, Jeeva (1990). "Geometry of quantum evolution".
3974:
Shanahan, B.; Chenu, A.; Margolus, N.; del Campo, A. (12 February 2018).
3864:
3793:
3779:
Lloyd, Seth (31 August 2000). "Ultimate physical limits to computation".
3484:
3195:
55:
4317:
3959:
2199:
with the ground-state defined so that it has energy zero at all times.
80:
3848:
Lloyd, Seth (24 May 2002). "Computational Capacity of the Universe".
3810:
47:
3522:
4564:
4515:
4428:
4371:
4051:
3994:
3931:
3732:
3590:(30 January 2013). "Quantum Speed Limits in Open System Dynamics".
3364:
1901:{\displaystyle \tau _{\perp }\geq {\frac {h}{4\langle E\rangle }},}
1229:
826:
4292:
4172:
3669:
3606:
3539:
3976:"Quantum Speed Limits across the Quantum-to-Classical Transition"
3194:
The quantum speed limit bounds establish an upper bound at which
3913:"Geometric quantum speed limits: a case for Wigner phase space"
1677:{\displaystyle \rho _{t}=|\psi _{t}\rangle \langle \psi _{t}|.}
3973:
3585:
1615:
This follows, as for a pure state, one has the density matrix
1228:
between the starting and ending states. The Bures distance is
4276:"Energy–time uncertainty relation for driven quantum systems"
4151:
3294:
2697:
This result establishes that the combined limits are strict:
2247:
3715:
155:{\displaystyle |\psi \rangle =\sum _{n}c_{n}|E_{n}\rangle .}
165:
The task is to provide a lower bound for the time interval
4346:
3208:
3146:
3111:
3065:
2982:
2940:
2913:
2843:
2809:
2706:
2623:
2587:
2564:
2534:
2495:
2460:
2410:
2377:
2262:
2209:
2078:
1943:
1917:
1860:
1796:
1690:
1621:
1561:
1538:
1448:
1365:
1337:
1317:
1297:
1274:
1238:
1203:
1097:
950:
917:
887:
835:
588:
505:
475:
434:
418:{\displaystyle \langle \psi _{0}|\psi _{t}\rangle =0}
377:
261:
248:. The time evolution of a pure state is given by the
226:
198:
171:
96:
3098:{\displaystyle E_{\text{max}}t_{\perp }=\pi \hbar }
857:; the distance along this curve is measured by the
4416:Journal of Physics A: Mathematical and Theoretical
4280:Journal of Physics A: Mathematical and Theoretical
4100:"Observing crossover between quantum speed limits"
4097:
3430:. Berlin, Heidelberg: Springer. pp. 115–123.
3304:. Imagining a computer operating at this limit, a
3278:
3172:
3132:
3097:
3048:
2957:
2926:
2896:
2826:
2792:
2689:
2609:
2573:
2547:
2516:
2481:
2446:
2396:
2360:
2222:
2188:
2049:
1929:
1900:
1843:when the initial and final states are orthogonal.
1835:
1782:
1684:The quantum angle (Fubini–Study distance) is then
1676:
1604:
1544:
1521:
1428:
1348:
1323:
1303:
1283:
1260:
1216:
1185:
1077:
933:
900:
841:
813:
567:
488:
453:
417:
357:
240:
212:
184:
154:
4598:
4469:Optimization through evolution and recombination
4353:"Quantum speed limits, coherence, and asymmetry"
3586:del Campo, A.; Egusquiza, I. L.; Plenio, M. B.;
2720:
4225:
3467:
3405:
3348:
4269:
4267:
3463:
3461:
3419:
2934:is the maximum energy eigenvalue appearing in
1088:The Mandelstam–Tamm limit then takes the form
4409:
4274:Deffner, Sebastian; Lutz, Eric (2013-08-23).
4034:
3133:{\displaystyle \left|\psi _{q}\right\rangle }
2447:{\displaystyle E_{1}=\pm \pi \hbar /\Delta t}
1991:
1969:
1950:
1944:
1924:
1918:
1889:
1883:
1772:
1744:
1653:
1650:
406:
378:
349:
277:
235:
207:
146:
105:
4273:
4264:
3652:
3458:
79:The speed limit theorems can be stated for
4410:Okuyama, Manaka; Ohzeki, Masayuki (2018).
2958:{\displaystyle \left|\psi \right\rangle .}
2827:{\displaystyle \left|\psi \right\rangle ,}
2756:
2752:
2060:
74:
4563:
4514:
4427:
4370:
4291:
4171:
4125:
4115:
4050:
4011:
3993:
3958:
3948:
3930:
3863:
3792:
3731:
3668:
3605:
3538:
3483:
3363:
2771:
2742:
2517:{\displaystyle \left|E_{1}\right\rangle }
2482:{\displaystyle \left|E_{0}\right\rangle }
1846:
555:
533:
4547:
4403:
4152:Lev B. Levitin; Tommaso Toffoli (2009),
2233:
464:
3910:
2610:{\displaystyle E_{\text{avg}}=\delta E}
2250:state. This is a two-level state in an
941:the variance of the energy is given by
14:
4599:
4498:
4147:
4145:
3105:is again attained for the qubit state
1836:{\displaystyle D_{B}=\arccos 0=\pi /2}
3847:
3778:
3401:
3399:
3344:
3342:
3183:
220:to evolve into a state orthogonal to
3173:{\displaystyle E_{\text{max}}=E_{1}}
4219:
4142:
868:
24:
4492:
3396:
3339:
3263:
3245:
2438:
66:QSL have been used to explore the
25:
4623:
3092:
3030:
2989:
2834:the average energy is bounded as
2763:
2734:
2664:
2637:
2430:
2144:
1930:{\displaystyle \langle E\rangle }
1141:
525:
329:
4475:
4460:
4351:; Zanardi, Paolo (2016-05-24).
4340:
4091:
4028:
3967:
3911:Deffner, S. (20 October 2017).
3904:
2069:. In this form, it is given by
861:. This is sometimes called the
368:Orthogonality is obtained when
192:required for the initial state
4582:10.1016/j.physleta.2017.12.042
4310:10.1088/1751-8113/46/33/335302
4190:10.1103/PhysRevLett.103.160502
4069:10.1103/PhysRevLett.120.070402
4013:10.1103/PhysRevLett.120.070401
3841:
3772:
3750:10.1103/PhysRevLett.114.233602
3709:
3687:10.1103/PhysRevLett.111.010402
3646:
3624:10.1103/PhysRevLett.110.050403
3579:
3557:10.1103/PhysRevLett.110.050402
3516:
3471:Physica D: Nonlinear Phenomena
3302:quantum-resistant cryptography
2967:quarter-pinched sphere theorem
2927:{\displaystyle E_{\text{max}}}
2183:
2157:
2131:
2127:
2104:
2095:
2024:
2008:
1984:
1976:
1776:
1758:
1740:
1727:
1701:
1667:
1636:
1510:
1504:
1423:
1406:
1261:{\displaystyle \sigma _{H}(t)}
1255:
1249:
1180:
1154:
1129:
1123:
1065:
1060:
1037:
1028:
1020:
1016:
988:
979:
972:
966:
802:
775:
765:
749:
738:
722:
683:
670:
662:
650:
635:
620:
599:
589:
428:and the minimum time interval
392:
335:
263:
241:{\displaystyle |\psi \rangle }
228:
213:{\displaystyle |\psi \rangle }
200:
132:
98:
13:
1:
4483:Quantum noise and information
3882:10.1103/PhysRevLett.88.237901
3502:10.1016/S0167-2789(98)00054-2
3332:
2969:in disguise, transported to
1579:
1455:
1383:
454:{\displaystyle t=t_{\perp }}
7:
4250:10.1103/PhysRevLett.65.1697
3436:10.1007/978-3-642-74626-0_8
3323:Bremermann–Bekenstein limit
2973:.) Thus, one has the bound
829:of the system's energy and
10:
4628:
4533:10.1103/PhysRevA.95.032305
4389:10.1103/PhysRevA.93.052331
3187:
934:{\displaystyle \rho _{t},}
489:{\displaystyle t_{\perp }}
185:{\displaystyle t_{\perp }}
2574:{\displaystyle \varphi .}
4481:Bremermann, H.J. (1965)
4466:Bremermann, H.J. (1962)
4446:10.1088/1751-8121/aacb90
3950:10.1088/1367-2630/aa83dc
3382:10.1088/1751-8121/aa86c6
3352:J. Phys. A: Math. Theor.
2971:complex projective space
855:projective Hilbert space
18:Margolus–Levitin theorem
4229:Physical Review Letters
4159:Physical Review Letters
4038:Physical Review Letters
3981:Physical Review Letters
3851:Physical Review Letters
3719:Physical Review Letters
3656:Physical Review Letters
3593:Physical Review Letters
3526:Physical Review Letters
3059:The strict lower bound
2397:{\displaystyle E_{0}=0}
2371:for energy eigenstates
2061:For time-varying states
1937:is the average energy,
1291:Equivalently, the time
879:classical probabilities
75:Preliminary definitions
4117:10.1126/sciadv.abj9119
3918:New Journal of Physics
3280:
3174:
3134:
3099:
3050:
2959:
2928:
2898:
2828:
2794:
2691:
2611:
2575:
2549:
2518:
2483:
2448:
2398:
2362:
2224:
2190:
2051:
1931:
1902:
1847:Margolus–Levitin limit
1837:
1784:
1678:
1606:
1546:
1523:
1430:
1350:
1349:{\displaystyle \rho '}
1325:
1305:
1285:
1262:
1218:
1187:
1079:
935:
902:
843:
815:
569:
490:
455:
419:
359:
242:
214:
186:
156:
3325:; it is saturated by
3281:
3175:
3135:
3100:
3051:
2960:
2929:
2899:
2829:
2795:
2692:
2612:
2576:
2550:
2548:{\displaystyle E_{1}}
2519:
2484:
2449:
2399:
2363:
2234:Levitin–Toffoli limit
2225:
2223:{\displaystyle H_{t}}
2191:
2052:
1932:
1903:
1838:
1790:and so one concludes
1785:
1679:
1607:
1547:
1545:{\displaystyle \tau }
1524:
1431:
1351:
1326:
1324:{\displaystyle \rho }
1311:taken to evolve from
1306:
1304:{\displaystyle \tau }
1286:
1263:
1219:
1217:{\displaystyle D_{B}}
1188:
1080:
936:
903:
901:{\displaystyle H_{t}}
844:
816:
570:
491:
465:Mandelstam–Tamm limit
456:
420:
360:
243:
215:
187:
157:
68:limits of computation
4612:Mathematical physics
3206:
3144:
3109:
3063:
2980:
2938:
2911:
2841:
2807:
2704:
2621:
2585:
2562:
2532:
2528:of the energy level
2493:
2458:
2408:
2375:
2260:
2207:
2076:
1941:
1915:
1858:
1794:
1688:
1619:
1559:
1536:
1446:
1363:
1335:
1315:
1295:
1272:
1236:
1201:
1095:
948:
915:
885:
851:Hamiltonian operator
833:
586:
503:
473:
432:
375:
259:
250:Schrödinger equation
224:
196:
169:
94:
60:open quantum systems
4574:2018PhLA..382..477S
4525:2017PhRvA..95c2305J
4438:2018JPhA...51E8001O
4381:2016PhRvA..93e2331M
4349:Spekkens, Robert W.
4302:2013JPhA...46G5302D
4242:1990PhRvL..65.1697A
4182:2009PhRvL.103p0502L
4061:2018PhRvL.120g0402O
4004:2018PhRvL.120g0401S
3941:2017NJPh...19j3018D
3874:2002PhRvL..88w7901L
3803:2000Natur.406.1047L
3787:(6799): 1047–1054.
3742:2015PhRvL.114w3602C
3679:2013PhRvL.111a0402D
3616:2013PhRvL.110e0403D
3549:2013PhRvL.110e0402T
3494:1998PhyD..120..188M
3374:2017JPhA...50S3001D
3308:to break a 128-bit
2252:equal superposition
2093:
1493:
1112:
1015:
965:
859:Fubini–Study metric
36:quantum speed limit
3306:brute-force search
3276:
3190:Bremermann's limit
3184:Bremermann's limit
3170:
3130:
3095:
3046:
2955:
2924:
2894:
2824:
2790:
2687:
2607:
2571:
2545:
2514:
2479:
2444:
2394:
2358:
2220:
2186:
2079:
2047:
2006:
1927:
1898:
1833:
1780:
1674:
1602:
1542:
1519:
1479:
1426:
1346:
1321:
1301:
1284:{\displaystyle t.}
1281:
1258:
1214:
1183:
1098:
1075:
1001:
951:
931:
898:
839:
811:
720:
565:
486:
451:
415:
355:
292:
238:
210:
182:
152:
120:
4607:Quantum mechanics
4551:Physics Letters A
4502:Physical Review A
4358:Physical Review A
4236:(14): 1697–1700.
3445:978-3-642-74628-4
3327:Hawking radiation
3222:
3154:
3073:
3044:
3041:
3003:
3000:
2921:
2892:
2886:
2871:
2859:
2853:
2783:
2779:
2750:
2658:
2595:
2555:and an arbitrary
2524:are unique up to
2294:
2293:
2238:A 2009 result by
2102:
1997:
1963:
1893:
1600:
1590:
1582:
1477:
1458:
1394:
1386:
1035:
986:
908:and time-varying
842:{\displaystyle H}
705:
703:
563:
541:
283:
111:
44:Leonid Mandelstam
32:quantum mechanics
16:(Redirected from
4619:
4593:
4567:
4544:
4518:
4486:
4479:
4473:
4464:
4458:
4457:
4431:
4407:
4401:
4400:
4374:
4344:
4338:
4337:
4295:
4271:
4262:
4261:
4223:
4217:
4216:
4175:
4149:
4140:
4139:
4129:
4119:
4110:(52): eabj9119.
4104:Science Advances
4095:
4089:
4088:
4054:
4032:
4026:
4025:
4015:
3997:
3971:
3965:
3964:
3962:
3952:
3934:
3908:
3902:
3901:
3867:
3865:quant-ph/0110141
3845:
3839:
3838:
3811:10.1038/35023282
3796:
3794:quant-ph/9908043
3776:
3770:
3769:
3735:
3713:
3707:
3706:
3672:
3650:
3644:
3643:
3609:
3583:
3577:
3576:
3542:
3520:
3514:
3513:
3487:
3485:quant-ph/9710043
3478:(1–2): 188–195.
3465:
3456:
3455:
3453:
3452:
3417:
3403:
3394:
3393:
3367:
3346:
3318:Bekenstein bound
3285:
3283:
3282:
3277:
3275:
3274:
3266:
3257:
3256:
3248:
3242:
3241:
3223:
3221:
3210:
3179:
3177:
3176:
3171:
3169:
3168:
3156:
3155:
3152:
3139:
3137:
3136:
3131:
3129:
3125:
3124:
3104:
3102:
3101:
3096:
3085:
3084:
3075:
3074:
3071:
3055:
3053:
3052:
3047:
3045:
3043:
3042:
3039:
3033:
3022:
3017:
3016:
3004:
3002:
3001:
2998:
2992:
2984:
2964:
2962:
2961:
2956:
2951:
2933:
2931:
2930:
2925:
2923:
2922:
2919:
2903:
2901:
2900:
2895:
2893:
2888:
2887:
2884:
2878:
2873:
2872:
2869:
2860:
2855:
2854:
2851:
2845:
2833:
2831:
2830:
2825:
2820:
2799:
2797:
2796:
2791:
2789:
2785:
2784:
2782:
2781:
2780:
2777:
2766:
2758:
2751:
2749:
2737:
2729:
2716:
2715:
2696:
2694:
2693:
2688:
2674:
2660:
2659:
2656:
2647:
2633:
2632:
2616:
2614:
2613:
2608:
2597:
2596:
2593:
2580:
2578:
2577:
2572:
2554:
2552:
2551:
2546:
2544:
2543:
2523:
2521:
2520:
2515:
2513:
2509:
2508:
2488:
2486:
2485:
2480:
2478:
2474:
2473:
2453:
2451:
2450:
2445:
2437:
2420:
2419:
2403:
2401:
2400:
2395:
2387:
2386:
2367:
2365:
2364:
2359:
2357:
2353:
2352:
2348:
2347:
2334:
2333:
2318:
2314:
2313:
2295:
2289:
2285:
2280:
2276:
2275:
2229:
2227:
2226:
2221:
2219:
2218:
2195:
2193:
2192:
2187:
2182:
2181:
2169:
2168:
2156:
2155:
2134:
2126:
2125:
2116:
2115:
2103:
2100:
2098:
2092:
2087:
2056:
2054:
2053:
2048:
2043:
2042:
2033:
2032:
2027:
2021:
2020:
2011:
2005:
1987:
1979:
1965:
1964:
1961:
1936:
1934:
1933:
1928:
1907:
1905:
1904:
1899:
1894:
1892:
1875:
1870:
1869:
1842:
1840:
1839:
1834:
1829:
1806:
1805:
1789:
1787:
1786:
1781:
1779:
1771:
1770:
1761:
1756:
1755:
1743:
1726:
1725:
1713:
1712:
1700:
1699:
1683:
1681:
1680:
1675:
1670:
1665:
1664:
1649:
1648:
1639:
1631:
1630:
1611:
1609:
1608:
1603:
1601:
1593:
1591:
1589:
1588:
1583:
1575:
1569:
1551:
1549:
1548:
1543:
1528:
1526:
1525:
1520:
1503:
1502:
1492:
1487:
1478:
1470:
1465:
1464:
1459:
1451:
1435:
1433:
1432:
1427:
1422:
1405:
1404:
1395:
1393:
1392:
1387:
1379:
1373:
1355:
1353:
1352:
1347:
1345:
1330:
1328:
1327:
1322:
1310:
1308:
1307:
1302:
1290:
1288:
1287:
1282:
1267:
1265:
1264:
1259:
1248:
1247:
1223:
1221:
1220:
1215:
1213:
1212:
1192:
1190:
1189:
1184:
1179:
1178:
1166:
1165:
1153:
1152:
1122:
1121:
1111:
1106:
1084:
1082:
1081:
1076:
1074:
1073:
1068:
1059:
1058:
1049:
1048:
1036:
1033:
1031:
1023:
1014:
1009:
1000:
999:
987:
984:
982:
964:
959:
940:
938:
937:
932:
927:
926:
907:
905:
904:
899:
897:
896:
869:For mixed states
848:
846:
845:
840:
820:
818:
817:
812:
810:
809:
800:
799:
787:
786:
774:
773:
768:
762:
761:
752:
747:
746:
741:
735:
734:
725:
719:
704:
696:
691:
690:
681:
677:
673:
665:
646:
642:
638:
633:
632:
623:
607:
606:
574:
572:
571:
566:
564:
562:
547:
542:
540:
528:
520:
515:
514:
495:
493:
492:
487:
485:
484:
460:
458:
457:
452:
450:
449:
424:
422:
421:
416:
405:
404:
395:
390:
389:
364:
362:
361:
356:
348:
347:
338:
333:
332:
328:
323:
322:
302:
301:
291:
276:
275:
266:
247:
245:
244:
239:
231:
219:
217:
216:
211:
203:
191:
189:
188:
183:
181:
180:
161:
159:
158:
153:
145:
144:
135:
130:
129:
119:
101:
21:
4627:
4626:
4622:
4621:
4620:
4618:
4617:
4616:
4597:
4596:
4495:
4493:Further reading
4490:
4489:
4480:
4476:
4465:
4461:
4408:
4404:
4347:Marvian, Iman;
4345:
4341:
4272:
4265:
4224:
4220:
4150:
4143:
4096:
4092:
4033:
4029:
3972:
3968:
3909:
3905:
3846:
3842:
3777:
3773:
3714:
3710:
3651:
3647:
3584:
3580:
3521:
3517:
3466:
3459:
3450:
3448:
3446:
3427:Selected Papers
3422:Peierls, Rudolf
3409:J. Phys. (USSR)
3404:
3397:
3347:
3340:
3335:
3267:
3262:
3261:
3249:
3244:
3243:
3237:
3233:
3214:
3209:
3207:
3204:
3203:
3192:
3186:
3164:
3160:
3151:
3147:
3145:
3142:
3141:
3120:
3116:
3112:
3110:
3107:
3106:
3080:
3076:
3070:
3066:
3064:
3061:
3060:
3038:
3034:
3023:
3021:
3012:
3008:
2997:
2993:
2985:
2983:
2981:
2978:
2977:
2941:
2939:
2936:
2935:
2918:
2914:
2912:
2909:
2908:
2883:
2879:
2877:
2868:
2864:
2850:
2846:
2844:
2842:
2839:
2838:
2810:
2808:
2805:
2804:
2776:
2772:
2767:
2759:
2757:
2738:
2730:
2728:
2727:
2723:
2711:
2707:
2705:
2702:
2701:
2670:
2655:
2651:
2643:
2628:
2624:
2622:
2619:
2618:
2592:
2588:
2586:
2583:
2582:
2563:
2560:
2559:
2539:
2535:
2533:
2530:
2529:
2504:
2500:
2496:
2494:
2491:
2490:
2469:
2465:
2461:
2459:
2456:
2455:
2433:
2415:
2411:
2409:
2406:
2405:
2382:
2378:
2376:
2373:
2372:
2343:
2339:
2335:
2326:
2322:
2309:
2305:
2301:
2300:
2296:
2284:
2271:
2267:
2263:
2261:
2258:
2257:
2244:Tommaso Toffoli
2236:
2214:
2210:
2208:
2205:
2204:
2177:
2173:
2164:
2160:
2151:
2147:
2130:
2121:
2117:
2111:
2107:
2099:
2094:
2088:
2083:
2077:
2074:
2073:
2063:
2038:
2034:
2028:
2023:
2022:
2016:
2012:
2007:
2001:
1983:
1975:
1960:
1956:
1942:
1939:
1938:
1916:
1913:
1912:
1879:
1874:
1865:
1861:
1859:
1856:
1855:
1849:
1825:
1801:
1797:
1795:
1792:
1791:
1775:
1766:
1762:
1757:
1751:
1747:
1739:
1721:
1717:
1708:
1704:
1695:
1691:
1689:
1686:
1685:
1666:
1660:
1656:
1644:
1640:
1635:
1626:
1622:
1620:
1617:
1616:
1592:
1584:
1574:
1573:
1568:
1560:
1557:
1556:
1537:
1534:
1533:
1498:
1494:
1488:
1483:
1469:
1460:
1450:
1449:
1447:
1444:
1443:
1415:
1400:
1396:
1388:
1378:
1377:
1372:
1364:
1361:
1360:
1338:
1336:
1333:
1332:
1316:
1313:
1312:
1296:
1293:
1292:
1273:
1270:
1269:
1243:
1239:
1237:
1234:
1233:
1208:
1204:
1202:
1199:
1198:
1174:
1170:
1161:
1157:
1148:
1144:
1117:
1113:
1107:
1102:
1096:
1093:
1092:
1069:
1064:
1063:
1054:
1050:
1044:
1040:
1032:
1027:
1019:
1010:
1005:
995:
991:
983:
978:
960:
955:
949:
946:
945:
922:
918:
916:
913:
912:
892:
888:
886:
883:
882:
871:
834:
831:
830:
805:
801:
795:
791:
782:
778:
769:
764:
763:
757:
753:
748:
742:
737:
736:
730:
726:
721:
709:
695:
686:
682:
669:
661:
657:
653:
634:
628:
624:
619:
615:
611:
602:
598:
587:
584:
583:
551:
546:
529:
521:
519:
510:
506:
504:
501:
500:
480:
476:
474:
471:
470:
467:
445:
441:
433:
430:
429:
400:
396:
391:
385:
381:
376:
373:
372:
343:
339:
334:
324:
318:
314:
307:
303:
297:
293:
287:
271:
267:
262:
260:
257:
256:
227:
225:
222:
221:
199:
197:
194:
193:
176:
172:
170:
167:
166:
140:
136:
131:
125:
121:
115:
97:
95:
92:
91:
77:
52:Norman Margolus
28:
23:
22:
15:
12:
11:
5:
4625:
4615:
4614:
4609:
4595:
4594:
4558:(7): 477–481.
4545:
4494:
4491:
4488:
4487:
4474:
4459:
4422:(31): 318001.
4402:
4339:
4286:(33): 335302.
4263:
4218:
4166:(16): 160502,
4141:
4090:
4027:
3966:
3925:(10): 103018.
3903:
3858:(23): 237901.
3840:
3771:
3726:(23): 233602.
3708:
3645:
3578:
3515:
3457:
3444:
3395:
3358:(45): 453001.
3337:
3336:
3334:
3331:
3310:encryption key
3287:
3286:
3273:
3270:
3265:
3260:
3255:
3252:
3247:
3240:
3236:
3232:
3229:
3226:
3220:
3217:
3213:
3188:Main article:
3185:
3182:
3167:
3163:
3159:
3150:
3128:
3123:
3119:
3115:
3094:
3091:
3088:
3083:
3079:
3069:
3057:
3056:
3037:
3032:
3029:
3026:
3020:
3015:
3011:
3007:
2996:
2991:
2988:
2954:
2950:
2947:
2944:
2917:
2905:
2904:
2891:
2882:
2876:
2867:
2863:
2858:
2849:
2823:
2819:
2816:
2813:
2801:
2800:
2788:
2775:
2770:
2765:
2762:
2755:
2748:
2745:
2741:
2736:
2733:
2726:
2722:
2719:
2714:
2710:
2686:
2683:
2680:
2677:
2673:
2669:
2666:
2663:
2654:
2650:
2646:
2642:
2639:
2636:
2631:
2627:
2606:
2603:
2600:
2591:
2570:
2567:
2542:
2538:
2512:
2507:
2503:
2499:
2477:
2472:
2468:
2464:
2443:
2440:
2436:
2432:
2429:
2426:
2423:
2418:
2414:
2393:
2390:
2385:
2381:
2369:
2368:
2356:
2351:
2346:
2342:
2338:
2332:
2329:
2325:
2321:
2317:
2312:
2308:
2304:
2299:
2292:
2288:
2283:
2279:
2274:
2270:
2266:
2240:Lev B. Levitin
2235:
2232:
2217:
2213:
2197:
2196:
2185:
2180:
2176:
2172:
2167:
2163:
2159:
2154:
2150:
2146:
2143:
2140:
2137:
2133:
2129:
2124:
2120:
2114:
2110:
2106:
2097:
2091:
2086:
2082:
2062:
2059:
2046:
2041:
2037:
2031:
2026:
2019:
2015:
2010:
2004:
2000:
1996:
1993:
1990:
1986:
1982:
1978:
1974:
1971:
1968:
1959:
1955:
1952:
1949:
1946:
1926:
1923:
1920:
1909:
1908:
1897:
1891:
1888:
1885:
1882:
1878:
1873:
1868:
1864:
1848:
1845:
1832:
1828:
1824:
1821:
1818:
1815:
1812:
1809:
1804:
1800:
1778:
1774:
1769:
1765:
1760:
1754:
1750:
1746:
1742:
1738:
1735:
1732:
1729:
1724:
1720:
1716:
1711:
1707:
1703:
1698:
1694:
1673:
1669:
1663:
1659:
1655:
1652:
1647:
1643:
1638:
1634:
1629:
1625:
1613:
1612:
1599:
1596:
1587:
1581:
1578:
1572:
1567:
1564:
1541:
1530:
1529:
1518:
1515:
1512:
1509:
1506:
1501:
1497:
1491:
1486:
1482:
1476:
1473:
1468:
1463:
1457:
1454:
1437:
1436:
1425:
1421:
1418:
1414:
1411:
1408:
1403:
1399:
1391:
1385:
1382:
1376:
1371:
1368:
1356:is bounded as
1344:
1341:
1320:
1300:
1280:
1277:
1257:
1254:
1251:
1246:
1242:
1226:Bures distance
1211:
1207:
1195:
1194:
1182:
1177:
1173:
1169:
1164:
1160:
1156:
1151:
1147:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1120:
1116:
1110:
1105:
1101:
1086:
1085:
1072:
1067:
1062:
1057:
1053:
1047:
1043:
1039:
1030:
1026:
1022:
1018:
1013:
1008:
1004:
998:
994:
990:
981:
977:
974:
971:
968:
963:
958:
954:
930:
925:
921:
910:density matrix
895:
891:
870:
867:
838:
823:
822:
808:
804:
798:
794:
790:
785:
781:
777:
772:
767:
760:
756:
751:
745:
740:
733:
729:
724:
718:
715:
712:
708:
702:
699:
694:
689:
685:
680:
676:
672:
668:
664:
660:
656:
652:
649:
645:
641:
637:
631:
627:
622:
618:
614:
610:
605:
601:
597:
594:
591:
577:
576:
561:
558:
554:
550:
545:
539:
536:
532:
527:
524:
518:
513:
509:
483:
479:
466:
463:
448:
444:
440:
437:
426:
425:
414:
411:
408:
403:
399:
394:
388:
384:
380:
366:
365:
354:
351:
346:
342:
337:
331:
327:
321:
317:
313:
310:
306:
300:
296:
290:
286:
282:
279:
274:
270:
265:
237:
234:
230:
209:
206:
202:
179:
175:
163:
162:
151:
148:
143:
139:
134:
128:
124:
118:
114:
110:
107:
104:
100:
76:
73:
26:
9:
6:
4:
3:
2:
4624:
4613:
4610:
4608:
4605:
4604:
4602:
4591:
4587:
4583:
4579:
4575:
4571:
4566:
4561:
4557:
4553:
4552:
4546:
4542:
4538:
4534:
4530:
4526:
4522:
4517:
4512:
4509:(3): 032305.
4508:
4504:
4503:
4497:
4496:
4484:
4478:
4471:
4470:
4463:
4455:
4451:
4447:
4443:
4439:
4435:
4430:
4425:
4421:
4417:
4413:
4406:
4398:
4394:
4390:
4386:
4382:
4378:
4373:
4368:
4365:(5): 052331.
4364:
4360:
4359:
4354:
4350:
4343:
4335:
4331:
4327:
4323:
4319:
4315:
4311:
4307:
4303:
4299:
4294:
4289:
4285:
4281:
4277:
4270:
4268:
4259:
4255:
4251:
4247:
4243:
4239:
4235:
4231:
4230:
4222:
4215:
4211:
4207:
4203:
4199:
4195:
4191:
4187:
4183:
4179:
4174:
4169:
4165:
4161:
4160:
4155:
4148:
4146:
4137:
4133:
4128:
4123:
4118:
4113:
4109:
4105:
4101:
4094:
4086:
4082:
4078:
4074:
4070:
4066:
4062:
4058:
4053:
4048:
4045:(7): 070402.
4044:
4040:
4039:
4031:
4023:
4019:
4014:
4009:
4005:
4001:
3996:
3991:
3988:(7): 070401.
3987:
3983:
3982:
3977:
3970:
3961:
3956:
3951:
3946:
3942:
3938:
3933:
3928:
3924:
3920:
3919:
3914:
3907:
3899:
3895:
3891:
3887:
3883:
3879:
3875:
3871:
3866:
3861:
3857:
3853:
3852:
3844:
3836:
3832:
3828:
3824:
3820:
3816:
3812:
3808:
3804:
3800:
3795:
3790:
3786:
3782:
3775:
3767:
3763:
3759:
3755:
3751:
3747:
3743:
3739:
3734:
3729:
3725:
3721:
3720:
3712:
3704:
3700:
3696:
3692:
3688:
3684:
3680:
3676:
3671:
3666:
3663:(1): 010402.
3662:
3658:
3657:
3649:
3641:
3637:
3633:
3629:
3625:
3621:
3617:
3613:
3608:
3603:
3600:(5): 050403.
3599:
3595:
3594:
3589:
3588:Huelga, S. F.
3582:
3574:
3570:
3566:
3562:
3558:
3554:
3550:
3546:
3541:
3536:
3533:(5): 050402.
3532:
3528:
3527:
3519:
3511:
3507:
3503:
3499:
3495:
3491:
3486:
3481:
3477:
3473:
3472:
3464:
3462:
3447:
3441:
3437:
3433:
3429:
3428:
3423:
3418:Reprinted as
3415:
3411:
3410:
3402:
3400:
3391:
3387:
3383:
3379:
3375:
3371:
3366:
3361:
3357:
3354:
3353:
3345:
3343:
3338:
3330:
3328:
3324:
3319:
3314:
3311:
3307:
3303:
3298:
3296:
3292:
3271:
3268:
3258:
3253:
3250:
3238:
3234:
3230:
3227:
3224:
3218:
3215:
3211:
3202:
3201:
3200:
3197:
3191:
3181:
3165:
3161:
3157:
3148:
3126:
3121:
3117:
3113:
3089:
3086:
3081:
3077:
3067:
3035:
3027:
3024:
3018:
3013:
3009:
3005:
2994:
2986:
2976:
2975:
2974:
2972:
2968:
2965:(This is the
2952:
2948:
2945:
2942:
2915:
2889:
2880:
2874:
2865:
2861:
2856:
2847:
2837:
2836:
2835:
2821:
2817:
2814:
2811:
2786:
2773:
2768:
2760:
2753:
2746:
2743:
2739:
2731:
2724:
2717:
2712:
2708:
2700:
2699:
2698:
2684:
2681:
2678:
2675:
2671:
2667:
2661:
2652:
2648:
2644:
2640:
2634:
2629:
2625:
2604:
2601:
2598:
2589:
2568:
2565:
2558:
2540:
2536:
2527:
2510:
2505:
2501:
2497:
2475:
2470:
2466:
2462:
2454:. The states
2441:
2434:
2427:
2424:
2421:
2416:
2412:
2391:
2388:
2383:
2379:
2354:
2349:
2344:
2340:
2336:
2330:
2327:
2323:
2319:
2315:
2310:
2306:
2302:
2297:
2290:
2286:
2281:
2277:
2272:
2268:
2264:
2256:
2255:
2254:
2253:
2249:
2245:
2241:
2231:
2215:
2211:
2200:
2178:
2174:
2170:
2165:
2161:
2152:
2148:
2141:
2138:
2135:
2122:
2118:
2112:
2108:
2089:
2084:
2080:
2072:
2071:
2070:
2068:
2058:
2044:
2039:
2035:
2029:
2017:
2013:
2002:
1998:
1994:
1988:
1980:
1972:
1966:
1957:
1953:
1947:
1921:
1895:
1886:
1880:
1876:
1871:
1866:
1862:
1854:
1853:
1852:
1844:
1830:
1826:
1822:
1819:
1816:
1813:
1810:
1807:
1802:
1798:
1767:
1763:
1752:
1748:
1736:
1733:
1730:
1722:
1718:
1714:
1709:
1705:
1696:
1692:
1671:
1661:
1657:
1645:
1641:
1632:
1627:
1623:
1597:
1594:
1585:
1576:
1570:
1565:
1562:
1555:
1554:
1553:
1539:
1516:
1513:
1507:
1499:
1495:
1489:
1484:
1480:
1474:
1471:
1466:
1461:
1452:
1442:
1441:
1440:
1419:
1416:
1412:
1409:
1401:
1397:
1389:
1380:
1374:
1369:
1366:
1359:
1358:
1357:
1342:
1339:
1318:
1298:
1278:
1275:
1252:
1244:
1240:
1231:
1227:
1209:
1205:
1175:
1171:
1167:
1162:
1158:
1149:
1145:
1138:
1135:
1132:
1126:
1118:
1114:
1108:
1103:
1099:
1091:
1090:
1089:
1070:
1055:
1051:
1045:
1041:
1024:
1011:
1006:
1002:
996:
992:
975:
969:
961:
956:
952:
944:
943:
942:
928:
923:
919:
911:
893:
889:
880:
876:
866:
864:
863:quantum angle
860:
856:
852:
836:
828:
806:
796:
792:
788:
783:
779:
770:
758:
754:
743:
731:
727:
716:
713:
710:
706:
700:
697:
692:
687:
678:
674:
666:
658:
654:
647:
643:
639:
629:
625:
616:
612:
608:
603:
595:
592:
582:
581:
580:
559:
556:
552:
548:
543:
537:
534:
530:
522:
516:
511:
507:
499:
498:
497:
481:
477:
462:
446:
442:
438:
435:
412:
409:
401:
397:
386:
382:
371:
370:
369:
352:
344:
340:
325:
319:
315:
311:
308:
304:
298:
294:
288:
284:
280:
272:
268:
255:
254:
253:
251:
232:
204:
177:
173:
149:
141:
137:
126:
122:
116:
112:
108:
102:
90:
89:
88:
86:
82:
72:
69:
64:
61:
57:
53:
49:
45:
41:
37:
33:
19:
4555:
4549:
4506:
4500:
4477:
4468:
4462:
4419:
4415:
4405:
4362:
4356:
4342:
4283:
4279:
4233:
4227:
4221:
4163:
4157:
4107:
4103:
4093:
4042:
4036:
4030:
3985:
3979:
3969:
3922:
3916:
3906:
3855:
3849:
3843:
3784:
3780:
3774:
3723:
3717:
3711:
3660:
3654:
3648:
3597:
3591:
3581:
3530:
3524:
3518:
3475:
3469:
3449:. Retrieved
3426:
3413:
3407:
3355:
3350:
3322:
3315:
3299:
3290:
3288:
3193:
3058:
2906:
2802:
2557:phase factor
2370:
2237:
2201:
2198:
2064:
1910:
1850:
1614:
1531:
1438:
1196:
1087:
875:Bures metric
872:
824:
578:
468:
427:
367:
164:
85:mixed states
78:
65:
39:
35:
29:
4318:11603/19394
3960:11603/19409
3196:computation
2067:mixed state
81:pure states
56:Lev Levitin
4601:Categories
4565:1701.05550
4516:1701.01175
4429:1802.00995
4372:1510.06474
4052:1710.03498
3995:1710.07335
3932:1704.03357
3733:1503.02591
3451:2024-04-06
3416:: 249–254.
3365:1705.08023
3333:References
2526:degeneracy
83:, and for
4541:118953874
4454:1751-8113
4397:2469-9926
4334:119313370
4326:1751-8113
4293:1104.5104
4198:0031-9007
4173:0905.3417
3819:1476-4687
3670:1302.5069
3607:1209.1737
3540:1209.0362
3269:−
3259:⋅
3251:−
3231:×
3219:π
3216:ℏ
3118:ψ
3093:ℏ
3090:π
3082:⊥
3031:ℏ
3028:π
3019:≤
3014:⊥
3006:≤
2990:ℏ
2987:π
2946:ψ
2875:≤
2862:≤
2815:ψ
2764:ℏ
2761:π
2744:δ
2735:ℏ
2732:π
2718:≥
2713:⊥
2679:δ
2668:π
2665:ℏ
2641:π
2638:ℏ
2630:⊥
2602:δ
2566:φ
2439:Δ
2431:ℏ
2428:π
2425:±
2331:φ
2269:ψ
2179:τ
2175:ρ
2162:ρ
2145:ℏ
2142:≥
2109:ρ
2090:τ
2081:∫
1999:∑
1992:⟩
1989:ψ
1973:ψ
1970:⟨
1951:⟩
1945:⟨
1925:⟩
1919:⟨
1890:⟩
1884:⟨
1872:≥
1867:⊥
1863:τ
1823:π
1814:
1773:⟩
1764:ψ
1749:ψ
1745:⟨
1737:
1719:ρ
1706:ρ
1658:ψ
1654:⟨
1651:⟩
1642:ψ
1624:ρ
1595:π
1580:¯
1577:σ
1571:ℏ
1566:≥
1563:τ
1540:τ
1496:σ
1490:τ
1481:∫
1475:τ
1456:¯
1453:σ
1417:ρ
1410:ρ
1384:¯
1381:σ
1375:ℏ
1370:≥
1367:τ
1340:ρ
1319:ρ
1299:τ
1241:σ
1176:τ
1172:ρ
1159:ρ
1142:ℏ
1139:≥
1115:σ
1109:τ
1100:∫
1042:ρ
1025:−
993:ρ
953:σ
920:ρ
789:−
707:∑
675:ψ
659:ψ
648:−
640:ψ
617:ψ
593:δ
557:δ
535:δ
526:ℏ
523:π
517:≥
512:⊥
482:⊥
447:⊥
407:⟩
398:ψ
383:ψ
379:⟨
350:⟩
330:ℏ
285:∑
278:⟩
269:ψ
236:⟩
233:ψ
208:⟩
205:ψ
178:⊥
147:⟩
113:∑
106:⟩
103:ψ
48:Igor Tamm
4590:55887738
4258:10042340
4214:36320152
4206:19905679
4136:34936463
4077:29542975
4022:29542956
3890:12059399
3827:10984064
3766:14904633
3758:26196802
3703:36711861
3695:23862985
3632:23414008
3573:38373815
3565:23414007
3424:(eds.).
3127:⟩
2949:⟩
2818:⟩
2511:⟩
2476:⟩
2350:⟩
2316:⟩
2278:⟩
1420:′
1343:′
1230:geodesic
827:variance
679:⟩
655:⟨
644:⟩
613:⟨
4570:Bibcode
4521:Bibcode
4434:Bibcode
4377:Bibcode
4298:Bibcode
4238:Bibcode
4178:Bibcode
4127:8694601
4085:4027745
4057:Bibcode
4000:Bibcode
3937:Bibcode
3898:6341263
3870:Bibcode
3799:Bibcode
3738:Bibcode
3675:Bibcode
3640:8362503
3612:Bibcode
3545:Bibcode
3490:Bibcode
3390:3477317
3370:Bibcode
2617:and so
1224:is the
849:is the
825:is the
4588:
4539:
4452:
4395:
4332:
4324:
4256:
4212:
4204:
4196:
4134:
4124:
4083:
4075:
4020:
3896:
3888:
3833:
3825:
3817:
3781:Nature
3764:
3756:
3701:
3693:
3638:
3630:
3571:
3563:
3510:468290
3508:
3442:
3388:
2907:where
1911:where
1811:arccos
1734:arccos
1439:where
1197:where
579:where
4586:S2CID
4560:arXiv
4537:S2CID
4511:arXiv
4424:arXiv
4367:arXiv
4330:S2CID
4288:arXiv
4210:S2CID
4168:arXiv
4081:S2CID
4047:arXiv
3990:arXiv
3927:arXiv
3894:S2CID
3860:arXiv
3835:75923
3831:S2CID
3789:arXiv
3762:S2CID
3728:arXiv
3699:S2CID
3665:arXiv
3636:S2CID
3602:arXiv
3569:S2CID
3535:arXiv
3506:S2CID
3480:arXiv
3386:S2CID
3360:arXiv
3295:joule
3140:with
2248:qubit
4450:ISSN
4393:ISSN
4322:ISSN
4254:PMID
4202:PMID
4194:ISSN
4132:PMID
4073:PMID
4018:PMID
3886:PMID
3823:PMID
3815:ISSN
3754:PMID
3691:PMID
3628:PMID
3561:PMID
3440:ISBN
3316:The
2489:and
2404:and
2242:and
54:and
46:and
34:, a
4578:doi
4556:382
4529:doi
4442:doi
4385:doi
4314:hdl
4306:doi
4246:doi
4186:doi
4164:103
4122:PMC
4112:doi
4065:doi
4043:120
4008:doi
3986:120
3955:hdl
3945:doi
3878:doi
3807:doi
3785:406
3746:doi
3724:114
3683:doi
3661:111
3620:doi
3598:110
3553:doi
3531:110
3498:doi
3476:120
3432:doi
3378:doi
3291:all
3153:max
3072:max
3040:max
2999:max
2920:max
2885:max
2870:avg
2852:max
2778:avg
2721:max
2657:avg
2594:avg
1962:avg
1331:to
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