671:
1184:
332:
886:
1948:
1759:
4353:
are just states that live on the higher dimensional equivalent of the Bloch sphere even for systems that are larger than a qubit. Separable states are the subset of the set of speudoseparable states, while for qubits the two sets coincide with each other. For systems larger than qubits, such quantum
983:
666:{\displaystyle \rho ^{T_{B}}:=(I\otimes T)(\rho )=\sum _{ijkl}p_{kl}^{ij}|i\rangle \langle j|\otimes (|k\rangle \langle l|)^{T}=\sum _{ijkl}p_{kl}^{ij}|i\rangle \langle j|\otimes |l\rangle \langle k|=\sum _{ijkl}p_{lk}^{ij}|i\rangle \langle j|\otimes |k\rangle \langle l|}
2994:
In higher dimensions, however, there exist maps that can't be decomposed in this fashion, and the criterion is no longer sufficient. Consequently, there are entangled states which have a positive partial transpose. Such states have the interesting property that they are
3342:
734:
4119:
2243:
3956:
are not necessarily physical pure density matrices since they can have negative eigenvalues. In this case, even entangled states can be written as a mixture of tensor products of single-party aphysical states, very similar to the form of
1770:
317:
1595:
2989:
1534:
1179:{\displaystyle \rho ^{T_{B}}={\begin{pmatrix}A_{11}^{T}&A_{12}^{T}&\dots &A_{1n}^{T}\\A_{21}^{T}&A_{22}^{T}&&\\\vdots &&\ddots &\\A_{n1}^{T}&&&A_{nn}^{T}\end{pmatrix}}}
2731:
2114:
205:
3767:
2789:
3597:
3668:
3420:
2669:
1429:
3167:
2409:
Showing that being PPT is also sufficient for the 2 X 2 and 3 X 2 (equivalently 2 X 3) cases is more involved. It was shown by the
Horodeckis that for every entangled state there exists an
3105:
For symmetric states of bipartite systems, the positivity of the partial transpose of the density matrix is related to the sign of certain two-body correlations. Here, symmetry means that
4240:
975:
932:
3927:
2922:
2453:
2619:
2579:
2537:
2497:
881:{\displaystyle \rho ={\begin{pmatrix}A_{11}&A_{12}&\dots &A_{1n}\\A_{21}&A_{22}&&\\\vdots &&\ddots &\\A_{n1}&&&A_{nn}\end{pmatrix}}}
3873:
2370:
2291:
719:
3245:
1576:
3506:
3045:-mode Gaussian states (see Ref. for a seemingly different but essentially equivalent approach). It was later found that Simon's condition is also necessary and sufficient for
2325:
2034:
4313:
2887:
2404:
1281:
1247:
2846:
2819:
4351:
4180:
3997:
2125:
3095:
3069:
3043:
1943:{\displaystyle \rho ^{T_{B}}={\frac {1}{4}}{\begin{pmatrix}1-p&0&0&-2p\\0&p+1&0&0\\0&0&p+1&0\\-2p&0&0&1-p\end{pmatrix}}}
1359:
1333:
1303:
1209:
1994:
3368:
3200:
3986:
3954:
3824:
3797:
3549:
3019:
formulated a particular version of the PPT criterion in terms of the second-order moments of canonical operators and showed that it is necessary and sufficient for
157:
46:
3097:-mode Gaussian states. Simon's condition can be generalized by taking into account the higher order moments of canonical operators or by using entropic measures.
212:
3623:
3463:
1583:
4142:
3688:
3526:
3440:
3240:
3220:
1754:{\displaystyle \rho ={\frac {1}{4}}{\begin{pmatrix}1-p&0&0&0\\0&p+1&-2p&0\\0&-2p&p+1&0\\0&0&0&1-p\end{pmatrix}}}
86:
66:
2927:
3991:
The concept of such pseudomixtures has been extended to non-symmetric states and to the multipartite case, by the definition of pseudoseparable states
1449:
4939:
Vitagliano, Giuseppe; Gühne, Otfried; Tóth, Géza (2024). "$ su(d)$ -squeezing and many-body entanglement geometry in finite-dimensional systems".
2674:
2456:
4790:
Walborn, S.; Taketani, B.; Salles, A.; Toscano, F.; de Matos Filho, R. (2009). "Entropic
Entanglement Criteria for Continuous Variables".
2050:
2855:
Loosely speaking, the transposition map is therefore the only one that can generate negative eigenvalues in these dimensions. So if
162:
3703:
2739:
3557:
127:
In higher dimensions, the test is inconclusive, and one should supplement it with more advanced tests, such as those based on
3628:
3373:
2624:
105:
5011:
4546:
Duan, Lu-Ming; Giedke, G.; Cirac, J. I.; Zoller, P. (2000). "Inseparability
Criterion for Continuous Variable Systems".
1367:
3111:
2924:
is positive for any Λ. Thus we conclude that the Peres–Horodecki criterion is also sufficient for separability when
3988:
are physical density matrices, which is consistent with the fact that for two qubits all PPT states are separable.
100:. In the 2×2 and 2×3 dimensional cases the condition is also sufficient. It is used to decide the separability of
4185:
937:
894:
3878:
2892:
2423:
3337:{\displaystyle (\vert n\rangle _{A}\vert m\rangle _{B}+\vert m\rangle _{A}\vert n\rangle _{B})/{\sqrt {2}}}
2584:
2544:
2502:
2462:
3829:
2330:
2251:
683:
1546:
3467:
2296:
3000:
1579:
4114:{\displaystyle \varrho =\sum _{k}p_{k}M_{k}^{(1)}\otimes M_{k}^{(2)}\otimes ...\otimes M_{k}^{(N)},}
2238:{\displaystyle \rho ^{T_{B}}=(I\otimes T)(\rho )=\sum p_{i}\rho _{i}^{A}\otimes (\rho _{i}^{B})^{T}}
1999:
4668:
Shchukin, E.; Vogel, W. (2005). "Inseparability
Criteria for Continuous Bipartite Quantum States".
4245:
2858:
2375:
1252:
1218:
2824:
2797:
4851:
Yichen Huang (October 2013). "Entanglement
Detection: Complexity and Shannon Entropic Criteria".
4437:
4318:
4147:
2414:
1309:. The converse of these statements is true if and only if the dimension of the product space is
3074:
3048:
3022:
1338:
1312:
1286:
1192:
1956:
3347:
3004:
312:{\displaystyle \rho =\sum _{ijkl}p_{kl}^{ij}|i\rangle \langle j|\otimes |k\rangle \langle l|}
101:
4485:
Simon, R. (2000). "Peres-Horodecki
Separability Criterion for Continuous Variable Systems".
3175:
4982:
4809:
4748:
4687:
4626:
4565:
4504:
3964:
3932:
3802:
3775:
3691:
3534:
2410:
1306:
142:
128:
31:
8:
4729:
Hillery, Mark; Zubairy, M. Suhail (2006). "Entanglement
Conditions for Two-Mode States".
2621:
can be decomposed into a sum of completely positive and completely copositive maps, when
4986:
4960:
4813:
4752:
4691:
4630:
4569:
4508:
3605:
3445:
113:
4921:
4895:
4868:
4833:
4799:
4772:
4738:
4711:
4677:
4650:
4616:
4589:
4555:
4528:
4494:
4467:
4449:
4418:
4384:
4372:
4127:
3673:
3511:
3425:
3225:
3205:
2996:
1540:
71:
51:
4463:
117:
4994:
4913:
4886:
Tóth, Géza; Gühne, Otfried (May 1, 2009). "Entanglement and
Permutational Symmetry".
4825:
4764:
4703:
4642:
4581:
4520:
4410:
4402:
4354:
states can be entangled, and in this case they can have PPT or non-PPT bipartitions.
121:
4925:
4837:
4776:
4715:
4654:
4532:
4471:
2406:
must also be positive semidefinite. This proves the necessity of the PPT criterion.
4990:
4944:
4909:
4905:
4872:
4860:
4821:
4817:
4756:
4695:
4634:
4593:
4573:
4512:
4459:
4422:
4394:
3958:
2984:{\displaystyle {\textrm {dim}}({\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B})\leq 6}
89:
4760:
4699:
2459:Λ is a necessary and sufficient condition for the separability of ρ, where Λ maps
728:
This definition can be seen more clearly if we write the state as a block matrix:
4964:
4638:
4577:
4516:
4398:
3015:
The Peres–Horodecki criterion has been extended to continuous variable systems.
1529:{\displaystyle \rho =p|\Psi ^{-}\rangle \langle \Psi ^{-}|+(1-p){\frac {I}{4}}}
680:
in the name implies that only part of the state is transposed. More precisely,
26:
4973:
Woronowicz, S. L. (1976). "Positive maps of low dimensional matrix algebras".
4948:
5005:
4864:
4406:
2852:
is the transposition map. This follows from the Størmer-Woronowicz theorem.
4917:
4829:
4768:
4707:
4646:
4585:
4524:
4414:
3016:
1440:
4743:
4682:
4621:
4560:
4499:
4454:
4389:
1212:
725:
applied to the A party and the transposition map applied to the B party.
109:
2726:{\displaystyle {\textrm {dim}}({\mathcal {H}}_{A})=2\;{\textrm {or}}\;3}
722:
4607:
Werner, R. F.; Wolf, M. M. (2001). "Bound
Entangled Gaussian States".
323:
1364:
The result is independent of the party that was transposed, because
4438:"Separability of mixed states: necessary and sufficient conditions"
4900:
4804:
2119:
In this case, the effect of the partial transposition is trivial:
2109:{\displaystyle \rho =\sum p_{i}\rho _{i}^{A}\otimes \rho _{i}^{B}}
4436:
Horodecki, Michał; Horodecki, Paweł; Horodecki, Ryszard (1996).
2420:
From the existence of entanglement witnesses, one can show that
2248:
As the transposition map preserves eigenvalues, the spectrum of
4789:
200:{\displaystyle {\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}}
4435:
3762:{\displaystyle \varrho =\sum _{k}p_{k}M_{k}\otimes M_{k},}
2784:{\displaystyle \Lambda =\Lambda _{1}+\Lambda _{2}\circ T,}
3242:. A full basis of the symmetric subspace is of the form
3592:{\displaystyle \langle M\otimes M\rangle _{\rho }\geq 0}
3202:
is the flip or swap operator exchanging the two parties
3663:{\displaystyle \langle M\otimes M\rangle _{\rho }<0}
3415:{\displaystyle \vert n\rangle _{A}\vert n\rangle _{B}.}
2413:. This is a result of geometric nature and invokes the
1809:
1620:
1012:
749:
108:
does not apply. The theorem was discovered in 1996 by
4545:
4321:
4248:
4188:
4150:
4130:
4000:
3967:
3935:
3881:
3832:
3805:
3778:
3706:
3697:
Moreover, bipartite symmetric PPT can be written as
3676:
3631:
3608:
3560:
3537:
3514:
3470:
3448:
3428:
3376:
3350:
3248:
3228:
3208:
3178:
3114:
3077:
3051:
3025:
2930:
2895:
2861:
2827:
2800:
2742:
2733:. In other words, every such map Λ can be written as
2677:
2664:{\displaystyle {\textrm {dim}}({\mathcal {H}}_{B})=2}
2627:
2587:
2547:
2505:
2465:
2426:
2378:
2333:
2299:
2254:
2128:
2053:
2002:
1959:
1773:
1598:
1549:
1452:
1370:
1341:
1315:
1289:
1255:
1221:
1195:
986:
940:
897:
737:
686:
335:
215:
165:
145:
74:
54:
34:
3071:-mode Gaussian states, but no longer sufficient for
4938:
4345:
4307:
4234:
4174:
4136:
4113:
3980:
3948:
3921:
3867:
3818:
3791:
3761:
3682:
3662:
3617:
3591:
3543:
3520:
3500:
3457:
3434:
3414:
3362:
3336:
3234:
3214:
3194:
3161:
3089:
3063:
3037:
2983:
2916:
2881:
2840:
2813:
2783:
2725:
2663:
2613:
2573:
2531:
2491:
2447:
2398:
2364:
2319:
2285:
2237:
2108:
2028:
1988:
1942:
1753:
1570:
1528:
1423:
1353:
1327:
1297:
1275:
1241:
1203:
1178:
969:
926:
880:
713:
665:
311:
199:
151:
80:
60:
40:
3551:has a positive partial transpose if and only if
2316:
1424:{\displaystyle \rho ^{T_{A}}=(\rho ^{T_{B}})^{T}}
1294:
1200:
934:, and each block is a square matrix of dimension
5003:
4728:
3929:However, for a subsystem larger than a qubit,
3162:{\displaystyle \rho F_{AB}=F_{AB}\rho =\rho ,}
3010:
4667:
4373:"Separability Criterion for Density Matrices"
4850:
3645:
3632:
3574:
3561:
3400:
3393:
3384:
3377:
3310:
3303:
3294:
3287:
3275:
3268:
3259:
3252:
1565:
1480:
1477:
652:
649:
627:
624:
562:
559:
537:
534:
462:
459:
434:
431:
326:(with respect to the B party) is defined as
298:
295:
273:
270:
4972:
4606:
2719:
2711:
2372:must still be positive semidefinite. Thus
2315:
1293:
1199:
4899:
4885:
4803:
4742:
4681:
4620:
4559:
4498:
4453:
4388:
4235:{\displaystyle {\rm {Tr}}(M_{k}^{(n)})=1}
970:{\displaystyle m=\dim {\mathcal {H}}_{B}}
927:{\displaystyle n=\dim {\mathcal {H}}_{A}}
3922:{\displaystyle {\rm {Tr}}(M_{k}^{2})=1.}
2917:{\displaystyle I\otimes \Lambda (\rho )}
2448:{\displaystyle I\otimes \Lambda (\rho )}
2044:If ρ is separable, it can be written as
1996:. Therefore, the state is entangled for
25:is a necessary condition, for the joint
4853:IEEE Transactions on Information Theory
16:Criterion in quantum information theory
5004:
4315:The single subsystem aphysical states
3531:It can be shown that for such states,
4484:
4370:
3528:is the dimension of the two parties.
2614:{\displaystyle B({\mathcal {H}}_{A})}
2574:{\displaystyle B({\mathcal {H}}_{B})}
2541:Furthermore, every positive map from
2532:{\displaystyle B({\mathcal {H}}_{A})}
2492:{\displaystyle B({\mathcal {H}}_{B})}
1249:are non-negative. In other words, if
3100:
3868:{\displaystyle {\rm {Tr}}(M_{k})=1}
2365:{\displaystyle (\rho _{i}^{B})^{T}}
2286:{\displaystyle (\rho _{i}^{B})^{T}}
714:{\displaystyle (I\otimes T)(\rho )}
13:
4969:, Cambridge University Press, 2006
4254:
4251:
4194:
4191:
3887:
3884:
3838:
3835:
2961:
2944:
2902:
2829:
2802:
2763:
2750:
2743:
2691:
2641:
2597:
2557:
2515:
2475:
2433:
1571:{\displaystyle |\Psi ^{-}\rangle }
1556:
1484:
1468:
956:
913:
186:
169:
48:of two quantum mechanical systems
14:
5023:
3690:then the state possesses non-PPT
3501:{\displaystyle 0\leq n,m\leq d-1}
2320:{\displaystyle \rho _{i}^{B}\;\!}
4371:Peres, Asher (August 19, 1996).
4144:is the number of subsystems and
2039:
1439:Consider this 2-qubit family of
977:. Then the partial transpose is
4932:
4879:
2293:is the same as the spectrum of
1582:, and the identity element, a
159:which acts on Hilbert space of
4910:10.1103/PhysRevLett.102.170503
4844:
4822:10.1103/PhysRevLett.103.160505
4783:
4722:
4661:
4600:
4539:
4478:
4429:
4364:
4338:
4332:
4296:
4287:
4281:
4275:
4262:
4259:
4223:
4218:
4212:
4199:
4167:
4161:
4103:
4097:
4067:
4061:
4043:
4037:
3910:
3892:
3856:
3843:
3319:
3249:
2972:
2938:
2911:
2905:
2702:
2685:
2652:
2635:
2608:
2591:
2568:
2551:
2526:
2509:
2486:
2469:
2442:
2436:
2353:
2334:
2274:
2255:
2226:
2207:
2170:
2164:
2161:
2149:
2029:{\displaystyle 1\geq p>1/3}
1975:
1960:
1551:
1513:
1501:
1494:
1463:
1412:
1391:
708:
702:
699:
687:
659:
642:
634:
617:
569:
552:
544:
527:
474:
469:
452:
448:
441:
424:
377:
371:
368:
356:
305:
288:
280:
263:
1:
4761:10.1103/PhysRevLett.96.050503
4700:10.1103/PhysRevLett.95.230502
4464:10.1016/S0375-9601(96)00706-2
4357:
4308:{\displaystyle {\rm {Tr}}=1.}
2882:{\displaystyle \rho ^{T_{B}}}
2399:{\displaystyle \rho ^{T_{B}}}
1276:{\displaystyle \rho ^{T_{B}}}
1242:{\displaystyle \rho ^{T_{B}}}
1189:The criterion states that if
134:
4995:10.1016/0034-4877(76)90038-0
2848:are completely positive and
2841:{\displaystyle \Lambda _{2}}
2814:{\displaystyle \Lambda _{1}}
7:
4639:10.1103/PhysRevLett.86.3658
4578:10.1103/PhysRevLett.84.2722
4517:10.1103/PhysRevLett.84.2726
4399:10.1103/PhysRevLett.77.1413
4346:{\displaystyle M_{k}^{(n)}}
4175:{\displaystyle M_{k}^{(n)}}
3011:Continuous variable systems
1283:has a negative eigenvalue,
139:If we have a general state
10:
5028:
5012:Quantum information theory
4966:Geometry of Quantum States
1764:and the partial transpose
1539:It can be regarded as the
1434:
1211:is separable then all the
112:and the Horodecki family (
98:positive partial transpose
4949:10.48550/arXiv.2406.13338
3090:{\displaystyle 2\oplus 2}
3064:{\displaystyle 1\oplus n}
3038:{\displaystyle 1\oplus 1}
1580:maximally entangled state
1354:{\displaystyle 2\times 3}
1328:{\displaystyle 2\times 2}
1298:{\displaystyle \rho \;\!}
1204:{\displaystyle \rho \;\!}
23:Peres–Horodecki criterion
4865:10.1109/TIT.2013.2257936
3602:holds for all operators
1989:{\displaystyle (1-3p)/4}
1953:Its least eigenvalue is
92:. It is also called the
4963:and Ingemar Bengtsson,
4888:Physical Review Letters
4792:Physical Review Letters
4731:Physical Review Letters
4670:Physical Review Letters
4609:Physical Review Letters
4548:Physical Review Letters
4487:Physical Review Letters
4377:Physical Review Letters
3363:{\displaystyle m\neq n}
2999:, i.e. they can not be
2455:being positive for all
2417:(see reference below).
4347:
4309:
4236:
4176:
4138:
4115:
3982:
3950:
3923:
3869:
3820:
3799:are probabilities and
3793:
3763:
3684:
3664:
3619:
3593:
3545:
3522:
3502:
3459:
3436:
3416:
3364:
3338:
3236:
3216:
3196:
3195:{\displaystyle F_{AB}}
3163:
3091:
3065:
3039:
2985:
2918:
2883:
2842:
2815:
2785:
2727:
2665:
2615:
2575:
2533:
2493:
2449:
2400:
2366:
2321:
2287:
2239:
2110:
2030:
1990:
1944:
1755:
1589:Its density matrix is
1572:
1530:
1425:
1355:
1329:
1299:
1277:
1243:
1205:
1180:
971:
928:
882:
715:
667:
313:
201:
153:
129:entanglement witnesses
82:
62:
42:
4348:
4310:
4237:
4177:
4139:
4116:
3983:
3981:{\displaystyle M_{k}}
3961:. In the qubit case,
3951:
3949:{\displaystyle M_{k}}
3924:
3870:
3821:
3819:{\displaystyle M_{k}}
3794:
3792:{\displaystyle p_{k}}
3764:
3685:
3665:
3620:
3594:
3546:
3544:{\displaystyle \rho }
3523:
3503:
3460:
3437:
3417:
3365:
3339:
3237:
3217:
3197:
3164:
3092:
3066:
3040:
3005:quantum communication
2986:
2919:
2884:
2843:
2816:
2786:
2728:
2666:
2616:
2576:
2534:
2494:
2450:
2401:
2367:
2322:
2288:
2240:
2111:
2031:
1991:
1945:
1756:
1584:maximally mixed state
1573:
1531:
1426:
1356:
1330:
1300:
1278:
1244:
1206:
1181:
972:
929:
883:
716:
668:
314:
202:
154:
152:{\displaystyle \rho }
106:Schmidt decomposition
83:
63:
43:
41:{\displaystyle \rho }
4319:
4246:
4186:
4148:
4128:
3998:
3965:
3933:
3879:
3830:
3803:
3776:
3704:
3674:
3629:
3606:
3558:
3535:
3512:
3468:
3446:
3426:
3374:
3348:
3246:
3226:
3206:
3176:
3112:
3075:
3049:
3023:
2928:
2893:
2859:
2825:
2798:
2740:
2675:
2625:
2585:
2545:
2503:
2463:
2424:
2411:entanglement witness
2376:
2331:
2327:, and in particular
2297:
2252:
2126:
2051:
2000:
1957:
1771:
1596:
1547:
1450:
1368:
1339:
1313:
1305:is guaranteed to be
1287:
1253:
1219:
1193:
984:
938:
895:
735:
684:
333:
213:
163:
143:
72:
52:
32:
4987:1976RpMP...10..165W
4814:2009PhRvL.103p0505W
4753:2006PhRvL..96e0503H
4692:2005PhRvL..95w0502S
4631:2001PhRvL..86.3658W
4570:2000PhRvL..84.2722D
4509:2000PhRvL..84.2726S
4342:
4285:
4222:
4171:
4107:
4071:
4047:
3909:
2415:Hahn–Banach theorem
2351:
2314:
2272:
2224:
2203:
2105:
2087:
1167:
1145:
1107:
1090:
1071:
1046:
1029:
615:
525:
422:
261:
4343:
4322:
4305:
4265:
4232:
4202:
4172:
4151:
4134:
4111:
4087:
4051:
4027:
4016:
3978:
3946:
3919:
3895:
3865:
3816:
3789:
3759:
3722:
3680:
3660:
3618:{\displaystyle M.}
3615:
3589:
3541:
3518:
3498:
3458:{\displaystyle m,}
3455:
3432:
3412:
3360:
3334:
3232:
3212:
3192:
3159:
3087:
3061:
3035:
2981:
2914:
2879:
2838:
2811:
2781:
2723:
2661:
2611:
2571:
2529:
2489:
2445:
2396:
2362:
2337:
2317:
2300:
2283:
2258:
2235:
2210:
2189:
2106:
2091:
2073:
2026:
1986:
1940:
1934:
1751:
1745:
1568:
1541:convex combination
1526:
1421:
1351:
1325:
1295:
1273:
1239:
1201:
1176:
1170:
1150:
1128:
1093:
1076:
1054:
1032:
1015:
967:
924:
878:
872:
711:
663:
595:
594:
505:
504:
402:
401:
309:
241:
240:
197:
149:
78:
58:
38:
4859:(10): 6774–6778.
4615:(16): 3658–3661.
4554:(12): 2722–2725.
4493:(12): 2726–2729.
4442:Physics Letters A
4137:{\displaystyle N}
4007:
3713:
3683:{\displaystyle M}
3521:{\displaystyle d}
3508:must hold, where
3435:{\displaystyle n}
3332:
3235:{\displaystyle B}
3215:{\displaystyle A}
3101:Symmetric systems
2935:
2716:
2682:
2632:
1802:
1613:
1524:
576:
486:
383:
222:
81:{\displaystyle B}
61:{\displaystyle A}
5019:
4998:
4961:Karol Życzkowski
4953:
4952:
4936:
4930:
4929:
4903:
4883:
4877:
4876:
4848:
4842:
4841:
4807:
4787:
4781:
4780:
4746:
4744:quant-ph/0507168
4726:
4720:
4719:
4685:
4683:quant-ph/0508132
4665:
4659:
4658:
4624:
4622:quant-ph/0009118
4604:
4598:
4597:
4563:
4561:quant-ph/9908056
4543:
4537:
4536:
4502:
4500:quant-ph/9909044
4482:
4476:
4475:
4457:
4455:quant-ph/9605038
4433:
4427:
4426:
4392:
4390:quant-ph/9604005
4383:(8): 1413–1415.
4368:
4352:
4350:
4349:
4344:
4341:
4330:
4314:
4312:
4311:
4306:
4295:
4294:
4284:
4273:
4258:
4257:
4241:
4239:
4238:
4233:
4221:
4210:
4198:
4197:
4181:
4179:
4178:
4173:
4170:
4159:
4143:
4141:
4140:
4135:
4120:
4118:
4117:
4112:
4106:
4095:
4070:
4059:
4046:
4035:
4026:
4025:
4015:
3987:
3985:
3984:
3979:
3977:
3976:
3959:separable states
3955:
3953:
3952:
3947:
3945:
3944:
3928:
3926:
3925:
3920:
3908:
3903:
3891:
3890:
3874:
3872:
3871:
3866:
3855:
3854:
3842:
3841:
3825:
3823:
3822:
3817:
3815:
3814:
3798:
3796:
3795:
3790:
3788:
3787:
3768:
3766:
3765:
3760:
3755:
3754:
3742:
3741:
3732:
3731:
3721:
3689:
3687:
3686:
3681:
3669:
3667:
3666:
3661:
3653:
3652:
3624:
3622:
3621:
3616:
3598:
3596:
3595:
3590:
3582:
3581:
3550:
3548:
3547:
3542:
3527:
3525:
3524:
3519:
3507:
3505:
3504:
3499:
3464:
3462:
3461:
3456:
3441:
3439:
3438:
3433:
3421:
3419:
3418:
3413:
3408:
3407:
3392:
3391:
3369:
3367:
3366:
3361:
3343:
3341:
3340:
3335:
3333:
3328:
3326:
3318:
3317:
3302:
3301:
3283:
3282:
3267:
3266:
3241:
3239:
3238:
3233:
3221:
3219:
3218:
3213:
3201:
3199:
3198:
3193:
3191:
3190:
3168:
3166:
3165:
3160:
3146:
3145:
3130:
3129:
3096:
3094:
3093:
3088:
3070:
3068:
3067:
3062:
3044:
3042:
3041:
3036:
2990:
2988:
2987:
2982:
2971:
2970:
2965:
2964:
2954:
2953:
2948:
2947:
2937:
2936:
2933:
2923:
2921:
2920:
2915:
2888:
2886:
2885:
2880:
2878:
2877:
2876:
2875:
2847:
2845:
2844:
2839:
2837:
2836:
2820:
2818:
2817:
2812:
2810:
2809:
2790:
2788:
2787:
2782:
2771:
2770:
2758:
2757:
2732:
2730:
2729:
2724:
2718:
2717:
2714:
2701:
2700:
2695:
2694:
2684:
2683:
2680:
2670:
2668:
2667:
2662:
2651:
2650:
2645:
2644:
2634:
2633:
2630:
2620:
2618:
2617:
2612:
2607:
2606:
2601:
2600:
2580:
2578:
2577:
2572:
2567:
2566:
2561:
2560:
2538:
2536:
2535:
2530:
2525:
2524:
2519:
2518:
2498:
2496:
2495:
2490:
2485:
2484:
2479:
2478:
2454:
2452:
2451:
2446:
2405:
2403:
2402:
2397:
2395:
2394:
2393:
2392:
2371:
2369:
2368:
2363:
2361:
2360:
2350:
2345:
2326:
2324:
2323:
2318:
2313:
2308:
2292:
2290:
2289:
2284:
2282:
2281:
2271:
2266:
2244:
2242:
2241:
2236:
2234:
2233:
2223:
2218:
2202:
2197:
2188:
2187:
2145:
2144:
2143:
2142:
2115:
2113:
2112:
2107:
2104:
2099:
2086:
2081:
2072:
2071:
2035:
2033:
2032:
2027:
2022:
1995:
1993:
1992:
1987:
1982:
1949:
1947:
1946:
1941:
1939:
1938:
1803:
1795:
1790:
1789:
1788:
1787:
1760:
1758:
1757:
1752:
1750:
1749:
1614:
1606:
1577:
1575:
1574:
1569:
1564:
1563:
1554:
1535:
1533:
1532:
1527:
1525:
1517:
1497:
1492:
1491:
1476:
1475:
1466:
1430:
1428:
1427:
1422:
1420:
1419:
1410:
1409:
1408:
1407:
1387:
1386:
1385:
1384:
1360:
1358:
1357:
1352:
1334:
1332:
1331:
1326:
1304:
1302:
1301:
1296:
1282:
1280:
1279:
1274:
1272:
1271:
1270:
1269:
1248:
1246:
1245:
1240:
1238:
1237:
1236:
1235:
1210:
1208:
1207:
1202:
1185:
1183:
1182:
1177:
1175:
1174:
1166:
1161:
1148:
1147:
1144:
1139:
1124:
1118:
1110:
1109:
1106:
1101:
1089:
1084:
1070:
1065:
1045:
1040:
1028:
1023:
1003:
1002:
1001:
1000:
976:
974:
973:
968:
966:
965:
960:
959:
933:
931:
930:
925:
923:
922:
917:
916:
887:
885:
884:
879:
877:
876:
869:
868:
855:
854:
852:
851:
836:
830:
822:
821:
819:
818:
807:
806:
793:
792:
773:
772:
761:
760:
721:is the identity
720:
718:
717:
712:
672:
670:
669:
664:
662:
645:
637:
620:
614:
606:
593:
572:
555:
547:
530:
524:
516:
503:
482:
481:
472:
455:
444:
427:
421:
413:
400:
352:
351:
350:
349:
318:
316:
315:
310:
308:
291:
283:
266:
260:
252:
239:
206:
204:
203:
198:
196:
195:
190:
189:
179:
178:
173:
172:
158:
156:
155:
150:
87:
85:
84:
79:
67:
65:
64:
59:
47:
45:
44:
39:
5027:
5026:
5022:
5021:
5020:
5018:
5017:
5016:
5002:
5001:
4975:Rep. Math. Phys
4957:
4956:
4937:
4933:
4884:
4880:
4849:
4845:
4788:
4784:
4727:
4723:
4666:
4662:
4605:
4601:
4544:
4540:
4483:
4479:
4434:
4430:
4369:
4365:
4360:
4331:
4326:
4320:
4317:
4316:
4290:
4286:
4274:
4269:
4250:
4249:
4247:
4244:
4243:
4211:
4206:
4190:
4189:
4187:
4184:
4183:
4160:
4155:
4149:
4146:
4145:
4129:
4126:
4125:
4096:
4091:
4060:
4055:
4036:
4031:
4021:
4017:
4011:
3999:
3996:
3995:
3972:
3968:
3966:
3963:
3962:
3940:
3936:
3934:
3931:
3930:
3904:
3899:
3883:
3882:
3880:
3877:
3876:
3850:
3846:
3834:
3833:
3831:
3828:
3827:
3810:
3806:
3804:
3801:
3800:
3783:
3779:
3777:
3774:
3773:
3750:
3746:
3737:
3733:
3727:
3723:
3717:
3705:
3702:
3701:
3675:
3672:
3671:
3670:holds for some
3648:
3644:
3630:
3627:
3626:
3607:
3604:
3603:
3577:
3573:
3559:
3556:
3555:
3536:
3533:
3532:
3513:
3510:
3509:
3469:
3466:
3465:
3447:
3444:
3443:
3427:
3424:
3423:
3403:
3399:
3387:
3383:
3375:
3372:
3371:
3349:
3346:
3345:
3327:
3322:
3313:
3309:
3297:
3293:
3278:
3274:
3262:
3258:
3247:
3244:
3243:
3227:
3224:
3223:
3207:
3204:
3203:
3183:
3179:
3177:
3174:
3173:
3138:
3134:
3122:
3118:
3113:
3110:
3109:
3103:
3076:
3073:
3072:
3050:
3047:
3046:
3024:
3021:
3020:
3013:
2997:bound entangled
2966:
2960:
2959:
2958:
2949:
2943:
2942:
2941:
2932:
2931:
2929:
2926:
2925:
2894:
2891:
2890:
2871:
2867:
2866:
2862:
2860:
2857:
2856:
2832:
2828:
2826:
2823:
2822:
2805:
2801:
2799:
2796:
2795:
2766:
2762:
2753:
2749:
2741:
2738:
2737:
2713:
2712:
2696:
2690:
2689:
2688:
2679:
2678:
2676:
2673:
2672:
2646:
2640:
2639:
2638:
2629:
2628:
2626:
2623:
2622:
2602:
2596:
2595:
2594:
2586:
2583:
2582:
2562:
2556:
2555:
2554:
2546:
2543:
2542:
2520:
2514:
2513:
2512:
2504:
2501:
2500:
2480:
2474:
2473:
2472:
2464:
2461:
2460:
2425:
2422:
2421:
2388:
2384:
2383:
2379:
2377:
2374:
2373:
2356:
2352:
2346:
2341:
2332:
2329:
2328:
2309:
2304:
2298:
2295:
2294:
2277:
2273:
2267:
2262:
2253:
2250:
2249:
2229:
2225:
2219:
2214:
2198:
2193:
2183:
2179:
2138:
2134:
2133:
2129:
2127:
2124:
2123:
2100:
2095:
2082:
2077:
2067:
2063:
2052:
2049:
2048:
2042:
2018:
2001:
1998:
1997:
1978:
1958:
1955:
1954:
1933:
1932:
1921:
1916:
1911:
1899:
1898:
1893:
1882:
1877:
1871:
1870:
1865:
1860:
1849:
1843:
1842:
1831:
1826:
1821:
1805:
1804:
1794:
1783:
1779:
1778:
1774:
1772:
1769:
1768:
1744:
1743:
1732:
1727:
1722:
1716:
1715:
1710:
1699:
1688:
1682:
1681:
1676:
1665:
1654:
1648:
1647:
1642:
1637:
1632:
1616:
1615:
1605:
1597:
1594:
1593:
1559:
1555:
1550:
1548:
1545:
1544:
1516:
1493:
1487:
1483:
1471:
1467:
1462:
1451:
1448:
1447:
1437:
1415:
1411:
1403:
1399:
1398:
1394:
1380:
1376:
1375:
1371:
1369:
1366:
1365:
1340:
1337:
1336:
1314:
1311:
1310:
1288:
1285:
1284:
1265:
1261:
1260:
1256:
1254:
1251:
1250:
1231:
1227:
1226:
1222:
1220:
1217:
1216:
1194:
1191:
1190:
1169:
1168:
1162:
1154:
1146:
1140:
1132:
1125:
1123:
1117:
1111:
1108:
1102:
1097:
1091:
1085:
1080:
1073:
1072:
1066:
1058:
1052:
1047:
1041:
1036:
1030:
1024:
1019:
1008:
1007:
996:
992:
991:
987:
985:
982:
981:
961:
955:
954:
953:
939:
936:
935:
918:
912:
911:
910:
896:
893:
892:
871:
870:
861:
857:
853:
844:
840:
837:
835:
829:
823:
820:
814:
810:
808:
802:
798:
795:
794:
785:
781:
779:
774:
768:
764:
762:
756:
752:
745:
744:
736:
733:
732:
685:
682:
681:
658:
641:
633:
616:
607:
599:
580:
568:
551:
543:
526:
517:
509:
490:
477:
473:
468:
451:
440:
423:
414:
406:
387:
345:
341:
340:
336:
334:
331:
330:
304:
287:
279:
262:
253:
245:
226:
214:
211:
210:
191:
185:
184:
183:
174:
168:
167:
166:
164:
161:
160:
144:
141:
140:
137:
96:criterion, for
73:
70:
69:
53:
50:
49:
33:
30:
29:
17:
12:
11:
5:
5025:
5015:
5014:
5000:
4999:
4981:(2): 165–183.
4970:
4955:
4954:
4931:
4894:(17): 170503.
4878:
4843:
4798:(16): 160505.
4782:
4721:
4676:(23): 230502.
4660:
4599:
4538:
4477:
4428:
4362:
4361:
4359:
4356:
4340:
4337:
4334:
4329:
4325:
4304:
4301:
4298:
4293:
4289:
4283:
4280:
4277:
4272:
4268:
4264:
4261:
4256:
4253:
4231:
4228:
4225:
4220:
4217:
4214:
4209:
4205:
4201:
4196:
4193:
4169:
4166:
4163:
4158:
4154:
4133:
4122:
4121:
4110:
4105:
4102:
4099:
4094:
4090:
4086:
4083:
4080:
4077:
4074:
4069:
4066:
4063:
4058:
4054:
4050:
4045:
4042:
4039:
4034:
4030:
4024:
4020:
4014:
4010:
4006:
4003:
3975:
3971:
3943:
3939:
3918:
3915:
3912:
3907:
3902:
3898:
3894:
3889:
3886:
3864:
3861:
3858:
3853:
3849:
3845:
3840:
3837:
3813:
3809:
3786:
3782:
3770:
3769:
3758:
3753:
3749:
3745:
3740:
3736:
3730:
3726:
3720:
3716:
3712:
3709:
3679:
3659:
3656:
3651:
3647:
3643:
3640:
3637:
3634:
3614:
3611:
3600:
3599:
3588:
3585:
3580:
3576:
3572:
3569:
3566:
3563:
3540:
3517:
3497:
3494:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3454:
3451:
3431:
3411:
3406:
3402:
3398:
3395:
3390:
3386:
3382:
3379:
3359:
3356:
3353:
3331:
3325:
3321:
3316:
3312:
3308:
3305:
3300:
3296:
3292:
3289:
3286:
3281:
3277:
3273:
3270:
3265:
3261:
3257:
3254:
3251:
3231:
3211:
3189:
3186:
3182:
3170:
3169:
3158:
3155:
3152:
3149:
3144:
3141:
3137:
3133:
3128:
3125:
3121:
3117:
3102:
3099:
3086:
3083:
3080:
3060:
3057:
3054:
3034:
3031:
3028:
3012:
3009:
2980:
2977:
2974:
2969:
2963:
2957:
2952:
2946:
2940:
2913:
2910:
2907:
2904:
2901:
2898:
2874:
2870:
2865:
2835:
2831:
2808:
2804:
2792:
2791:
2780:
2777:
2774:
2769:
2765:
2761:
2756:
2752:
2748:
2745:
2722:
2710:
2707:
2704:
2699:
2693:
2687:
2660:
2657:
2654:
2649:
2643:
2637:
2610:
2605:
2599:
2593:
2590:
2570:
2565:
2559:
2553:
2550:
2528:
2523:
2517:
2511:
2508:
2488:
2483:
2477:
2471:
2468:
2444:
2441:
2438:
2435:
2432:
2429:
2391:
2387:
2382:
2359:
2355:
2349:
2344:
2340:
2336:
2312:
2307:
2303:
2280:
2276:
2270:
2265:
2261:
2257:
2246:
2245:
2232:
2228:
2222:
2217:
2213:
2209:
2206:
2201:
2196:
2192:
2186:
2182:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
2151:
2148:
2141:
2137:
2132:
2117:
2116:
2103:
2098:
2094:
2090:
2085:
2080:
2076:
2070:
2066:
2062:
2059:
2056:
2041:
2038:
2025:
2021:
2017:
2014:
2011:
2008:
2005:
1985:
1981:
1977:
1974:
1971:
1968:
1965:
1962:
1951:
1950:
1937:
1931:
1928:
1925:
1922:
1920:
1917:
1915:
1912:
1910:
1907:
1904:
1901:
1900:
1897:
1894:
1892:
1889:
1886:
1883:
1881:
1878:
1876:
1873:
1872:
1869:
1866:
1864:
1861:
1859:
1856:
1853:
1850:
1848:
1845:
1844:
1841:
1838:
1835:
1832:
1830:
1827:
1825:
1822:
1820:
1817:
1814:
1811:
1810:
1808:
1801:
1798:
1793:
1786:
1782:
1777:
1762:
1761:
1748:
1742:
1739:
1736:
1733:
1731:
1728:
1726:
1723:
1721:
1718:
1717:
1714:
1711:
1709:
1706:
1703:
1700:
1698:
1695:
1692:
1689:
1687:
1684:
1683:
1680:
1677:
1675:
1672:
1669:
1666:
1664:
1661:
1658:
1655:
1653:
1650:
1649:
1646:
1643:
1641:
1638:
1636:
1633:
1631:
1628:
1625:
1622:
1621:
1619:
1612:
1609:
1604:
1601:
1567:
1562:
1558:
1553:
1537:
1536:
1523:
1520:
1515:
1512:
1509:
1506:
1503:
1500:
1496:
1490:
1486:
1482:
1479:
1474:
1470:
1465:
1461:
1458:
1455:
1436:
1433:
1418:
1414:
1406:
1402:
1397:
1393:
1390:
1383:
1379:
1374:
1350:
1347:
1344:
1324:
1321:
1318:
1292:
1268:
1264:
1259:
1234:
1230:
1225:
1198:
1187:
1186:
1173:
1165:
1160:
1157:
1153:
1149:
1143:
1138:
1135:
1131:
1127:
1126:
1122:
1119:
1116:
1113:
1112:
1105:
1100:
1096:
1092:
1088:
1083:
1079:
1075:
1074:
1069:
1064:
1061:
1057:
1053:
1051:
1048:
1044:
1039:
1035:
1031:
1027:
1022:
1018:
1014:
1013:
1011:
1006:
999:
995:
990:
964:
958:
952:
949:
946:
943:
921:
915:
909:
906:
903:
900:
889:
888:
875:
867:
864:
860:
856:
850:
847:
843:
839:
838:
834:
831:
828:
825:
824:
817:
813:
809:
805:
801:
797:
796:
791:
788:
784:
780:
778:
775:
771:
767:
763:
759:
755:
751:
750:
748:
743:
740:
710:
707:
704:
701:
698:
695:
692:
689:
676:Note that the
674:
673:
661:
657:
654:
651:
648:
644:
640:
636:
632:
629:
626:
623:
619:
613:
610:
605:
602:
598:
592:
589:
586:
583:
579:
575:
571:
567:
564:
561:
558:
554:
550:
546:
542:
539:
536:
533:
529:
523:
520:
515:
512:
508:
502:
499:
496:
493:
489:
485:
480:
476:
471:
467:
464:
461:
458:
454:
450:
447:
443:
439:
436:
433:
430:
426:
420:
417:
412:
409:
405:
399:
396:
393:
390:
386:
382:
379:
376:
373:
370:
367:
364:
361:
358:
355:
348:
344:
339:
320:
319:
307:
303:
300:
297:
294:
290:
286:
282:
278:
275:
272:
269:
265:
259:
256:
251:
248:
244:
238:
235:
232:
229:
225:
221:
218:
194:
188:
182:
177:
171:
148:
136:
133:
77:
57:
37:
27:density matrix
15:
9:
6:
4:
3:
2:
5024:
5013:
5010:
5009:
5007:
4996:
4992:
4988:
4984:
4980:
4976:
4971:
4968:
4967:
4962:
4959:
4958:
4950:
4946:
4942:
4935:
4927:
4923:
4919:
4915:
4911:
4907:
4902:
4897:
4893:
4889:
4882:
4874:
4870:
4866:
4862:
4858:
4854:
4847:
4839:
4835:
4831:
4827:
4823:
4819:
4815:
4811:
4806:
4801:
4797:
4793:
4786:
4778:
4774:
4770:
4766:
4762:
4758:
4754:
4750:
4745:
4740:
4737:(5): 050503.
4736:
4732:
4725:
4717:
4713:
4709:
4705:
4701:
4697:
4693:
4689:
4684:
4679:
4675:
4671:
4664:
4656:
4652:
4648:
4644:
4640:
4636:
4632:
4628:
4623:
4618:
4614:
4610:
4603:
4595:
4591:
4587:
4583:
4579:
4575:
4571:
4567:
4562:
4557:
4553:
4549:
4542:
4534:
4530:
4526:
4522:
4518:
4514:
4510:
4506:
4501:
4496:
4492:
4488:
4481:
4473:
4469:
4465:
4461:
4456:
4451:
4447:
4443:
4439:
4432:
4424:
4420:
4416:
4412:
4408:
4404:
4400:
4396:
4391:
4386:
4382:
4378:
4374:
4367:
4363:
4355:
4335:
4327:
4323:
4302:
4299:
4291:
4278:
4270:
4266:
4229:
4226:
4215:
4207:
4203:
4164:
4156:
4152:
4131:
4108:
4100:
4092:
4088:
4084:
4081:
4078:
4075:
4072:
4064:
4056:
4052:
4048:
4040:
4032:
4028:
4022:
4018:
4012:
4008:
4004:
4001:
3994:
3993:
3992:
3989:
3973:
3969:
3960:
3941:
3937:
3916:
3913:
3905:
3900:
3896:
3862:
3859:
3851:
3847:
3811:
3807:
3784:
3780:
3756:
3751:
3747:
3743:
3738:
3734:
3728:
3724:
3718:
3714:
3710:
3707:
3700:
3699:
3698:
3695:
3693:
3677:
3657:
3654:
3649:
3641:
3638:
3635:
3612:
3609:
3586:
3583:
3578:
3570:
3567:
3564:
3554:
3553:
3552:
3538:
3529:
3515:
3495:
3492:
3489:
3486:
3483:
3480:
3477:
3474:
3471:
3452:
3449:
3429:
3409:
3404:
3396:
3388:
3380:
3357:
3354:
3351:
3329:
3323:
3314:
3306:
3298:
3290:
3284:
3279:
3271:
3263:
3255:
3229:
3209:
3187:
3184:
3180:
3172:holds, where
3156:
3153:
3150:
3147:
3142:
3139:
3135:
3131:
3126:
3123:
3119:
3115:
3108:
3107:
3106:
3098:
3084:
3081:
3078:
3058:
3055:
3052:
3032:
3029:
3026:
3018:
3008:
3006:
3002:
2998:
2992:
2978:
2975:
2967:
2955:
2950:
2908:
2899:
2896:
2889:is positive,
2872:
2868:
2863:
2853:
2851:
2833:
2806:
2778:
2775:
2772:
2767:
2759:
2754:
2746:
2736:
2735:
2734:
2720:
2708:
2705:
2697:
2658:
2655:
2647:
2603:
2588:
2563:
2548:
2539:
2521:
2506:
2481:
2466:
2458:
2457:positive maps
2439:
2430:
2427:
2418:
2416:
2412:
2407:
2389:
2385:
2380:
2357:
2347:
2342:
2338:
2310:
2305:
2301:
2278:
2268:
2263:
2259:
2230:
2220:
2215:
2211:
2204:
2199:
2194:
2190:
2184:
2180:
2176:
2173:
2167:
2158:
2155:
2152:
2146:
2139:
2135:
2130:
2122:
2121:
2120:
2101:
2096:
2092:
2088:
2083:
2078:
2074:
2068:
2064:
2060:
2057:
2054:
2047:
2046:
2045:
2040:Demonstration
2037:
2023:
2019:
2015:
2012:
2009:
2006:
2003:
1983:
1979:
1972:
1969:
1966:
1963:
1935:
1929:
1926:
1923:
1918:
1913:
1908:
1905:
1902:
1895:
1890:
1887:
1884:
1879:
1874:
1867:
1862:
1857:
1854:
1851:
1846:
1839:
1836:
1833:
1828:
1823:
1818:
1815:
1812:
1806:
1799:
1796:
1791:
1784:
1780:
1775:
1767:
1766:
1765:
1746:
1740:
1737:
1734:
1729:
1724:
1719:
1712:
1707:
1704:
1701:
1696:
1693:
1690:
1685:
1678:
1673:
1670:
1667:
1662:
1659:
1656:
1651:
1644:
1639:
1634:
1629:
1626:
1623:
1617:
1610:
1607:
1602:
1599:
1592:
1591:
1590:
1587:
1585:
1581:
1560:
1542:
1521:
1518:
1510:
1507:
1504:
1498:
1488:
1472:
1459:
1456:
1453:
1446:
1445:
1444:
1442:
1441:Werner states
1432:
1416:
1404:
1400:
1395:
1388:
1381:
1377:
1372:
1362:
1348:
1345:
1342:
1322:
1319:
1316:
1308:
1290:
1266:
1262:
1257:
1232:
1228:
1223:
1214:
1196:
1171:
1163:
1158:
1155:
1151:
1141:
1136:
1133:
1129:
1120:
1114:
1103:
1098:
1094:
1086:
1081:
1077:
1067:
1062:
1059:
1055:
1049:
1042:
1037:
1033:
1025:
1020:
1016:
1009:
1004:
997:
993:
988:
980:
979:
978:
962:
950:
947:
944:
941:
919:
907:
904:
901:
898:
873:
865:
862:
858:
848:
845:
841:
832:
826:
815:
811:
803:
799:
789:
786:
782:
776:
769:
765:
757:
753:
746:
741:
738:
731:
730:
729:
726:
724:
705:
696:
693:
690:
679:
655:
646:
638:
630:
621:
611:
608:
603:
600:
596:
590:
587:
584:
581:
577:
573:
565:
556:
548:
540:
531:
521:
518:
513:
510:
506:
500:
497:
494:
491:
487:
483:
478:
465:
456:
445:
437:
428:
418:
415:
410:
407:
403:
397:
394:
391:
388:
384:
380:
374:
365:
362:
359:
353:
346:
342:
337:
329:
328:
327:
325:
301:
292:
284:
276:
267:
257:
254:
249:
246:
242:
236:
233:
230:
227:
223:
219:
216:
209:
208:
207:
192:
180:
175:
146:
132:
130:
125:
123:
119:
115:
111:
107:
103:
99:
95:
91:
75:
55:
35:
28:
24:
19:
4978:
4974:
4965:
4940:
4934:
4891:
4887:
4881:
4856:
4852:
4846:
4795:
4791:
4785:
4734:
4730:
4724:
4673:
4669:
4663:
4612:
4608:
4602:
4551:
4547:
4541:
4490:
4486:
4480:
4448:(1–2): 1–8.
4445:
4441:
4431:
4380:
4376:
4366:
4123:
3990:
3771:
3696:
3692:entanglement
3601:
3530:
3171:
3104:
3017:Rajiah Simon
3014:
2993:
2854:
2849:
2793:
2540:
2419:
2408:
2247:
2118:
2043:
1952:
1763:
1588:
1538:
1438:
1363:
1188:
890:
727:
677:
675:
322:Its partial
321:
138:
126:
104:, where the
102:mixed states
97:
93:
22:
20:
18:
1213:eigenvalues
110:Asher Peres
4358:References
3625:Hence, if
3007:purposes.
135:Definition
4901:0812.4453
4805:0909.0147
4407:0031-9007
4085:⊗
4073:⊗
4049:⊗
4009:∑
4002:ϱ
3826:fulfill
3744:⊗
3715:∑
3708:ϱ
3650:ρ
3646:⟩
3639:⊗
3633:⟨
3584:≥
3579:ρ
3575:⟩
3568:⊗
3562:⟨
3539:ρ
3493:−
3487:≤
3475:≤
3422:Here for
3401:⟩
3385:⟩
3355:≠
3311:⟩
3295:⟩
3276:⟩
3260:⟩
3154:ρ
3148:ρ
3116:ρ
3082:⊕
3056:⊕
3030:⊕
3001:distilled
2976:≤
2956:⊗
2909:ρ
2903:Λ
2900:⊗
2864:ρ
2830:Λ
2803:Λ
2773:∘
2764:Λ
2751:Λ
2744:Λ
2440:ρ
2434:Λ
2431:⊗
2381:ρ
2339:ρ
2302:ρ
2260:ρ
2212:ρ
2205:⊗
2191:ρ
2177:∑
2168:ρ
2156:⊗
2131:ρ
2093:ρ
2089:⊗
2075:ρ
2061:∑
2055:ρ
2007:≥
1967:−
1927:−
1903:−
1834:−
1816:−
1776:ρ
1738:−
1691:−
1668:−
1627:−
1600:ρ
1566:⟩
1561:−
1557:Ψ
1508:−
1489:−
1485:Ψ
1481:⟨
1478:⟩
1473:−
1469:Ψ
1454:ρ
1396:ρ
1373:ρ
1346:×
1320:×
1307:entangled
1291:ρ
1258:ρ
1224:ρ
1197:ρ
1121:⋱
1115:⋮
1050:…
989:ρ
951:
908:
833:⋱
827:⋮
777:…
739:ρ
706:ρ
694:⊗
653:⟨
650:⟩
639:⊗
628:⟨
625:⟩
578:∑
563:⟨
560:⟩
549:⊗
538:⟨
535:⟩
488:∑
463:⟨
460:⟩
446:⊗
435:⟨
432:⟩
385:∑
375:ρ
363:⊗
338:ρ
324:transpose
299:⟨
296:⟩
285:⊗
274:⟨
271:⟩
224:∑
217:ρ
181:⊗
147:ρ
90:separable
36:ρ
5006:Category
4926:43527866
4918:19518768
4838:10523704
4830:19905682
4777:43756465
4769:16486912
4716:28595936
4708:16384285
4655:20897950
4647:11328047
4586:11017309
4533:11664720
4525:11017310
4472:10580997
4415:10063072
4182:fulfill
88:, to be
4983:Bibcode
4873:7149863
4810:Bibcode
4749:Bibcode
4688:Bibcode
4627:Bibcode
4594:9948874
4566:Bibcode
4505:Bibcode
4423:5246518
1435:Example
678:partial
122:Ryszard
4924:
4916:
4871:
4836:
4828:
4775:
4767:
4714:
4706:
4653:
4645:
4592:
4584:
4531:
4523:
4470:
4421:
4413:
4405:
4124:where
3772:where
3344:with
2794:where
891:Where
120:, and
114:Michał
4941:Arxiv
4922:S2CID
4896:arXiv
4869:S2CID
4834:S2CID
4800:arXiv
4773:S2CID
4739:arXiv
4712:S2CID
4678:arXiv
4651:S2CID
4617:arXiv
4590:S2CID
4556:arXiv
4529:S2CID
4495:arXiv
4468:S2CID
4450:arXiv
4419:S2CID
4385:arXiv
118:Paweł
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