3445:
in their paper "Geometrical aspects of entanglement" describe the problem and study the geometry of the separable states as a subset of the general state matrices. This subset have some intersection with the subset of states holding Peres-Horodecki criterion. In this paper, Leinaas et al. also give a
3295:
in many cases and is believed to be so in general. Some appreciation for this difficulty can be obtained if one attempts to solve the problem by employing the direct brute force approach, for a fixed dimension. The problem quickly becomes intractable, even for low dimensions. Thus more sophisticated
3454:
Testing for separability in the general case is an NP-hard problem. Leinaas et al. formulated an iterative, probabilistic algorithm for testing if a given state is separable. When the algorithm is successful, it gives an explicit, random, representation of the given state as a separable state.
1600:
37:
are multipartite quantum states that can be written as a tensor product of states in each space. The physical intuition behind these definitions is that product states have no correlation between the different degrees of freedom, while separable states might have correlations, but all such
711:
1075:
3175:
3270:
2119:
1451:
849:
2684:
Notice that, again from the definition of the tensor product, any density matrix, indeed any matrix acting on the composite state space, can be trivially written in the desired form, if we drop the requirement that
2522:
3330:
also applies. Specifically, Simon formulated a particular version of the Peres-Horodecki criterion in terms of the second-order moments of canonical operators and showed that it is necessary and sufficient for
2263:
2191:
1328:
1249:
1800:
483:
323:
249:
542:
1744:
3059:
2995:
2033:
2904:
1696:
Physically, this means that it is not possible to assign a definite (pure) state to the subsystems, which instead ought to be described as statistical ensembles of pure states, that is, as
1991:
1949:
1907:
1865:
3545:
Gurvits, L., Classical deterministic complexity of
Edmonds’ problem and quantum entanglement, in Proceedings of the 35th ACM Symposium on Theory of Computing, ACM Press, New York, 2003.
534:
1115:
3093:
2803:
2572:
959:
2334:
884:
417:
110:
3357:-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
2759:
2721:
2671:
2633:
2443:
2405:
1691:
1443:
1410:
1170:
951:
775:
2367:
1663:
3557:
Sevag
Gharibian, Strong NP-Hardness of the Quantum Separability Problem, Quantum Information and Computation, Vol. 10, No. 3&4, pp. 343-360, 2010. arXiv:0810.4507.
3407:
3381:
3355:
1630:
1142:
747:
2855:
2595:
2294:
1382:
1355:
377:
350:
167:
140:
3409:-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.
3101:
3190:
2038:
1595:{\displaystyle \rho _{A}\equiv \operatorname {Tr} _{B}(|\psi \rangle \!\langle \psi |)=\sum _{k=1}^{r_{\psi }}p_{k}\,|u_{k}\rangle \!\langle u_{k}|.}
784:
2451:
4253:
4127:
Walborn, S.; Taketani, B.; Salles, A.; Toscano, F.; de Matos Filho, R. (2009). "Entropic
Entanglement Criteria for Continuous Variables".
3567:
Hofmann, Holger F.; Takeuchi, Shigeki (September 22, 2003). "Violation of local uncertainty relations as a signature of entanglement".
2196:
2124:
1254:
1175:
536:. From the very definition of the tensor product, any vector of norm 1, i.e. a pure state of the composite system, can be written as
1753:
3005:
The above discussion generalizes easily to the case of a quantum system consisting of more than two subsystems. Let a system have
422:
254:
180:
3315:
is actually a necessary and sufficient condition for separability. Other separability criteria include (but not limited to) the
3275:
Or, in the infinite-dimensional case, ρ is separable if it can be approximated in the trace norm by states of the above form.
706:{\displaystyle |\psi \rangle =\sum _{i,j}c_{i,j}(|a_{i}\rangle \otimes |b_{j}\rangle )=\sum _{i,j}c_{i,j}|a_{i}b_{j}\rangle ,}
52:
if it is not separable. In general, determining if a state is separable is not straightforward and the problem is classed as
2805:
If these requirements are satisfied, then we can interpret the total state as a probability distribution over uncorrelated
1703:
3323:, and those based on uncertainty relations. See Ref. for a review of separability criteria in discrete variable systems.
3012:
2924:
3883:
Duan, Lu-Ming; Giedke, G.; Cirac, J. I.; Zoller, P. (2000). "Inseparability
Criterion for Continuous Variable Systems".
2827:
operators with trace 1, and a state is separable if it can be approximated, in trace norm, by states of the above form.
1996:
3288:
2860:
918:
1954:
1912:
1870:
1815:
1809:. That is, a quantum-mechanical pure state is separable if and only if it is in the image of the Segre embedding.
3327:
3312:
69:
3418:
38:
correlations can be explained as due to a classical random variable, as opposed as being due to entanglement.
488:
4258:
1070:{\displaystyle |\psi \rangle =\sum _{k=1}^{r_{\psi }}{\sqrt {p_{k}}}(|u_{k}\rangle \otimes |v_{k}\rangle ),}
1083:
3681:
Gühne, Otfried; Lewenstein, Maciej (August 24, 2004). "Entropic uncertainty relations and entanglement".
3064:
2764:
2533:
2677:
of the appropriate subsystems. It is clear from the definition that the family of separable states is a
2299:
922:
854:
382:
75:
2726:
2688:
2638:
2600:
2410:
2372:
4146:
4085:
4024:
3963:
3902:
3841:
3788:
3743:
3700:
3639:
3586:
3507:
3464:
49:
2597:
is called an entangled state. We can assume without loss of generality in the above expression that
1668:
1415:
1387:
1147:
928:
752:
4005:
Shchukin, E.; Vogel, W. (2005). "Inseparability
Criteria for Continuous Bipartite Quantum States".
3734:
Huang, Yichen (July 29, 2010). "Entanglement criteria via concave-function uncertainty relations".
3430:
3422:
3320:
2915:
1747:
4205:
4188:
Yichen Huang (October 2013). "Entanglement
Detection: Complexity and Shannon Entropic Criteria".
4170:
4136:
4109:
4075:
4048:
4014:
3987:
3953:
3926:
3892:
3865:
3831:
3804:
3778:
3716:
3690:
3663:
3629:
3602:
3576:
3525:
3497:
3455:
Otherwise it gives the distance of the given state from the nearest separable state it can find.
3442:
2339:
1635:
3386:
3360:
3334:
1608:
1120:
4162:
4101:
4040:
3979:
3918:
3857:
3655:
3434:
22:
719:
4197:
4154:
4093:
4032:
3971:
3910:
3849:
3822:
Simon, R. (2000). "Peres-Horodecki
Separability Criterion for Continuous Variable Systems".
3796:
3751:
3708:
3647:
3594:
3515:
3170:{\displaystyle |\psi \rangle =|\psi _{1}\rangle \otimes \cdots \otimes |\psi _{n}\rangle .}
2833:
2580:
2279:
1360:
1333:
355:
328:
145:
118:
45:
the definition simplifies: a pure state is separable if and only if it is a product state.
4238:
3620:
Gühne, Otfried (March 18, 2004). "Characterizing
Entanglement via Uncertainty Relations".
3426:
3316:
2823:
When the state spaces are infinite-dimensional, density matrices are replaced by positive
2813:
1806:
4150:
4089:
4066:
Hillery, Mark; Zubairy, M.Suhail (2006). "Entanglement
Conditions for Two-Mode States".
4028:
3967:
3906:
3845:
3792:
3747:
3704:
3643:
3590:
3511:
3303:
is a necessary condition a state must satisfy to be separable. In the low-dimensional (
2274:
2273:
Consider the mixed state case. A mixed state of the composite system is described by a
1697:
910:
states coincide for pure states, they do not in the more general case of mixed states.
3296:
formulations are required. The separability problem is a subject of current research.
3265:{\displaystyle \rho =\sum _{k}p_{k}\rho _{1}^{k}\otimes \cdots \otimes \rho _{n}^{k}.}
1805:
Formally, the embedding of a product of states into the product space is given by the
4247:
3808:
3720:
3529:
3520:
3485:
2806:
914:
113:
30:
4174:
4113:
4052:
3991:
3869:
3606:
3429:. In the bipartite case, a quantum state is separable if and only if it lies in the
3283:
The problem of deciding whether a state is separable in general is sometimes called
2114:{\displaystyle |\psi \rangle \equiv {\sqrt {1/3}}|0\rangle +{\sqrt {2/3}}|1\rangle }
4209:
4158:
3930:
3667:
4097:
4036:
3800:
3651:
3975:
3914:
3853:
3486:"Fine-structure classification of multiqubit entanglement by algebraic geometry"
3438:
2824:
42:
3755:
3712:
3598:
68:. By a postulate of quantum mechanics these can be described as vectors in the
2678:
4201:
64:
Consider first composite states with two degrees of freedom, referred to as
4166:
4105:
4044:
3983:
3922:
3861:
3659:
844:{\displaystyle |\psi \rangle =|\psi _{1}\rangle \otimes |\psi _{2}\rangle }
4225:"Geometrical aspects of entanglement", Physical Review A 74, 012313 (2006)
4080:
4019:
3958:
3897:
3836:
3695:
3634:
3581:
3484:
Gharahi, Masoud; Mancini, Stefano; Ottaviani, Giorgio (October 1, 2020).
3292:
53:
3944:
Werner, R. F.; Wolf, M. M. (2001). "Bound
Entangled Gaussian States".
3291:. It is considered to be a difficult problem. It has been shown to be
2517:{\displaystyle \rho =\sum _{k}p_{k}\rho _{1}^{k}\otimes \rho _{2}^{k}}
1632:
is pure --- that is, is projection with unit-rank --- if and only if
3502:
4141:
3783:
3446:
numerical approach to test for separability in the general case.
2258:{\displaystyle {\sqrt {1/3}}|01\rangle +{\sqrt {2/3}}|10\rangle }
2186:{\displaystyle {\sqrt {1/2}}|00\rangle +{\sqrt {1/2}}|11\rangle }
1323:{\displaystyle \{|v_{k}\rangle \}_{k=1}^{r_{\psi }}\subset H_{2}}
1244:{\displaystyle \{|u_{k}\rangle \}_{k=1}^{r_{\psi }}\subset H_{1}}
2817:
33:
that can be written as a convex combination of product states.
2816:, a separable state can be created from any other state using
2445:
which are mixed states of the respective subsystems such that
1795:{\displaystyle \rho _{A}\equiv \operatorname {Tr} _{B}(\rho )}
3769:
Gühne, Otfried; Tóth, Géza (2009). "Entanglement detection".
478:{\displaystyle \{|{a_{i}}\rangle \otimes |{b_{j}}\rangle \}}
318:{\displaystyle \{|{b_{j}}\rangle \}_{j=1}^{m}\subset H_{2}}
244:{\displaystyle \{|{a_{i}}\rangle \}_{i=1}^{n}\subset H_{1}}
1993:, are all product (and thus separable) pure states, as is
2914:. One property of the product state is that in terms of
2863:
112:. In this discussion we will focus on the case of the
3389:
3363:
3337:
3193:
3104:
3067:
3015:
2927:
2836:
2767:
2729:
2691:
2641:
2603:
2583:
2536:
2454:
2413:
2375:
2342:
2302:
2282:
2199:
2127:
2041:
1999:
1957:
1915:
1873:
1818:
1756:
1706:
1671:
1638:
1611:
1454:
1418:
1390:
1363:
1336:
1257:
1178:
1150:
1123:
1086:
962:
931:
857:
787:
755:
722:
545:
491:
425:
385:
358:
331:
257:
183:
148:
121:
78:
3383:-mode Gaussian states, but no longer sufficient for
2673:
are all rank-1 projections, that is, they represent
1739:{\displaystyle \rho =|\psi \rangle \!\langle \psi |}
3054:{\displaystyle H=H_{1}\otimes \cdots \otimes H_{n}}
2990:{\displaystyle S(\rho )=S(\rho _{1})+S(\rho _{2}).}
1445:. At the same time, the partial state has the form
3401:
3375:
3349:
3264:
3169:
3087:
3053:
2989:
2898:
2849:
2797:
2753:
2715:
2665:
2627:
2589:
2566:
2516:
2437:
2399:
2361:
2328:
2288:
2257:
2185:
2113:
2027:
1985:
1943:
1901:
1859:
1794:
1738:
1685:
1657:
1624:
1594:
1437:
1404:
1376:
1349:
1322:
1243:
1164:
1136:
1109:
1069:
945:
878:
843:
769:
741:
705:
528:
477:
411:
371:
344:
317:
243:
161:
134:
104:
1724:
1570:
1495:
4221:
4219:
2028:{\displaystyle |0\rangle \otimes |\psi \rangle }
913:Pure states are entangled if and only if their
2899:{\textstyle \rho =\rho _{1}\otimes \rho _{2},}
8:
3161:
3134:
3113:
3076:
2748:
2730:
2710:
2692:
2660:
2642:
2622:
2604:
2432:
2414:
2394:
2376:
2252:
2223:
2180:
2151:
2108:
2079:
2050:
2022:
2008:
1986:{\displaystyle |1\rangle \otimes |1\rangle }
1980:
1966:
1944:{\displaystyle |0\rangle \otimes |1\rangle }
1938:
1924:
1902:{\displaystyle |0\rangle \otimes |0\rangle }
1896:
1882:
1860:{\displaystyle H_{1}=H_{2}=\mathbb {C} ^{2}}
1725:
1721:
1680:
1571:
1567:
1496:
1492:
1399:
1280:
1276:
1258:
1201:
1197:
1179:
1159:
1058:
1037:
971:
940:
873:
838:
817:
796:
764:
697:
631:
610:
554:
523:
520:
492:
472:
469:
446:
426:
282:
278:
258:
208:
204:
184:
3553:
3551:
3541:
3539:
2857:, then the state can be expressed just as
2768:
2537:
4140:
4079:
4018:
3957:
3896:
3835:
3782:
3694:
3633:
3580:
3519:
3501:
3388:
3362:
3336:
3253:
3248:
3229:
3224:
3214:
3204:
3192:
3155:
3146:
3128:
3119:
3105:
3103:
3068:
3066:
3045:
3026:
3014:
2975:
2953:
2926:
2887:
2874:
2862:
2841:
2835:
2818:local actions and classical communication
2783:
2773:
2766:
2742:
2737:
2728:
2704:
2699:
2690:
2654:
2649:
2640:
2616:
2611:
2602:
2582:
2552:
2542:
2535:
2508:
2503:
2490:
2485:
2475:
2465:
2453:
2426:
2421:
2412:
2388:
2383:
2374:
2347:
2341:
2320:
2307:
2301:
2281:
2244:
2234:
2229:
2215:
2205:
2200:
2198:
2172:
2162:
2157:
2143:
2133:
2128:
2126:
2100:
2090:
2085:
2071:
2061:
2056:
2042:
2040:
2014:
2000:
1998:
1972:
1958:
1956:
1930:
1916:
1914:
1888:
1874:
1872:
1851:
1847:
1846:
1836:
1823:
1817:
1812:For example, in a two-qubit space, where
1774:
1761:
1755:
1731:
1713:
1705:
1672:
1670:
1643:
1637:
1616:
1610:
1584:
1578:
1561:
1552:
1551:
1545:
1533:
1528:
1517:
1502:
1484:
1472:
1459:
1453:
1423:
1417:
1391:
1389:
1368:
1362:
1341:
1335:
1314:
1299:
1294:
1283:
1270:
1261:
1256:
1235:
1220:
1215:
1204:
1191:
1182:
1177:
1151:
1149:
1128:
1122:
1093:
1087:
1085:
1052:
1043:
1031:
1022:
1011:
1005:
997:
992:
981:
963:
961:
932:
930:
867:
858:
856:
832:
823:
811:
802:
788:
786:
756:
754:
727:
721:
691:
681:
672:
660:
644:
625:
616:
604:
595:
580:
564:
546:
544:
514:
504:
495:
490:
462:
457:
452:
439:
434:
429:
424:
403:
390:
384:
363:
357:
336:
330:
309:
296:
285:
271:
266:
261:
256:
235:
222:
211:
197:
192:
187:
182:
153:
147:
126:
120:
96:
83:
77:
902:. Note that, even though the notions of
4190:IEEE Transactions on Information Theory
3476:
3417:Quantum mechanics may be modelled on a
3413:Characterization via algebraic geometry
529:{\displaystyle \{|a_{i}b_{j}\rangle \}}
3180:Similarly, a mixed state ρ acting on
1746:is thus entangled if and only if the
16:Quantum states that are not entangled
7:
3326:In continuous variable systems, the
1110:{\displaystyle {\sqrt {p_{k}}}>0}
3184:is separable if it is a convex sum
3088:{\displaystyle |\psi \rangle \in H}
2830:If there is only a single non-zero
2798:{\displaystyle \;\sum _{k}p_{k}=1.}
2567:{\displaystyle \;\sum _{k}p_{k}=1.}
3095:is separable if it takes the form
3001:Extending to the multipartite case
2329:{\displaystyle H_{1}\otimes H_{2}}
1330:are sets of orthonormal states in
879:{\displaystyle |\psi _{i}\rangle }
412:{\displaystyle H_{1}\otimes H_{2}}
105:{\displaystyle H_{1}\otimes H_{2}}
14:
2820:while an entangled state cannot.
2754:{\displaystyle \{\rho _{2}^{k}\}}
2716:{\displaystyle \{\rho _{1}^{k}\}}
2666:{\displaystyle \{\rho _{2}^{k}\}}
2628:{\displaystyle \{\rho _{1}^{k}\}}
2438:{\displaystyle \{\rho _{2}^{k}\}}
2400:{\displaystyle \{\rho _{1}^{k}\}}
2121:. On the other hand, states like
60:Separability of bipartite systems
3521:10.1103/PhysRevResearch.2.043003
3009:subsystems and have state space
2336:. ρ is separable if there exist
4159:10.1103/PhysRevLett.103.160505
3147:
3120:
3106:
3069:
2981:
2968:
2959:
2946:
2937:
2931:
2245:
2216:
2173:
2144:
2101:
2072:
2043:
2015:
2001:
1973:
1959:
1931:
1917:
1889:
1875:
1789:
1783:
1732:
1714:
1686:{\displaystyle |\psi \rangle }
1673:
1585:
1553:
1507:
1503:
1485:
1481:
1438:{\displaystyle r_{\psi }>1}
1405:{\displaystyle |\psi \rangle }
1392:
1262:
1183:
1165:{\displaystyle |\psi \rangle }
1152:
1061:
1044:
1023:
1019:
964:
946:{\displaystyle |\psi \rangle }
933:
890:-th space, it is said to be a
859:
824:
803:
789:
770:{\displaystyle |\psi \rangle }
757:
673:
634:
617:
596:
592:
547:
496:
485:, or in more compact notation
453:
430:
262:
188:
1:
4098:10.1103/PhysRevLett.96.050503
4037:10.1103/PhysRevLett.95.230502
3801:10.1016/j.physrep.2009.02.004
3652:10.1103/PhysRevLett.92.117903
1412:is entangled if and only if
379:, respectively. A basis for
4254:Quantum information science
3976:10.1103/PhysRevLett.86.3658
3915:10.1103/PhysRevLett.84.2722
3854:10.1103/PhysRevLett.84.2726
2362:{\displaystyle p_{k}\geq 0}
1658:{\displaystyle r_{\psi }=1}
1117:are positive real numbers,
4275:
3756:10.1103/PhysRevA.82.012335
3713:10.1103/PhysRevA.70.022316
3599:10.1103/PhysRevA.68.032103
3425:of two such spaces is the
3289:quantum information theory
2761:are themselves states and
1384:, respectively. The state
169:being finite-dimensional.
3402:{\displaystyle 2\oplus 2}
3376:{\displaystyle 1\oplus n}
3350:{\displaystyle 1\oplus 1}
3328:Peres-Horodecki criterion
3313:Peres-Horodecki criterion
1665:, which is equivalent to
1625:{\displaystyle \rho _{A}}
1137:{\displaystyle r_{\psi }}
921:. To see this, write the
898:. Otherwise it is called
325:be orthonormal bases for
4239:"StateSeparator" web-app
4202:10.1109/TIT.2013.2257936
3490:Physical Review Research
3450:Testing for separability
3433:of the Segre embedding.
3419:projective Hilbert space
3285:the separability problem
4129:Physical Review Letters
4068:Physical Review Letters
4007:Physical Review Letters
3946:Physical Review Letters
3885:Physical Review Letters
3824:Physical Review Letters
3622:Physical Review Letters
1144:is the Schmidt rank of
781:, that is, in the form
742:{\displaystyle c_{i,j}}
41:In the special case of
3403:
3377:
3351:
3301:separability criterion
3279:Separability criterion
3266:
3171:
3089:
3055:
2991:
2900:
2851:
2799:
2755:
2717:
2667:
2629:
2591:
2568:
2518:
2439:
2401:
2363:
2330:
2290:
2259:
2187:
2115:
2029:
1987:
1945:
1903:
1861:
1796:
1740:
1687:
1659:
1626:
1596:
1540:
1439:
1406:
1378:
1351:
1324:
1245:
1166:
1138:
1111:
1071:
1004:
947:
894:, and, in particular,
880:
845:
771:
743:
707:
530:
479:
413:
373:
346:
319:
245:
163:
136:
106:
48:A state is said to be
3404:
3378:
3352:
3267:
3172:
3090:
3056:
2992:
2901:
2852:
2850:{\displaystyle p_{k}}
2800:
2756:
2718:
2668:
2630:
2592:
2590:{\displaystyle \rho }
2569:
2519:
2440:
2402:
2364:
2331:
2291:
2289:{\displaystyle \rho }
2260:
2188:
2116:
2030:
1988:
1946:
1904:
1862:
1797:
1750:of the partial state
1741:
1688:
1660:
1627:
1597:
1513:
1440:
1407:
1379:
1377:{\displaystyle H_{2}}
1352:
1350:{\displaystyle H_{1}}
1325:
1246:
1167:
1139:
1112:
1072:
977:
948:
923:Schmidt decomposition
881:
846:
772:
744:
708:
531:
480:
414:
374:
372:{\displaystyle H_{2}}
347:
345:{\displaystyle H_{1}}
320:
246:
164:
162:{\displaystyle H_{2}}
137:
135:{\displaystyle H_{1}}
107:
3465:Entanglement witness
3387:
3361:
3335:
3191:
3102:
3065:
3013:
2925:
2861:
2834:
2765:
2727:
2689:
2639:
2601:
2581:
2534:
2452:
2411:
2373:
2340:
2300:
2280:
2197:
2125:
2039:
1997:
1955:
1913:
1871:
1816:
1754:
1704:
1669:
1636:
1609:
1452:
1416:
1388:
1361:
1334:
1255:
1176:
1148:
1121:
1084:
960:
929:
886:a pure state in the
855:
785:
777:can be written as a
753:
720:
543:
489:
423:
383:
356:
329:
255:
181:
146:
119:
76:
4151:2009PhRvL.103p0505W
4090:2006PhRvL..96e0503H
4029:2005PhRvL..95w0502S
3968:2001PhRvL..86.3658W
3907:2000PhRvL..84.2722D
3846:2000PhRvL..84.2726S
3793:2009PhR...474....1G
3748:2010PhRvA..82a2335H
3705:2004PhRvA..70b2316G
3644:2004PhRvL..92k7903G
3591:2003PhRvA..68c2103H
3512:2020PhRvR...2d3003G
3423:categorical product
3321:reduction criterion
3258:
3234:
2747:
2709:
2659:
2621:
2513:
2495:
2431:
2393:
2265:are not separable.
1748:von Neumann entropy
1306:
1227:
749:is a constant. If
301:
227:
3399:
3373:
3347:
3262:
3244:
3220:
3209:
3167:
3085:
3051:
2987:
2896:
2847:
2795:
2778:
2751:
2733:
2713:
2695:
2663:
2645:
2625:
2607:
2587:
2564:
2547:
2514:
2499:
2481:
2470:
2435:
2417:
2397:
2379:
2359:
2326:
2286:
2255:
2183:
2111:
2025:
1983:
1941:
1899:
1857:
1792:
1736:
1683:
1655:
1622:
1592:
1435:
1402:
1374:
1347:
1320:
1279:
1241:
1200:
1162:
1134:
1107:
1067:
943:
876:
841:
767:
739:
703:
655:
575:
526:
475:
409:
369:
342:
315:
281:
241:
207:
159:
132:
102:
4196:(10): 6774–6778.
3952:(16): 3658–3661.
3891:(12): 2722–2725.
3830:(12): 2726–2729.
3736:Physical Review A
3683:Physical Review A
3569:Physical Review A
3435:Jon Magne Leinaas
3200:
2769:
2538:
2461:
2242:
2213:
2170:
2141:
2098:
2069:
1693:being separable.
1099:
1017:
640:
560:
29:are multipartite
23:quantum mechanics
4266:
4226:
4223:
4214:
4213:
4185:
4179:
4178:
4144:
4124:
4118:
4117:
4083:
4081:quant-ph/0507168
4063:
4057:
4056:
4022:
4020:quant-ph/0508132
4002:
3996:
3995:
3961:
3959:quant-ph/0009118
3941:
3935:
3934:
3900:
3898:quant-ph/9908056
3880:
3874:
3873:
3839:
3837:quant-ph/9909044
3819:
3813:
3812:
3786:
3766:
3760:
3759:
3731:
3725:
3724:
3698:
3696:quant-ph/0403219
3678:
3672:
3671:
3637:
3635:quant-ph/0306194
3617:
3611:
3610:
3584:
3582:quant-ph/0212090
3564:
3558:
3555:
3546:
3543:
3534:
3533:
3523:
3505:
3481:
3408:
3406:
3405:
3400:
3382:
3380:
3379:
3374:
3356:
3354:
3353:
3348:
3271:
3269:
3268:
3263:
3257:
3252:
3233:
3228:
3219:
3218:
3208:
3176:
3174:
3173:
3168:
3160:
3159:
3150:
3133:
3132:
3123:
3109:
3094:
3092:
3091:
3086:
3072:
3060:
3058:
3057:
3052:
3050:
3049:
3031:
3030:
2996:
2994:
2993:
2988:
2980:
2979:
2958:
2957:
2908:simply separable
2905:
2903:
2902:
2897:
2892:
2891:
2879:
2878:
2856:
2854:
2853:
2848:
2846:
2845:
2814:quantum channels
2804:
2802:
2801:
2796:
2788:
2787:
2777:
2760:
2758:
2757:
2752:
2746:
2741:
2722:
2720:
2719:
2714:
2708:
2703:
2672:
2670:
2669:
2664:
2658:
2653:
2634:
2632:
2631:
2626:
2620:
2615:
2596:
2594:
2593:
2588:
2573:
2571:
2570:
2565:
2557:
2556:
2546:
2523:
2521:
2520:
2515:
2512:
2507:
2494:
2489:
2480:
2479:
2469:
2444:
2442:
2441:
2436:
2430:
2425:
2406:
2404:
2403:
2398:
2392:
2387:
2368:
2366:
2365:
2360:
2352:
2351:
2335:
2333:
2332:
2327:
2325:
2324:
2312:
2311:
2295:
2293:
2292:
2287:
2264:
2262:
2261:
2256:
2248:
2243:
2238:
2230:
2219:
2214:
2209:
2201:
2192:
2190:
2189:
2184:
2176:
2171:
2166:
2158:
2147:
2142:
2137:
2129:
2120:
2118:
2117:
2112:
2104:
2099:
2094:
2086:
2075:
2070:
2065:
2057:
2046:
2034:
2032:
2031:
2026:
2018:
2004:
1992:
1990:
1989:
1984:
1976:
1962:
1950:
1948:
1947:
1942:
1934:
1920:
1908:
1906:
1905:
1900:
1892:
1878:
1866:
1864:
1863:
1858:
1856:
1855:
1850:
1841:
1840:
1828:
1827:
1801:
1799:
1798:
1793:
1779:
1778:
1766:
1765:
1745:
1743:
1742:
1737:
1735:
1717:
1698:density matrices
1692:
1690:
1689:
1684:
1676:
1664:
1662:
1661:
1656:
1648:
1647:
1631:
1629:
1628:
1623:
1621:
1620:
1605:It follows that
1601:
1599:
1598:
1593:
1588:
1583:
1582:
1566:
1565:
1556:
1550:
1549:
1539:
1538:
1537:
1527:
1506:
1488:
1477:
1476:
1464:
1463:
1444:
1442:
1441:
1436:
1428:
1427:
1411:
1409:
1408:
1403:
1395:
1383:
1381:
1380:
1375:
1373:
1372:
1356:
1354:
1353:
1348:
1346:
1345:
1329:
1327:
1326:
1321:
1319:
1318:
1305:
1304:
1303:
1293:
1275:
1274:
1265:
1250:
1248:
1247:
1242:
1240:
1239:
1226:
1225:
1224:
1214:
1196:
1195:
1186:
1171:
1169:
1168:
1163:
1155:
1143:
1141:
1140:
1135:
1133:
1132:
1116:
1114:
1113:
1108:
1100:
1098:
1097:
1088:
1076:
1074:
1073:
1068:
1057:
1056:
1047:
1036:
1035:
1026:
1018:
1016:
1015:
1006:
1003:
1002:
1001:
991:
967:
952:
950:
949:
944:
936:
885:
883:
882:
877:
872:
871:
862:
850:
848:
847:
842:
837:
836:
827:
816:
815:
806:
792:
776:
774:
773:
768:
760:
748:
746:
745:
740:
738:
737:
712:
710:
709:
704:
696:
695:
686:
685:
676:
671:
670:
654:
630:
629:
620:
609:
608:
599:
591:
590:
574:
550:
535:
533:
532:
527:
519:
518:
509:
508:
499:
484:
482:
481:
476:
468:
467:
466:
456:
445:
444:
443:
433:
418:
416:
415:
410:
408:
407:
395:
394:
378:
376:
375:
370:
368:
367:
351:
349:
348:
343:
341:
340:
324:
322:
321:
316:
314:
313:
300:
295:
277:
276:
275:
265:
250:
248:
247:
242:
240:
239:
226:
221:
203:
202:
201:
191:
168:
166:
165:
160:
158:
157:
141:
139:
138:
133:
131:
130:
111:
109:
108:
103:
101:
100:
88:
87:
66:bipartite states
27:separable states
4274:
4273:
4269:
4268:
4267:
4265:
4264:
4263:
4244:
4243:
4235:
4230:
4229:
4224:
4217:
4187:
4186:
4182:
4126:
4125:
4121:
4065:
4064:
4060:
4004:
4003:
3999:
3943:
3942:
3938:
3882:
3881:
3877:
3821:
3820:
3816:
3771:Physics Reports
3768:
3767:
3763:
3733:
3732:
3728:
3680:
3679:
3675:
3619:
3618:
3614:
3566:
3565:
3561:
3556:
3549:
3544:
3537:
3483:
3482:
3478:
3473:
3461:
3452:
3427:Segre embedding
3415:
3385:
3384:
3359:
3358:
3333:
3332:
3317:range criterion
3281:
3210:
3189:
3188:
3151:
3124:
3100:
3099:
3063:
3062:
3061:. A pure state
3041:
3022:
3011:
3010:
3003:
2971:
2949:
2923:
2922:
2883:
2870:
2859:
2858:
2837:
2832:
2831:
2779:
2763:
2762:
2725:
2724:
2687:
2686:
2637:
2636:
2599:
2598:
2579:
2578:
2548:
2532:
2531:
2471:
2450:
2449:
2409:
2408:
2371:
2370:
2343:
2338:
2337:
2316:
2303:
2298:
2297:
2278:
2277:
2271:
2195:
2194:
2123:
2122:
2037:
2036:
1995:
1994:
1953:
1952:
1911:
1910:
1869:
1868:
1845:
1832:
1819:
1814:
1813:
1807:Segre embedding
1770:
1757:
1752:
1751:
1702:
1701:
1700:. A pure state
1667:
1666:
1639:
1634:
1633:
1612:
1607:
1606:
1574:
1557:
1541:
1529:
1468:
1455:
1450:
1449:
1419:
1414:
1413:
1386:
1385:
1364:
1359:
1358:
1337:
1332:
1331:
1310:
1295:
1266:
1253:
1252:
1231:
1216:
1187:
1174:
1173:
1146:
1145:
1124:
1119:
1118:
1089:
1082:
1081:
1048:
1027:
1007:
993:
958:
957:
927:
926:
863:
853:
852:
828:
807:
783:
782:
751:
750:
723:
718:
717:
687:
677:
656:
621:
600:
576:
541:
540:
510:
500:
487:
486:
458:
435:
421:
420:
399:
386:
381:
380:
359:
354:
353:
332:
327:
326:
305:
267:
253:
252:
231:
193:
179:
178:
175:
149:
144:
143:
122:
117:
116:
92:
79:
74:
73:
62:
17:
12:
11:
5:
4272:
4270:
4262:
4261:
4259:Quantum states
4256:
4246:
4245:
4242:
4241:
4234:
4233:External links
4231:
4228:
4227:
4215:
4180:
4135:(16): 160505.
4119:
4058:
4013:(23): 230502.
3997:
3936:
3875:
3814:
3761:
3726:
3673:
3628:(11): 117903.
3612:
3559:
3547:
3535:
3475:
3474:
3472:
3469:
3468:
3467:
3460:
3457:
3451:
3448:
3414:
3411:
3398:
3395:
3392:
3372:
3369:
3366:
3346:
3343:
3340:
3280:
3277:
3273:
3272:
3261:
3256:
3251:
3247:
3243:
3240:
3237:
3232:
3227:
3223:
3217:
3213:
3207:
3203:
3199:
3196:
3178:
3177:
3166:
3163:
3158:
3154:
3149:
3145:
3142:
3139:
3136:
3131:
3127:
3122:
3118:
3115:
3112:
3108:
3084:
3081:
3078:
3075:
3071:
3048:
3044:
3040:
3037:
3034:
3029:
3025:
3021:
3018:
3002:
2999:
2998:
2997:
2986:
2983:
2978:
2974:
2970:
2967:
2964:
2961:
2956:
2952:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2906:and is called
2895:
2890:
2886:
2882:
2877:
2873:
2869:
2866:
2844:
2840:
2807:product states
2794:
2791:
2786:
2782:
2776:
2772:
2750:
2745:
2740:
2736:
2732:
2712:
2707:
2702:
2698:
2694:
2675:pure ensembles
2662:
2657:
2652:
2648:
2644:
2624:
2619:
2614:
2610:
2606:
2586:
2575:
2574:
2563:
2560:
2555:
2551:
2545:
2541:
2525:
2524:
2511:
2506:
2502:
2498:
2493:
2488:
2484:
2478:
2474:
2468:
2464:
2460:
2457:
2434:
2429:
2424:
2420:
2416:
2396:
2391:
2386:
2382:
2378:
2358:
2355:
2350:
2346:
2323:
2319:
2315:
2310:
2306:
2285:
2275:density matrix
2270:
2267:
2254:
2251:
2247:
2241:
2237:
2233:
2228:
2225:
2222:
2218:
2212:
2208:
2204:
2182:
2179:
2175:
2169:
2165:
2161:
2156:
2153:
2150:
2146:
2140:
2136:
2132:
2110:
2107:
2103:
2097:
2093:
2089:
2084:
2081:
2078:
2074:
2068:
2064:
2060:
2055:
2052:
2049:
2045:
2024:
2021:
2017:
2013:
2010:
2007:
2003:
1982:
1979:
1975:
1971:
1968:
1965:
1961:
1940:
1937:
1933:
1929:
1926:
1923:
1919:
1898:
1895:
1891:
1887:
1884:
1881:
1877:
1854:
1849:
1844:
1839:
1835:
1831:
1826:
1822:
1791:
1788:
1785:
1782:
1777:
1773:
1769:
1764:
1760:
1734:
1730:
1727:
1723:
1720:
1716:
1712:
1709:
1682:
1679:
1675:
1654:
1651:
1646:
1642:
1619:
1615:
1603:
1602:
1591:
1587:
1581:
1577:
1573:
1569:
1564:
1560:
1555:
1548:
1544:
1536:
1532:
1526:
1523:
1520:
1516:
1512:
1509:
1505:
1501:
1498:
1494:
1491:
1487:
1483:
1480:
1475:
1471:
1467:
1462:
1458:
1434:
1431:
1426:
1422:
1401:
1398:
1394:
1371:
1367:
1344:
1340:
1317:
1313:
1309:
1302:
1298:
1292:
1289:
1286:
1282:
1278:
1273:
1269:
1264:
1260:
1238:
1234:
1230:
1223:
1219:
1213:
1210:
1207:
1203:
1199:
1194:
1190:
1185:
1181:
1161:
1158:
1154:
1131:
1127:
1106:
1103:
1096:
1092:
1078:
1077:
1066:
1063:
1060:
1055:
1051:
1046:
1042:
1039:
1034:
1030:
1025:
1021:
1014:
1010:
1000:
996:
990:
987:
984:
980:
976:
973:
970:
966:
942:
939:
935:
915:partial states
875:
870:
866:
861:
840:
835:
831:
826:
822:
819:
814:
810:
805:
801:
798:
795:
791:
766:
763:
759:
736:
733:
730:
726:
714:
713:
702:
699:
694:
690:
684:
680:
675:
669:
666:
663:
659:
653:
650:
647:
643:
639:
636:
633:
628:
624:
619:
615:
612:
607:
603:
598:
594:
589:
586:
583:
579:
573:
570:
567:
563:
559:
556:
553:
549:
525:
522:
517:
513:
507:
503:
498:
494:
474:
471:
465:
461:
455:
451:
448:
442:
438:
432:
428:
406:
402:
398:
393:
389:
366:
362:
339:
335:
312:
308:
304:
299:
294:
291:
288:
284:
280:
274:
270:
264:
260:
238:
234:
230:
225:
220:
217:
214:
210:
206:
200:
196:
190:
186:
174:
171:
156:
152:
129:
125:
114:Hilbert spaces
99:
95:
91:
86:
82:
70:tensor product
61:
58:
35:Product states
31:quantum states
15:
13:
10:
9:
6:
4:
3:
2:
4271:
4260:
4257:
4255:
4252:
4251:
4249:
4240:
4237:
4236:
4232:
4222:
4220:
4216:
4211:
4207:
4203:
4199:
4195:
4191:
4184:
4181:
4176:
4172:
4168:
4164:
4160:
4156:
4152:
4148:
4143:
4138:
4134:
4130:
4123:
4120:
4115:
4111:
4107:
4103:
4099:
4095:
4091:
4087:
4082:
4077:
4074:(5): 050503.
4073:
4069:
4062:
4059:
4054:
4050:
4046:
4042:
4038:
4034:
4030:
4026:
4021:
4016:
4012:
4008:
4001:
3998:
3993:
3989:
3985:
3981:
3977:
3973:
3969:
3965:
3960:
3955:
3951:
3947:
3940:
3937:
3932:
3928:
3924:
3920:
3916:
3912:
3908:
3904:
3899:
3894:
3890:
3886:
3879:
3876:
3871:
3867:
3863:
3859:
3855:
3851:
3847:
3843:
3838:
3833:
3829:
3825:
3818:
3815:
3810:
3806:
3802:
3798:
3794:
3790:
3785:
3780:
3777:(1–6): 1–75.
3776:
3772:
3765:
3762:
3757:
3753:
3749:
3745:
3742:(1): 012335.
3741:
3737:
3730:
3727:
3722:
3718:
3714:
3710:
3706:
3702:
3697:
3692:
3689:(2): 022316.
3688:
3684:
3677:
3674:
3669:
3665:
3661:
3657:
3653:
3649:
3645:
3641:
3636:
3631:
3627:
3623:
3616:
3613:
3608:
3604:
3600:
3596:
3592:
3588:
3583:
3578:
3575:(3): 032103.
3574:
3570:
3563:
3560:
3554:
3552:
3548:
3542:
3540:
3536:
3531:
3527:
3522:
3517:
3513:
3509:
3504:
3499:
3496:(4): 043003.
3495:
3491:
3487:
3480:
3477:
3470:
3466:
3463:
3462:
3458:
3456:
3449:
3447:
3444:
3440:
3436:
3432:
3428:
3424:
3420:
3412:
3410:
3396:
3393:
3390:
3370:
3367:
3364:
3344:
3341:
3338:
3329:
3324:
3322:
3318:
3314:
3311:) cases, the
3310:
3306:
3302:
3297:
3294:
3290:
3286:
3278:
3276:
3259:
3254:
3249:
3245:
3241:
3238:
3235:
3230:
3225:
3221:
3215:
3211:
3205:
3201:
3197:
3194:
3187:
3186:
3185:
3183:
3164:
3156:
3152:
3143:
3140:
3137:
3129:
3125:
3116:
3110:
3098:
3097:
3096:
3082:
3079:
3073:
3046:
3042:
3038:
3035:
3032:
3027:
3023:
3019:
3016:
3008:
3000:
2984:
2976:
2972:
2965:
2962:
2954:
2950:
2943:
2940:
2934:
2928:
2921:
2920:
2919:
2917:
2913:
2912:product state
2909:
2893:
2888:
2884:
2880:
2875:
2871:
2867:
2864:
2842:
2838:
2828:
2826:
2821:
2819:
2815:
2810:
2808:
2792:
2789:
2784:
2780:
2774:
2770:
2743:
2738:
2734:
2705:
2700:
2696:
2682:
2680:
2676:
2655:
2650:
2646:
2617:
2612:
2608:
2584:
2561:
2558:
2553:
2549:
2543:
2539:
2530:
2529:
2528:
2509:
2504:
2500:
2496:
2491:
2486:
2482:
2476:
2472:
2466:
2462:
2458:
2455:
2448:
2447:
2446:
2427:
2422:
2418:
2389:
2384:
2380:
2356:
2353:
2348:
2344:
2321:
2317:
2313:
2308:
2304:
2283:
2276:
2268:
2266:
2249:
2239:
2235:
2231:
2226:
2220:
2210:
2206:
2202:
2177:
2167:
2163:
2159:
2154:
2148:
2138:
2134:
2130:
2105:
2095:
2091:
2087:
2082:
2076:
2066:
2062:
2058:
2053:
2047:
2019:
2011:
2005:
1977:
1969:
1963:
1935:
1927:
1921:
1893:
1885:
1879:
1867:, the states
1852:
1842:
1837:
1833:
1829:
1824:
1820:
1810:
1808:
1803:
1786:
1780:
1775:
1771:
1767:
1762:
1758:
1749:
1728:
1718:
1710:
1707:
1699:
1694:
1677:
1652:
1649:
1644:
1640:
1617:
1613:
1589:
1579:
1575:
1562:
1558:
1546:
1542:
1534:
1530:
1524:
1521:
1518:
1514:
1510:
1499:
1489:
1478:
1473:
1469:
1465:
1460:
1456:
1448:
1447:
1446:
1432:
1429:
1424:
1420:
1396:
1369:
1365:
1342:
1338:
1315:
1311:
1307:
1300:
1296:
1290:
1287:
1284:
1271:
1267:
1236:
1232:
1228:
1221:
1217:
1211:
1208:
1205:
1192:
1188:
1156:
1129:
1125:
1104:
1101:
1094:
1090:
1064:
1053:
1049:
1040:
1032:
1028:
1012:
1008:
998:
994:
988:
985:
982:
978:
974:
968:
956:
955:
954:
937:
924:
920:
916:
911:
909:
905:
901:
897:
893:
892:product state
889:
868:
864:
833:
829:
820:
812:
808:
799:
793:
780:
779:simple tensor
761:
734:
731:
728:
724:
700:
692:
688:
682:
678:
667:
664:
661:
657:
651:
648:
645:
641:
637:
626:
622:
613:
605:
601:
587:
584:
581:
577:
571:
568:
565:
561:
557:
551:
539:
538:
537:
515:
511:
505:
501:
463:
459:
449:
440:
436:
404:
400:
396:
391:
387:
364:
360:
337:
333:
310:
306:
302:
297:
292:
289:
286:
272:
268:
236:
232:
228:
223:
218:
215:
212:
198:
194:
172:
170:
154:
150:
127:
123:
115:
97:
93:
89:
84:
80:
71:
67:
59:
57:
55:
51:
46:
44:
39:
36:
32:
28:
24:
19:
4193:
4189:
4183:
4132:
4128:
4122:
4071:
4067:
4061:
4010:
4006:
4000:
3949:
3945:
3939:
3888:
3884:
3878:
3827:
3823:
3817:
3774:
3770:
3764:
3739:
3735:
3729:
3686:
3682:
3676:
3625:
3621:
3615:
3572:
3568:
3562:
3493:
3489:
3479:
3453:
3416:
3325:
3308:
3304:
3300:
3298:
3284:
3282:
3274:
3181:
3179:
3006:
3004:
2911:
2907:
2829:
2822:
2812:In terms of
2811:
2683:
2674:
2576:
2526:
2272:
2269:Mixed states
1811:
1804:
1802:is nonzero.
1695:
1604:
1079:
912:
907:
903:
899:
895:
891:
887:
778:
715:
176:
65:
63:
47:
40:
34:
26:
20:
18:
3443:Eirik Ovrum
3439:Jan Myrheim
2825:trace class
173:Pure states
43:pure states
4248:Categories
3503:1910.09665
3471:References
3421:, and the
2679:convex set
2577:Otherwise
2296:acting on
4142:0909.0147
3809:119288569
3784:0811.2803
3721:118952931
3530:204824024
3394:⊕
3368:⊕
3342:⊕
3246:ρ
3242:⊗
3239:⋯
3236:⊗
3222:ρ
3202:∑
3195:ρ
3162:⟩
3153:ψ
3144:⊗
3141:⋯
3138:⊗
3135:⟩
3126:ψ
3114:⟩
3111:ψ
3080:∈
3077:⟩
3074:ψ
3039:⊗
3036:⋯
3033:⊗
2973:ρ
2951:ρ
2935:ρ
2885:ρ
2881:⊗
2872:ρ
2865:ρ
2771:∑
2735:ρ
2697:ρ
2647:ρ
2609:ρ
2585:ρ
2540:∑
2501:ρ
2497:⊗
2483:ρ
2463:∑
2456:ρ
2419:ρ
2381:ρ
2354:≥
2314:⊗
2284:ρ
2253:⟩
2224:⟩
2181:⟩
2152:⟩
2109:⟩
2080:⟩
2054:≡
2051:⟩
2048:ψ
2023:⟩
2020:ψ
2012:⊗
2009:⟩
1981:⟩
1970:⊗
1967:⟩
1939:⟩
1928:⊗
1925:⟩
1897:⟩
1886:⊗
1883:⟩
1787:ρ
1781:
1768:≡
1759:ρ
1729:ψ
1726:⟨
1722:⟩
1719:ψ
1708:ρ
1681:⟩
1678:ψ
1645:ψ
1614:ρ
1572:⟨
1568:⟩
1535:ψ
1515:∑
1500:ψ
1497:⟨
1493:⟩
1490:ψ
1479:
1466:≡
1457:ρ
1425:ψ
1400:⟩
1397:ψ
1308:⊂
1301:ψ
1277:⟩
1229:⊂
1222:ψ
1198:⟩
1160:⟩
1157:ψ
1130:ψ
1059:⟩
1041:⊗
1038:⟩
999:ψ
979:∑
972:⟩
969:ψ
941:⟩
938:ψ
908:separable
900:entangled
896:separable
874:⟩
865:ψ
839:⟩
830:ψ
821:⊗
818:⟩
809:ψ
797:⟩
794:ψ
765:⟩
762:ψ
698:⟩
642:∑
632:⟩
614:⊗
611:⟩
562:∑
555:⟩
552:ψ
521:⟩
470:⟩
450:⊗
447:⟩
397:⊗
303:⊂
279:⟩
229:⊂
205:⟩
90:⊗
50:entangled
4175:10523704
4167:19905682
4114:43756465
4106:16486912
4053:28595936
4045:16384285
3992:20897950
3984:11328047
3923:11017309
3870:11664720
3862:11017310
3660:15089173
3607:54893300
3459:See also
917:are not
419:is then
4210:7149863
4147:Bibcode
4086:Bibcode
4025:Bibcode
3964:Bibcode
3931:9948874
3903:Bibcode
3842:Bibcode
3789:Bibcode
3744:Bibcode
3701:Bibcode
3668:5696147
3640:Bibcode
3587:Bibcode
3508:Bibcode
3293:NP-hard
2916:entropy
904:product
54:NP-hard
4208:
4173:
4165:
4112:
4104:
4051:
4043:
3990:
3982:
3929:
3921:
3868:
3860:
3807:
3719:
3666:
3658:
3605:
3528:
2527:where
1172:, and
1080:where
716:where
72:space
4206:S2CID
4171:S2CID
4137:arXiv
4110:S2CID
4076:arXiv
4049:S2CID
4015:arXiv
3988:S2CID
3954:arXiv
3927:S2CID
3893:arXiv
3866:S2CID
3832:arXiv
3805:S2CID
3779:arXiv
3717:S2CID
3691:arXiv
3664:S2CID
3630:arXiv
3603:S2CID
3577:arXiv
3526:S2CID
3498:arXiv
3431:image
3309:2 X 3
3305:2 X 2
2035:with
851:with
4163:PMID
4102:PMID
4041:PMID
3980:PMID
3919:PMID
3858:PMID
3656:PMID
3441:and
3307:and
2723:and
2635:and
2407:and
1430:>
1357:and
1251:and
1102:>
919:pure
906:and
352:and
251:and
177:Let
142:and
4198:doi
4155:doi
4133:103
4094:doi
4033:doi
3972:doi
3911:doi
3850:doi
3797:doi
3775:474
3752:doi
3709:doi
3648:doi
3595:doi
3516:doi
3287:in
2910:or
2193:or
953:as
925:of
21:In
4250::
4218:^
4204:.
4194:59
4192:.
4169:.
4161:.
4153:.
4145:.
4131:.
4108:.
4100:.
4092:.
4084:.
4072:96
4070:.
4047:.
4039:.
4031:.
4023:.
4011:95
4009:.
3986:.
3978:.
3970:.
3962:.
3950:86
3948:.
3925:.
3917:.
3909:.
3901:.
3889:84
3887:.
3864:.
3856:.
3848:.
3840:.
3828:84
3826:.
3803:.
3795:.
3787:.
3773:.
3750:.
3740:82
3738:.
3715:.
3707:.
3699:.
3687:70
3685:.
3662:.
3654:.
3646:.
3638:.
3626:92
3624:.
3601:.
3593:.
3585:.
3573:68
3571:.
3550:^
3538:^
3524:.
3514:.
3506:.
3492:.
3488:.
3437:,
3319:,
3299:A
2918:,
2809:.
2793:1.
2681:.
2562:1.
2369:,
2250:10
2221:01
2178:11
2149:00
1951:,
1909:,
1772:Tr
1470:Tr
56:.
25:,
4212:.
4200::
4177:.
4157::
4149::
4139::
4116:.
4096::
4088::
4078::
4055:.
4035::
4027::
4017::
3994:.
3974::
3966::
3956::
3933:.
3913::
3905::
3895::
3872:.
3852::
3844::
3834::
3811:.
3799::
3791::
3781::
3758:.
3754::
3746::
3723:.
3711::
3703::
3693::
3670:.
3650::
3642::
3632::
3609:.
3597::
3589::
3579::
3532:.
3518::
3510::
3500::
3494:2
3397:2
3391:2
3371:n
3365:1
3345:1
3339:1
3260:.
3255:k
3250:n
3231:k
3226:1
3216:k
3212:p
3206:k
3198:=
3182:H
3165:.
3157:n
3148:|
3130:1
3121:|
3117:=
3107:|
3083:H
3070:|
3047:n
3043:H
3028:1
3024:H
3020:=
3017:H
3007:n
2985:.
2982:)
2977:2
2969:(
2966:S
2963:+
2960:)
2955:1
2947:(
2944:S
2941:=
2938:)
2932:(
2929:S
2894:,
2889:2
2876:1
2868:=
2843:k
2839:p
2790:=
2785:k
2781:p
2775:k
2749:}
2744:k
2739:2
2731:{
2711:}
2706:k
2701:1
2693:{
2661:}
2656:k
2651:2
2643:{
2623:}
2618:k
2613:1
2605:{
2559:=
2554:k
2550:p
2544:k
2510:k
2505:2
2492:k
2487:1
2477:k
2473:p
2467:k
2459:=
2433:}
2428:k
2423:2
2415:{
2395:}
2390:k
2385:1
2377:{
2357:0
2349:k
2345:p
2322:2
2318:H
2309:1
2305:H
2246:|
2240:3
2236:/
2232:2
2227:+
2217:|
2211:3
2207:/
2203:1
2174:|
2168:2
2164:/
2160:1
2155:+
2145:|
2139:2
2135:/
2131:1
2106:1
2102:|
2096:3
2092:/
2088:2
2083:+
2077:0
2073:|
2067:3
2063:/
2059:1
2044:|
2016:|
2006:0
2002:|
1978:1
1974:|
1964:1
1960:|
1936:1
1932:|
1922:0
1918:|
1894:0
1890:|
1880:0
1876:|
1853:2
1848:C
1843:=
1838:2
1834:H
1830:=
1825:1
1821:H
1790:)
1784:(
1776:B
1763:A
1733:|
1715:|
1711:=
1674:|
1653:1
1650:=
1641:r
1618:A
1590:.
1586:|
1580:k
1576:u
1563:k
1559:u
1554:|
1547:k
1543:p
1531:r
1525:1
1522:=
1519:k
1511:=
1508:)
1504:|
1486:|
1482:(
1474:B
1461:A
1433:1
1421:r
1393:|
1370:2
1366:H
1343:1
1339:H
1316:2
1312:H
1297:r
1291:1
1288:=
1285:k
1281:}
1272:k
1268:v
1263:|
1259:{
1237:1
1233:H
1218:r
1212:1
1209:=
1206:k
1202:}
1193:k
1189:u
1184:|
1180:{
1153:|
1126:r
1105:0
1095:k
1091:p
1065:,
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1054:k
1050:v
1045:|
1033:k
1029:u
1024:|
1020:(
1013:k
1009:p
995:r
989:1
986:=
983:k
975:=
965:|
934:|
888:i
869:i
860:|
834:2
825:|
813:1
804:|
800:=
790:|
758:|
735:j
732:,
729:i
725:c
701:,
693:j
689:b
683:i
679:a
674:|
668:j
665:,
662:i
658:c
652:j
649:,
646:i
638:=
635:)
627:j
623:b
618:|
606:i
602:a
597:|
593:(
588:j
585:,
582:i
578:c
572:j
569:,
566:i
558:=
548:|
524:}
516:j
512:b
506:i
502:a
497:|
493:{
473:}
464:j
460:b
454:|
441:i
437:a
431:|
427:{
405:2
401:H
392:1
388:H
365:2
361:H
338:1
334:H
311:2
307:H
298:m
293:1
290:=
287:j
283:}
273:j
269:b
263:|
259:{
237:1
233:H
224:n
219:1
216:=
213:i
209:}
199:i
195:a
189:|
185:{
155:2
151:H
128:1
124:H
98:2
94:H
85:1
81:H
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