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Cybernetics & Systems

OPEN QUANTUM SYSTEMS AND COMPLETE POSITIVITY

Roberto Floreanini B

Published Apr 1, 2001• 7 citations
Publisher

Abstract

The dynamics of an open quantum system is usually assumed to be generated by time-evolution equations of Lindblad form. It follows that the time evolutors are completely positive. Far from being just a useful technicality, complete positivity is one of the many aspects of quantum entanglement and is necessary to avoid physical inconsistencies. This issue is addressed by examining the open dynamics of two level systems in situations typical of chemical physics and quantum optics.

DOI
10.1080/01969720151033490

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The dynamics of an open quantum system is usually assumed to be generated by time-evolution equations of Lindblad form. It follows that the time evolutors are completely positive. Far from being just a useful technicality, complete positivity is one of the many aspects of quantum entanglement and is necessary to avoid physical inconsistencies. This issue is addressed by examining the open dynamics of two level systems in situations typical of chemical physics and quantum optics.
We review the standard treatment of open quantum systems in relation to quantum entanglement, analyzing, in particular, the behavior of bipartite systems immersed in the same environment. We first focus upon the notion of complete positivity, a physically motivated algebraic constraint on the quantum dynamics, in relation to quantum entanglement, i.e. the existence of statistical correlations which can not be accounted for by classical probability. We then study the entanglement power of heat baths versus their decohering properties, a topic of increasing importance in the framework of the fast developing fields of quantum information, communication and computation. The presentation is self contained and, through several examples, it offers a detailed survey of the physics and of the most relevant and used techniques relative to both quantum open system dynamics and quantum entanglement.
The title may be a bit misleading. Perhaps, "On the Complete Positivity of Reduced Quantum Dynamics," would be a more fitting title. Determining whether or not completely positive (CP) maps are required to describe open system quantum dynamics is an extremely important issue concerning the fundamental mathematical foundations of QM, as well as many other areas of physics. it had been typically believed that only CP maps actually describe the dynamical evolution of open quantum systems, however there has been speculation as to whether this is a strict constraint on the mathematical and physical structure of stochastic quantum dynamical maps. The objective of this thesis is to demonstrate that given a particular unitary operator, an initial system state, a final system state, and the dimension of the environment state, there exists no CP map with a composite system-environment, product initial state that is compatible with the given constraints on the reduced quantum dynamics of the system under investigation.
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.
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.
The majority of quantum open system models in the literature are simplistic in the sense that they only explicitly account for that part of the environment that directly interacts with the system of interest. A quantum open system with an open environment is examined using the projection operator method in the weak coupling limit. The openness of environment is modelled by nonunitary evolution of the Lindblad form. Under certain conditions, the resulting master equation for the system is insensitive to the initial state of the environment and to initial entanglements between the system and environment for time scales greater than the environment relaxation timescales. For the particular case of an environment consisting of a harmonic oscillator bath, the resulting master equations are demonstrated to have the algebraic form for completely positive evolution. The open environment model is illustrated for the particular case of a system linearly coupled to an oscillator bath.
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.