A property (C) for permutation pairs is introduced. It is shown that if a pair {π1, π2} of permutations of (1, 2, …, n) has property (C), then the D-type map Φπ1, π2 on n×n complex matrices constructed from {π1, π2} is positive. A necessary and sufficient condition is obtained for a pair {π1, π2} to have property (C), and an easily checked necessary and sufficient condition for the pairs of the form {πp, πq} to have property (C) is given, where π is the permutation defined by π(i)=i + 1 mod n and 1 ≤ p < q ≤ n.
Jinchuan HOU
,
Haili ZHAO
. POSITIVE MAPS CONSTRUCTED FROM PERMUTATION PAIRS[J]. Acta mathematica scientia, Series B, 2019
, 39(1)
: 148
-164
.
DOI: 10.1007/s10473-019-0113-x
[1] Chefles A, Jozsa R. Winter A, On the existence of physical transformations between sets of quantum states. International J Quantum Information, 2004, 2:11-21
[2] Horn R A, Johnson C R. Matrix Analysis. New York:Cambridge Univ Press, 1985
[3] Hou J C, Li C K, Poon Y T, et al. A new criterion and a special class of k-positiv maps. Lin Alg Appl, 2015, 470:51-69
[4] Huang Z J, Li C K, Poon E, et al. Physical transformation between quantum states. arXiv:1203.5547
[5] Kraus K. States, Effects, and Operations:Fundamental Notions of Quantun Theory. Lecture Notes in Physics, 190. Berlin:Spring-Verlag, 1983
[6] Albert P, Uhlmanm A. A problem relating to positive linear maps on matrix algebras. Rep Math Phys, 1980, 18:163
[7] Augusiak R, Bae J, Czekaj L, et al. On structural physical approximations and entanglement breaking maps. J Phys A:Math Theor, 2011, 44:185308
[8] Choi M D. Completely Positive Linear Maps on Complex Matrix. Lin Alg Appl, 1975, 10:285-290
[9] Chruściński D, Kossakowski A. Spectral conditions for positive maps. Comm Math Phys, 2009, 290:1051
[10] Hou J C. A characterization of positive linear maps and criteria for entangled quantum states. J Phys A:Math Theor, 2010, 43:385201
[11] Hou J C. Acharacterization of positive elementary operators. J Operator Theory, 1996, 39:43-58
[12] Li C K, Poon Y T. Interpolation by Completely Positive Maps. Linear Multilinear Algebra, 2011, 59:1159-1170
[13] Nielsen M A, Chuang I L. Quantum Computation and Quantum Information. Cambridge:Cambridge University Press, 2000
[14] Horodecki M, Horodecki P, Horodecki R. Separability of mixed states:necessary and sufficient conditions. Phys Lett A, 1996, 223:1
[15] Qi X F, Hou J C. Positive finite rank elementary operators and characterizing entanglement of states. J Phys A:Math Theor, 2011, 44:215305
[16] Qi X F, Hou J C. Characterization of optimal entanglement witnesses. Phy Rev A, 2012, 85:022334
[17] Yan S Q, Hou J C. LPP elementary operator criterion of full separability for states in multipartite quantum systems. J Phys A:Math Theor, 2012, 45:435303
[18] Li X S, Gao X H, Fei S M. Lower bound of concurrence based on positive maps. Phys Rev A, 2011, 83:034303
[19] Qin H H, Fei S M. Lower bound of concurrence based on generalized positive maps. Commun Theor Phys, 2013, 60(12):663-666
[20] Qi X F, Hou J C. Optimality of entanglement witnesses constructed from arbitrary permutations. Quantum Information Processing, 2015, 14:2499-2515
[21] Qi X F, Hou J C. Indecomposability of entanglement witnesses constructed from any permutations. Quantum Information and Computation, 2015, 15(5/6):0478-0488
[22] Yamagami S. Cyclic inequalities. Proc Amer Math Soc, 1993, 118:521-527
[23] Duan Z B, Niu L F, Zhao H L. Lower bound of concurrence based on D-type poeitive maps. preprint.
[24] Zhao H L, Hou J C. A necessary and sufficient condition for positivity of linear maps on M4 constructed from permutation pairs. Operators and Matrices, 2015, 9(3):597-617
