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AIP conference proceedings

Complete positivity in quantum open system dynamics

Fabio Benatti

Published Jan 1, 2001• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.1388685

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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.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, a description of the continuous-variable entanglement for a system consisting of two independent harmonic oscillators interacting with a general environment was given. Using Peres-Simon necessary and sufficient criterion for separability of two-mode Gaussian states, the generation and evolution of entanglement in terms of the covariance matrix for an arbitrary Gaussian input state was described. For some values of diffusion and dissipation coefficients describing the environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation, entanglement sudden death or a periodic collapse and revival of entanglement take place. The time evolution of the logarithmic negativity, which characterizes the degree of entanglement of the quantum state was analyzed.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we study the continuous variable entanglement for a system consisting of two independent harmonic oscillators interacting with a general environment. We solve the Kossakowski-Lindblad master equation for the time evolution of the considered system and describe the entanglement in terms of the covariance matrix for an arbitrary Gaussian input state. Using Peres–Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we show that for certain values of diffusion and dissipation coefficients describing the environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation, entanglement sudden death or a periodic collapse and revival of entanglement take place. We analyze also the time evolution of the logarithmic negativity, which characterizes the degree of entanglement of the quantum state.
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.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the dynamics of entanglement for a system consisting of two uncoupled harmonic oscillators interacting with a thermal environment. Using Peres-Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we describe the evolution of entanglement in terms of the covariance matrix for a Gaussian input state. For some values of the temperature of environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation, entanglement sudden death or a repeated collapse and revival of entanglement take place. We analyze also the time evolution of the logarithmic negativity, which characterizes the degree of entanglement of the quantum state.