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Communications in Theoretical Physics

Entropy of Quantum Dynamical Systems and Sufficient Families in Orthomodular Lattices with Bayessian State

Mona Khare, Shraddha Roy

Published Sep 1, 2008• 17 citations
Publisher

Abstract

The purpose of the present paper is to study the entropy h s (Φ) of a quantum dynamical systems Φ = (L,s,ϕ), where s is a bayessian state on an orthomodular lattice L. Having introduced the notion of entropy h s (ϕ, ) of partition of a Boolean algebra B with respect to a state s and a state preserving homomorphism ϕ, we prove a few results on that, define the entropy of a dynamical system h s (Φ), and show its invariance. The concept of sufficient families is also given and we establish that h s (Φ) comes out to be equal to the supremum of h s (ϕ, ), where varies over any sufficient family. The present theory has then been extended to the quantum dynamical system (L,s,ϕ), which as an effect of the theory of commutators and Bell inequalities can equivalently be replaced by the dynamical system (B,s 0 ,ϕ), where B is a Boolean algebra and s 0 is a state on B.

DOI
10.1088/0253-6102/50/3/02

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The purpose of the present paper is to study the entropy h s (Φ) of a quantum dynamical systems Φ = (L,s,ϕ), where s is a bayessian state on an orthomodular lattice L. Having introduced the notion of entropy h s (ϕ, ) of partition of a Boolean algebra B with respect to a state s and a state preserving homomorphism ϕ, we prove a few results on that, define the entropy of a dynamical system h s (Φ), and show its invariance. The concept of sufficient families is also given and we establish that h s (Φ) comes out to be equal to the supremum of h s (ϕ, ), where varies over any sufficient family. The present theory has then been extended to the quantum dynamical system (L,s,ϕ), which as an effect of the theory of commutators and Bell inequalities can equivalently be replaced by the dynamical system (B,s 0 ,ϕ), where B is a Boolean algebra and s 0 is a state on B.
Abstract States of quantum systems correspond to vectors in a Hilbert space and observations to closed subspaces. Hence, this logic corresponds to the algebra of closed subspaces of a Hilbert space. This can be considered as a complete lattice with orthocomplementation, but it is not distributive. It satisfies a weaker condition, the so-called orthomodularity. Later on, it was recognized that joins in this structure need not exist provided the subspaces are not orthogonal. Hence, the resulting structure need not be a lattice but a so-called orthomodular poset, more generally an orthoposet only. For orthoposets, we introduce a binary relation Δ\mathrel \DeltaΔ and a binary operator d(x,y)d(x,y)d(x,y) that are generalizations of the binary relation C\textrm{C}C and the commutator c(x,y)c(x,y)c(x,y), respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations Δ\mathrel \DeltaΔ and ↔\leftrightarrow↔ (the latter was introduced by P. Pták and S. Pulmannová) and the operator d(x,y)d(x,y)d(x,y). In addition, we investigate certain orthomodular posets of subsets of a finite set. In particular, we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of Δ\DeltaΔ-blocks with respect to Boolean subalgebras and distributive subposets they include.
In this exposition we prove an existence theorem for a quantum mechanical dynamical entropy based on von-Neumann’s measurement theory. To that end we introduced a Shannon type of information associated with a quantum channel or measurement based on Araki’s relative entropy. This is an invariance for the dynamics which generalizes Kolmogorov-Sinai ’s notion of dynamical entropy of a measure preserving transformation. In this context we also introduce a natural notion of channel capacity as a generalization of Shannon’s capacity of a channel and finds its relation with A.S. Holevo’s notion of capacity. 2 A measurement in classical dynamics gives rise to a measurable partition ζ = (ζi) of the configuration space ( measure space) (M, B, µ). For a par-µ(E∩ζi) tition ζ, we can associate a family of measure µζi(E) =
In this manuscript, we study discrete-time dynamics of systems that arise in physics and information theory, and the measure of disorder in these systems known as dy- namical entropy. The study of dynamics in classical systems is done from two distinct viewpoints: random walks and dynamical systems. Random walks are probabilistic in nature and are described by stochastic processes. On the other hand, dynami- cal systems are described algebraically and deterministic in nature. The measure of disorder from either viewpoint is known as dynamical entropy. Entropy is an essential notion in physics and information theory. Motivated by the study of disorder for the positions and velocities of gas molecules, the notion of entropy was first introduced mathematically by Boltzmann near the end of the 19th Century and gives rise to the second law of thermodynamics. Almost eighty years after Boltzmann, Shannon became the father of the new field of information theory when he produced his groundbreaking works where he used entropy as a measure of information transfer between two sources. In the last two years of the 1950’s, Kolmogorov and Sinai extended the notions of Boltzmann to a dynamical entropy. The Kolmogorov-Sinai dynamical entropy gives a measure for the disorder of a system of particles (e.g. gas molecules) averaged over time, quantifying the uncertainty in the dynamics of a system. The advent of quantum mechanics and its pervasiveness in nature has required the development of non-commutative generalizations of dynamics and dynamical entropy to the quantum regime. Many of each have been proposed. In particular, we recall the definitions of quantum random walks, dynamical systems and Markov chains. We motivate each generalization by relating to its classical counterpart. Quantum dynamical entropy (QDE) is a generalization of the Kolmogorov-Sinai dynamical entropy to quantum mechanics. There have been numerous definitions for QDE beginning with that of Connes, Narnhofer and Thirring in 1987. We focus on the semi- classical approach given by Słomczyński and Życzkowski in 1994 and the quantum Markov chain approach which started with Accardi, Ohya and Watanabe in 1997. Linearity of a dynamical entropy means that the dynamical entropy of the n-fold composition of a dynamical map with itself is equal to n times the dynamical entropy of the map for every positive integer n. We show that the quantum dynamical entropy introduced by Słomczyński and Życzkowski is nonlinear in the time interval between successive measurements of a quantum dynamical system. This is in contrast to Kolmogorov-Sinai dynamical entropy for classical dynamical systems, which is linear in time. We also compute the exact values of quantum dynamical entropy for the Hadamard walk with varying Lüders-von Neumann instruments and partitions. In 1948, Shannon proved the Source Coding Theorem which gives upper and lower bounds on the minimal expected codeword length in terms of the entropy of a ran- dom variable. This theorem can be leveraged to give the minimal expected average codeword length for a string of symbols in terms of entropy rate, which can be in- terpreted as the dynamical entropy of a stochastic process. In 1994, Schumacher defined indeterminate-length quantum codes and proved a Quantum Source Coding Theorem. We introduce the notion of stochastic ensembles of pure states and give a novel representation in terms of quantum Markov chains. Moreover, this representa- tion allows us to extend the Quantum Source Coding Theorem, giving the minimal expected average codeword length of an indeterminate-length quantum code in terms of quantum dynamical entropy. This manuscript includes joint work with Dr. George Androulakis (Mathematics, University of South Carolina).
Sachs showed that a Boolean algebra is determined by its lattice of subalgebras. We establish the corresponding result for orthomodular lattices. We show that an orthomodular lattice L is determined by its lattice of subalgebras Sub(L), as well as by its poset of Boolean subalgebras BSub(L). The domain BSub(L) has recently found use in an approach to the foundations of quantum mechanics initiated by Butterfield and Isham, at least in the case where L is the orthomodular lattice of projections of a Hilbert space, or von Neumann algebra. The results here may add some additional perspective to this line of work.