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Foundations of Physics

Probability Description and Entropy of Classical and Quantum Systems

Margarita A. Man’ko, Vladimir I. Man’ko

Published Dec 29, 2009• 63 citations
Publisher
DOI
10.1007/s10701-009-9403-9

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We consider the description of open quantum systems with probability sinks (or sources) in terms of general non-Hermitian Hamiltonians. Within such a framework, we study novel possible definitions of the quantum linear entropy as an indicator of the flow of information during the dynamics. Such linear entropy functionals are necessary in the case of a partially Wigner-transformed non-Hermitian Hamiltonian (which is typically useful within a mixed quantum-classical representation). Both the case of a system represented by a pure non-Hermitian Hamiltonian as well as that of the case of non-Hermitian dynamics in a classical bath are explicitly considered.
We present a classical probability model appropriate to the description of quantum randomness. This tool, that we have called stochastic gauge system, constitutes a contextual scheme in which the Kolmogorov probability space depends upon the experimental setup, in accordance with quantum mechanics. Therefore, the probability space behaves like a gauge parameter. We discuss the technical issues of this theory and apply the concept to classically emulate quantum entangled states and even `super-quantum' systems. We exhibit bipartite examples leading to maximum violation of Bell-CHSH inequalities like EPR pairs or exceeding the Tsirelson bound like PR-boxes, as well as tripartite cases simulating GHZ or W-states. We address also the question of partially correlated systems and multipartite entanglements. In this model, the classical equivalent of the entanglement entropy is identified with the Kullback-Leibler divergence. Hence, we propose a natural generalisation of this function to multipartite systems, leading to a simple evaluation of the degree of entanglement and determining the bounds of maximum entanglement. Finally, we obtain a constructive necessary and sufficient condition of multipartite entanglement.
Understanding wave-packet deformation and spreading is central to the description of transport and localization phenomena in both classical and quantum systems. In this work, we introduce information-theoretic and complexity-based indicators. We analyze the evolution of the probability distribution using Shannon and linear entropies, together with the inverse participation ratio, to quantify localization–delocalization processes and their associated fluctuations. These measures naturally motivate the use of statistical complexity definitions, including the López–Mancini–Calbet (CLMC) and Shiner–Davison–Landsberg (CSDL) complexities. In addition, we compute the probability current for both single-particle and two-particle configurations, providing a dynamical description of transport in terms of particle motion. The influence of an external field on entropy production, complexity, and probability currents is also examined, highlighting the interplay between information spreading, localization, and transport in lattice systems.