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A characterization of positive linear maps and criteria of entanglement for quantum states. (English) Zbl 1198.81055

Summary: Let \(H\) and \(K\) be (finite- or infinite-dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from \(\mathcal B(H)\) into \(\mathcal B(K)\) is given, which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras. This characterization is then applied to give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion for separability is given which shows that a state \(\rho \) on \(H \otimes K\) is separable if and only if (\(\Phi \otimes I)\rho \geqslant 0\) for all positive finite-rank elementary operators \(\Phi \). Examples of NCP and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the PPT criterion and the realignment criterion.

MSC:

81P40 Quantum coherence, entanglement, quantum correlations
81P15 Quantum measurement theory, state operations, state preparations
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[1] Bengtsson I and Zyczkowski K 2006 (Cambridge: Cambridge University Press)
[2] Nielsen M A and Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press)
[3] Werner R F 1989 Phys. Rev. A 40 4277 · Zbl 1371.81145
[4] Bennett C H, Brassard G, Crépeau C, Jozsa R, Peres A and Wootters W K 1993 Phys. Rev. Lett.70 1895 · Zbl 1051.81505
[5] Bouwmeester D, Pan J W, Mattle K, Eibl M, Weinfurter H and Zeilinger A 1997 Nature390 575 · Zbl 1369.81006
[6] Deutsch D, Ekert A, Jozsa R, Macchiavello C, Popescu S and Sanpera A 1996 Phys. Rev. Lett.77 2818
[7] Deutsch D, Ekert A, Jozsa R, Macchiavello C, Popescu S and Sanpera A 1998 Phys. Rev. Lett.80 2022
[8] Shor P W 1995 Phys. Rev. A 52 2493
[9] Horodecki M, Horodecki P and Horodecki R 1996 Phys. Lett. A 223 1-8 · Zbl 1037.81501
[10] Peres A 1996 Phys. Lett. A 202 16 · Zbl 1020.81540
[11] Horodecki M, Horodecki P and Horodecki R 1998 Phys. Lett. A 80 5239 · Zbl 0947.81005
[12] Chen K and Wu L A 2003 Quantum Inf. Comput.3 193
[13] Horodecki M and Horodecki P 1999 Phys. Rev. A 59 4206
[14] Cerf N J, Adami C and Gingrich R M 1999 Phys. Rev. A 60 893
[15] Horodecki R, Horodecki P and Horodecki M 2009 Rev. Mod. Phys.81 865 · Zbl 1205.81012
[16] Hou J C and Qi X F 2010 Phys. Rev. A 81 062351
[17] Salgado D and Sánchez-Gómez J L 2005 Open Syst. Inf. Dyn.12 55-64 · Zbl 1067.81529
[18] Ł Skowronek, Størmer E and Życzkowski K 2009 J. Math. Phys.50 062106 (doi:10.1063/1.3155378) · Zbl 1216.46052
[19] Grabowski J, Kuś M and Marmo G 2007 Open. Syst. Inf. Dyn.14 355-70 · Zbl 1137.81005
[20] Kye S-H 2003 Trends in Mathematics Inf. Center Math. Sci.6 83-91
[21] Kadison R V and Ringrose J R 1983 Fundamentals of the Theory of Operator Algebras II(Graduate Studies in Math. vol 16) (New York: Academic)
[22] Størmer E 2008 J. Funct. Anal.254 2304-13 · Zbl 1143.46033
[23] Dixmier J 1981 Von Neumann Algebras (Amsterdan: North-Holland)
[24] Chruściński D and Kossakowski A 2007 Open Syst. Inf. Dyn.14 275 · Zbl 1129.81019
[25] Chruściński D and Kossakowski A 2008 J. Phys. A: Math. Theor.41 145301 · Zbl 1136.81005
[26] Augusiak R and Stasi¡änska J 2008 Phys. Rev. A 77 010303
[27] Choi M D 1975 Linear Alg. Appl.10 285-90 · Zbl 0327.15018
[28] Choi M D 1980 J. Operator Theory4 271-85
[29] Choi M D 1982 Proc. Symp. Pure Math.38 583-90
[30] Depillis J 1967 Pac. J. Math.23 129-37 · Zbl 0166.30003
[31] Hou J C 1998 J. Operator Theory39 43-58
[32] Paulsen V 2002 Completely Bounded Maps and Operator Algebras(Cambridge Studies in Advanced Mathematics vol 78) (Cambridge: Cambridge University Press)
[33] Stinespring W F 1955 Proc. Am. Math. Soc.6 211-6
[34] Hou J C 1989 Sci. China A 32 929-40
[35] Mathieu M 1989 Math. Ann.284 223-44 · Zbl 0648.46052
[36] Effros E G and Ruan Z-J 2000 Operator Spaces (Oxford: Clarendon)
[37] Hou J C 1993 Sci. China A 36 1025-35
[38] Sperling J and Vogel W 2009 Phys. Rev. A 79 052313
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