A Note on Positive Maps and Classification of States

The relation between positive maps on Mn and subsets of self-adjoint elements from Mn ⊗ Mn is investigated. It has been shown that there exists the isomorphism between the set of all positive normalized maps on Mn and the subset of separable states on Mn ⊗ Mn.

[1]  Jun Tomiyama,et al.  Indecomposable Positive Maps in Matrix Algebras , 1988, Canadian Mathematical Bulletin.

[2]  A. Robertson,et al.  Positive extensions of automorphisms of spin factors , 1983, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

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

[4]  A. Robertson.  Positive Projections on C*-Algebras and an Extremal Positive Map , 1985 .

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

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

[7]  Karol Zyczkowski,et al.  On Duality between Quantum Maps and Quantum States , 2004, Open Syst. Inf. Dyn..

[8]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[9]  K. Lendi,et al.  Quantum Dynamical Semigroups and Applications , 1987 .

[10]  A. Jamiołkowski.  An effective method of investigation of positive maps on the set of positive definite operators , 1974 .

[11]  H. Osaka.  Indecomposable positive maps in low dimensional matrix algebras , 1991 .

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

[13]  P. M. Mathews,et al.  STOCHASTIC DYNAMICS OF QUANTUM-MECHANICAL SYSTEMS , 1961 .

[14]  Kil-Chan Ha.  A class of atomic positive linear maps in matrix algebras , 2003 .

[15]  Man-Duen Choi.  Completely positive linear maps on complex matrices , 1975 .

[16]  E. Sudarshan,et al.  Extreme affine transformations , 1976 .

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

[18]  G. Kimura.  The Bloch Vector for N-Level Systems , 2003, quant-ph/0301152.

[19]  B. M. Fulk.  MATH , 1992 .

[20]  S. Woronowicz.  Nonextendible positive maps , 1976 .

[21]  A. Guyan Robertson.  Schwarz inequalities and the decomposition of positive maps on C *-algebras , 1983 .

[22]  Kadison,et al.  Isometries of Operator Algebras , 1951 .

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

[24]  B. Terhal.  A family of indecomposable positive linear maps based on entangled quantum states , 1998, quant-ph/9810091.

[25]  Seung-Hyeok Kye,et al.  DUALITY FOR POSITIVE LINEAR MAPS IN MATRIX ALGEBRAS , 2000 .

[26]  H. Osaka.  A Class of Extremal Positive Maps in 3×3 Matrix Algebras , 1992 .

[27]  J. Pillis.  Linear transformations which preserve hermitian and positive semidefinite operators. , 1967 .

[28]  Werner,et al.  Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. , 1989, Physical review. A, General physics.

[29]  Erling Størmer,et al.  Extension of positive maps into B (H) , 1986 .

[30]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[31]  Seung-Hyeok Kye,et al.  Generalized Choi maps in three-dimensional matrix algebra , 1992 .

[32]  Andrzej Jamiolkowski.  Errata: On a Stroboscopic Approach to Quantum Tomography of Qudits Governed by Gaussian Semigroups - Open Sys. Information Dyn. 11, 63 (2003) , 2004, Open Syst. Inf. Dyn..

[33]  Masanori Ohya,et al.  How Can We Observe and Describe Chaos? , 2003, Open Syst. Inf. Dyn..

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

[35]  E. Størmer.  On the Jordan structure of *-algebras , 1965 .

[36]  Kil-Chan Ha,et al.  Atomic positive linear maps in matrix algebras , 1998 .

[37]  Erling Størmer.  Decomposition of positive projections onC*-algebras , 1980 .

[38]  L. Gurvits,et al.  Largest separable balls around the maximally mixed bipartite quantum state , 2002, quant-ph/0204159.

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

[40]  Hiroyuki Osaka,et al.  A series of absolutely indecomposable positive maps in matrix algebras , 1993 .

[41]  Man-Duen Choi.  Positive semidefinite biquadratic forms , 1975 .

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