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Separability of \(n\)-particle mixed states: necessary and sufficient conditions in terms of linear maps. (English) Zbl 0984.81007
Summary: We provide necessary and sufficient conditions for separability of mixed states of \(n\)-particle systems. The conditions are formulated in terms of maps which are positive on product states of \(n-1\) particles. The method of construction of the maps on the basis of unextendible product bases is provided. The three-qubit problem is discussed in more detail, involving Bloch sphere parameters of qubit state.

MSC:
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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[1] Einstein, A; Podolsky, B; Rosen, N, Phys. rev., 47, 777, (1935)
[2] Schrödinger, E, Naturwissenschaften, 23, 807, (1935)
[3] Peres, A, Phys. rev. lett., 77, 1413, (1996)
[4] Horodecki, M; Horodecki, P; Horodecki, R, Phys. lett. A, 223, 8, (1996)
[5] Horodecki, P, Phys. lett. A, 232, 333, (1997)
[6] Choi, M.D, Proc. sympos. pure math., 38, 583, (1982)
[7] Lewenstein, M; Kraus, B; Cirac, J.I; Horodecki, P, Phys. rev. A, 62, 052310, (2000)
[8] Lewenstein, M; Kraus, B; Horodecki, P; Cirac, J.I
[9] Horodecki, M; Horodecki, P; Horodecki, R, Phys. rev. lett., 78, 574, (1997)
[10] Horodecki, M; Horodecki, P; Horodecki, R, Phys. rev. lett., 80, 5239, (1998)
[11] Horodecki, P; Horodecki, M; Horodecki, R, Phys. rev. lett., 82, 1046, (1999)
[12] Terhal, B.M
[13] Vedral, V; Plenio, M.B; Rippin, M.A; Knight, P.L, Phys. rev. lett., 78, 2275, (1997)
[14] Vedral, V; Plenio, M.B; Jacobs, K; Knight, P.L, Phys. rev. A, 57, 4452, (1997)
[15] Werner, R.F, Phys. rev. A, 40, 4277, (1989) · Zbl 1371.81145
[16] Murao, M; Plenio, M.B; Popescu, S; Vedral, V; Knight, P.L, Phys. rev. A, 57, R4075, (1998)
[17] Dür, W; Cirac, J.I; Tarrach, R, Phys. rev. lett., 83, 3562, (1999)
[18] Steane, A, Rep. prog. phys., 61, 117, (1998)
[19] Życzkowski, K; Horodecki, P; Sanpera, A; Lewenstein, M, Phys. rev. A, 58, 833, (1998)
[20] Vidal, G; Tarrach, R, Phys. rev. A, 59, 141, (1999)
[21] Braunstein, S; Caves, C.M; Jozsa, R; Lindena, N; Popescu, S, Phys. rev. lett., 83, 1054, (1999)
[22] Schack, R; Caves, C, Phys. rev. A, 60, 4354, (1999)
[23] Linden, N; Popescu, S
[24] Linden, N; Popescu, S, Fortschr. phys., 49, 567, (1998)
[25] Linden, N; Popescu, S; Sudbery, A, Phys. rev. lett., 83, 243, (1999)
[26] Kempe, J, Phys. rev. A, 60, 910, (1999)
[27] Bennett, C.H; Popescu, S; Rohlich, D; Smolin, J; Thaplyial, A
[28] Linden, N; Popescu, S; Schumacher, B; Westmoreland, W
[29] Dür, W
[30] Galvaro, E.F; Plenio, M.B; Virmani, S
[31] Acin, A; Adrianov, A; Jané, E; Tarrach, R
[32] Brassard, G; Mor, T, (), 1-9
[33] Bennett, C.H; DiVincenzo, D.P; Mor, T; Shor, P.W; Smolin, J.A; Terhal, B.M, Phys. rev. lett., 83, 3081, (1999)
[34] DiVincenzo, D.P; Mor, T; Shor, P.W; Smolin, J.A; Terhal, B.M
[35] Thapliyal, A, Phys. rev. A, 59, 3336, (1999)
[36] Dür, W; Cirac, J.I, Phys. rev. A, 61, 042314, (2000)
[37] Murao, M; Jonathan, D; Plenio, M.B; Vedral, V, Phys. rev. A, 59, 156, (1999)
[38] Janzing, D; Beth, T
[39] Aharonov, D
[40] Eggeling, T; Werner, R.F
[41] Smolin, J.A
[42] Shor, P.W; Smolin, J.A; Thaplyial, A.W
[43] Murao, M; Vedral, V, Phys. rev. lett., 86, 352, (2001)
[44] Terhal, B.M, Phys. lett. A, 271, 319, (2000)
[45] Jamiołkowski, A, Rep. math. phys., 3, 275, (1972)
[46] Woronowicz, S.L, Rep. math. phys., 10, 165, (1976)
[47] B. Terhal, private communication
[48] Wu, S; Chen, X; Zhang, Y
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