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Information Transmission in Quantum Open Systems

Masanori Ohya, Hiroki Suyari

Published May 20, 1996• 0 citations
OpenAlex

Abstract

There are several damping phenomena in quantum optics.Such phenomena have been usually explained by open systems.In statistical physics, open system dynamics have been used to study the irreversibility and the approach to equilibrium.In this paper the dynamical change of the mutual entropy is discussed for a model of quantum Markov chain.The concrete Stinespring's expression for this model is obtained and applied to the derivation of the mutual entropy, and some computational results are presented.

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Mutual entropy was introduced in classical communication theories to measure how much information can be transmitted from an input system to an output system. It has been expanded to quantum systems and applied to various physical models. By contrast, quantum Markov chains, introduced as an extension of classical Markov chains, have been used to explain the irreversibility of observed systems. In [1], a physical model used frequently in quantum optics was applied to a quantum Markov chain. The Stinespring-Kraus expression was derived to discuss mutual entropy on quantum Markov chains. Mutual entropy was then computed and irreversibility was examined using numerical experiments. To further the ongoing research, mean mutual entropy is applied here to the same quantum Markov chain as the previous one; necessary discussion is included. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(1): 104–112, 1997

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There are several damping phenomena in quantum optics.Such phenomena have been usually explained by open systems.In statistical physics, open system dynamics have been used to study the irreversibility and the approach to equilibrium.In this paper the dynamical change of the mutual entropy is discussed for a model of quantum Markov chain.The concrete Stinespring's expression for this model is obtained and applied to the derivation of the mutual entropy, and some computational results are presented.
Mutual entropy was introduced in classical communication theories to measure how much information can be transmitted from an input system to an output system. It has been expanded to quantum systems and applied to various physical models. By contrast, quantum Markov chains, introduced as an extension of classical Markov chains, have been used to explain the irreversibility of observed systems. In [1], a physical model used frequently in quantum optics was applied to a quantum Markov chain. The Stinespring-Kraus expression was derived to discuss mutual entropy on quantum Markov chains. Mutual entropy was then computed and irreversibility was examined using numerical experiments. To further the ongoing research, mean mutual entropy is applied here to the same quantum Markov chain as the previous one; necessary discussion is included. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(1): 104–112, 1997
In classical information theory, one of the most important theorems are the coding theorems, which were discussed by calculating the mean entropy and the mean mutual entropy defined by the classical dynamical entropy (Kolmogorov-Sinai). The quantum dynamical entropy was first studied by Emch [13] and Connes-Stormer [11]. After that, several approaches for introducing the quantum dynamical entropy are done [10, 3, 8, 39, 15, 44, 9, 27, 28, 2, 19, 45]. The efficiency of information transmission for the quantum processes is investigated by using the von Neumann entropy [22] and the Ohya mutual entropy [24]. These entropies were extended to S- mixing entropy by Ohya [26, 27] in general quantum systems. The mean entropy and the mean mutual entropy for the quantum dynamical systems were introduced based on the S- mixing entropy. In this paper, we discuss the efficiency of information transmission to calculate the mean mutual entropy with respect to the modulated initial states and the connected channel for the quantum dynamical systems.
Analyzing the dynamics of open quantum systems has a long history in mathematics and physics. Depending on the system at hand, basic physical phenomena that one would like to explain are, for example, convergence to equilibrium, the dynamics of quantum coherences (decoherence) and quantum correlations (entanglement), or the emergence of heat and particle fluxes in non-equilibrium situations. From the mathematical physics perspective, one of the main challenges is to derive the irreversible dynamics of the open system, starting from a unitary dynamics of the system and its environment. The repeated interactions systems considered in these notes are models of non-equilibrium quantum statistical mechanics. They are relevant in quantum optics, and more generally, serve as a relatively well treatable approximation of a more difficult quantum dynamics. In particular, the repeated interaction models allow to determine the large time (stationary) asymptotics of quantum systems out of equilibrium.
Open quantum systems are often encountered in many different physical situations. From quantum optics to statistical mechanics, they are fundamental in the understanding of a great variety of different phenomena. Some of the most common examples are the relaxation to equilibrium, the existence of nonequilibrium stationary states, and the dynamics of atoms in interaction with electromagnetic fields. A crucial step in the analysis is to consider the quantum open system and its environment as the two mutually interacting components of a larger isolated system. Thereafter, the so‐called Markov approximation is often considered, which consists on assuming that the time scales associated to the dynamics of the quantum open system are larger than those of the environment. It is the interplay of the different time scales associated with the system and the environment what determines the validity of the different approximations made. In this paper we will discuss the dynamics of a open quantum system in contact with a reservoir when the Markov approximation is not valid, and we have to include some non‐Markovian or memory effects.
Review of the irreversibility problem in modern physics with new researches is given. Some characteristics of the Markov chains are specified and the important property of monotonicity of a probability is formulated. Using one thin inequality, the behavior of relative entropy in the classical case is considered. Further we pass to studying of the irreversibility phenomena in quantum problems. By new method is received the Lindblad’s equation and its physical essence is explained. Deep analogy between the classical Markov processes and development described by the Lindblad’s equation is conducted. Using method of comparison of the Lind-blad’s equation with the linear Langevin equation we receive a system of differential equations, which are more general, than the Caldeira-Leggett equation. Here we consider quantum systems without inverse influ-ence on a surrounding background with high temperature. Quantum diffusion of a single particle is consid-ered and possible ways of the permission of the Schrödinger’s cat paradox and the role of an external world for the phenomena with quantum irreversibility are discussed. In spite of previous opinion we conclude that in the equilibrium environment is not necessary to postulate the processes with collapses of wave functions. Besides, we draw attention to the fact that the Heisenberg’s uncertainty relation does not always mean the restriction is usually the product of the average values of commuting variables. At last, some prospects in the problem of quantum irreversibility are discussed.