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arXiv:quant-ph/0310151

Complete positivity and dissipative factorized dynamics: some comments

Fabio Benatti, Roberto Floreanini B, Marco Piani, Romano, R.

Published Oct 24, 2003• 4 citations
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Abstract

After reviewing the main properties of time-evolutions of open quantum systems, some considerations about the positivity of factorized Markovian dynamics for bipartite systems are made. In particular, it is shown that the positivity of the whole time-evolution in general does not ask for the complete positivity of the single system time-evolutions, if they are allowed to differ. However, they must be completely positive if one is a small perturbation of the other, which is the typical situation for open systems in a heat bath.

DOI
10.48550/arxiv.quant-ph/0310151

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The statistical interpretation of quantum mechanics requires that, while evolving in time, states of quantum systems keep a positive spectrum of eigenvalues. That is, the time-evolution must be positivity preserving or, in short, positive. When dealing with dissipative quantum systems, positivity is no longer sufficient. The presence of entangled states makes it necessary that the time-evolution be completely positive. In this paper we discuss this latter notion in relation to simple positivity and quantum entanglement.
We study the concepts of complete positivity and positivity in a two-level open quantum system whose dynamics are governed by a time-local quantum master equation. We establish necessary and sufficient conditions on the time-dependent relaxation rates to ensure complete positivity and positivity of the dynamical map. We discuss their relations with the non-Markovian behavior of the open system. We also analyze the long-time asymptotic behavior of the dynamics as a function of the rates. We show under which conditions on the rates the system tends to an equilibrium state. Different examples illustrate this general study.