Positive maps, states, entanglement and all that; some old and new problems

We outline a new approach to the characterization as well as to the classification of positive maps. This approach is based on the facial structures of the set of states and of the cone of positive maps. In particular, the equivalence between Schroedinger's and Heisenberg's pictures is reviewed in this more general setting. Furthermore, we discuss in detail the structure of positive maps for two and three dimensional systems. In particular, the explicit form of decomposition of a positive map and the uniqueness of this decomposition for extremal positive maps for 2 dimensional case are described. The difference of the structure of positive maps between 2 dimensional and 3 dimensional cases is clarified. The resulting characterization of positive maps is applied to the study of quantum correlations and entanglement

[1]  W. Stinespring.  Positive functions on *-algebras , 1955 .

[2]  E. Størmer.  Positive linear maps of operator algebras , 2012 .

[3]  Richard V. Kadison,et al.  Transformations of states in operator theory and dynamics , 1965 .

[4]  P. Dirac.  QUANTUM ELECTRODYNAMICS WITHOUT DEAD WOOD , 1965 .

[5]  William Arveson,et al.  Subalgebras ofC*-algebras , 1969 .

[6]  W. Arveson.  On subalgebras of $C^*$-algebras , 1969 .

[7]  A. Jamiołkowski.  Linear transformations which preserve trace and positive semidefiniteness of operators , 1972 .

[8]  Man-Duen Choi.  A schwarz inequality for positive linear maps on $C^{\ast}$-algebras , 1974 .

[9]  A. Connes.  Caractérisation des espaces vectoriels ordonnés sous-jacents aux algèbres de von Neumann , 1974 .

[10]  S. Woronowicz.  Positive maps of low dimensional matrix algebras , 1976 .

[11]  C*-Algebras and Applications to Physics , 1978 .

[12]  E. Effros.  Aspects of non-commutative order , 1978 .

[13]  O. Bratteli.  Operator Algebras And Quantum Statistical Mechanics , 1979 .

[14]  T. Andô.  Concavity of certain maps on positive definite matrices and applications to Hadamard products , 1979 .

[15]  Some remarks on C∗-convexity , 1981 .

[16]  J. Ward,et al.  The geometric structure of generalized state spaces , 1981 .

[17]  E. Størmer.  DECOMPOSABLE POSITIVE MAPS ON C*-ALGEBRAS , 1982 .

[18]  R. Kadison,et al.  Fundamentals of the Theory of Operator Algebras , 1983 .

[19]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[20]  P. Muhly,et al.  On Isometries of Operator Algebras , 1994 .

[21]  Pérès.  Separability Criterion for Density Matrices. , 1996, Physical review letters.

[22]  Seung-Hyeok Kye.  BOUNDARIES OF THE CONE OF POSITIVE LINEAR MAPS AND ITS SUBCONES IN MATRIX ALGEBRAS , 1996 .

[23]  M. Horodecki,et al.  Separability of mixed states: necessary and sufficient conditions , 1996, quant-ph/9605038.

[24]  Seung-Hyeok Kye.  Facial Structures for the Positive Linear Maps Between Matrix Algebras , 1996, Canadian Mathematical Bulletin.

[25]  Does quantum chaos exist? (A quantum Lyapunov exponents approach.) , 1998, quant-ph/9805068.

[26]  W. Majewski,et al.  On a characterization of positive maps , 2001 .

[27]  F. Shultz,et al.  State Spaces of Operator Algebras , 2001 .

[28]  W. Majewski.  On entanglement of states and quantum correlations , 2002, math-ph/0202030.

[29]  F. Shultz,et al.  Geometry Of State Spaces Of Operator Algebras , 2002 .

[30]  M. Keyl.  Fundamentals of quantum information theory , 2002, quant-ph/0202122.

[31]  Kil-Chan Ha,et al.  Entangled states with positive partial transposes arising from indecomposable positive linear maps , 2003, quant-ph/0305005.

[32]  W. Majewski,et al.  On k-decomposability of positive maps , 2003, quant-ph/0411035.

[33]  Andrzej Kossakowski,et al.  A Class of Linear Positive Maps in Matrix Algebras , 2003, Open Syst. Inf. Dyn..

[34]  On Quantum Correlations and Positive Maps , 2004, math-ph/0403024.

[35]  Wladyslaw A. Majewski.  On Positive Maps, Entanglement and Quantization , 2004, Open Syst. Inf. Dyn..

[36]  Andrzej Kossakowski,et al.  A Class of Linear Positive Maps in Matrix Algebras II , 2004, Open Syst. Inf. Dyn..

[37]  Facial Structures for Decomposable Positive Linear Maps in Matrix Algebras , 2005 .

[38]  K. R. Parthasarathy,et al.  Mathematical Foundation of Quantum Mechanics , 2005 .

[39]  Decomposability of extremal positive maps on $M_2$ , 2005 .