E C G Sudarshan and Anil Shaji 2003 J. Phys. A: Math. Gen. 36 5073 doi:10.1088/0305-4470/36/18/312
E C G Sudarshan1 and Anil Shaji2
Show affiliationsThe most general evolution of the density matrix of a quantum system with a finite-dimensional state space is by stochastic maps which take a density matrix linearly into the set of density matrices. These dynamical stochastic maps form a linear convex set that may be viewed as supermatrices. The property of Hermiticity of density matrices renders an associated supermatrix Hermitian and hence diagonalizable. The positivity of the density matrix does not make the associated supermatrix positive though. If the map itself is positive, it is called completely positive and it has a simple parametrization. This is extended to all positive (not completely positive) maps. A general dynamical map that does not preserve the norm of the density matrices it acts on can be thought of as the contraction of a norm-preserving map of an extended system. The reconstruction of such extended dynamics is also given.
03.65.Ta Foundations of quantum mechanics; measurement theory
15A57 Other types of matrices (Hermitian, skew-Hermitian, etc.)
81S25 Quantum stochastic calculus
15A48 Positive matrices and their generalizations; cones of matrices
Issue 18 (9 May 2003)
Received 9 January 2003, in final form 5 March 2003
Published 23 April 2003
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