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arXiv:quant-ph/9806013

Volume of classical and quantum ensembles: geometric approach to entropy and information

Michael J. W. Hall

Published Jun 4, 1998• 0 citations
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Abstract

It is shown for classical and quantum ensembles that there is a unique quantity which has the properties of a "volume". This quantity is a function of the ensemble entropy, and hence provides a geometric interpretation for the latter. It further provides a geometric picture for deriving and unifying a number of results in classical and quantum information theory, and for discussing entropic uncertainty relations.

DOI
10.48550/arxiv.quant-ph/9806013

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It is shown for classical and quantum ensembles that there is a unique quantity which has the properties of a "volume". This quantity is a function of the ensemble entropy, and hence provides a geometric interpretation for the latter. It further provides a geometric picture for deriving and unifying a number of results in classical and quantum information theory, and for discussing entropic uncertainty relations.
It is shown that a unique measure of volume is associated with any statistical ensemble, which directly quantifies the inherent spread or localization of the ensemble. It is applicable whether the ensemble is classical or quantum, continuous or discrete, and may be derived from a small number of theory-independent geometric postulates. Remarkably, this unique ensemble volume is proportional to the exponential of the ensemble entropy, and hence provides an interesting geometric characterization of the latter quantity. Applications include unified volume-based derivations of results in quantum and classical information theory, a precise geometric interpretation of thermodynamic entropy for equilibrium ensembles, a geometric derivation of semiclassical uncertainty relations, a means for defining classical and quantum localization for arbitrary evolution processes, and a proposed definition for the spot size of an optical beam. Advantages of ensemble volume over other measures of localization (root-mean-square deviation, Renyi entropies, and inverse participation ratio) are discussed.
Let SSS be the von Neumann entropy of a finite ensemble E\mathcal{E}E of pure quantum states. We show that SSS may be naturally viewed as a function of a set of geometrical volumes in Hilbert space defined by the states and that SSS is monotonically increasing in each of these variables. Since SSS is the Schumacher compression limit of E\mathcal{E}E, this monotonicity property suggests a geometrical interpretation of the quantum redundancy involved in the compression process. It provides clarification of previous work in which it was shown that SSS may be increased while increasing the overlap of each pair of states in the ensemble. As a by-product, our mathematical techniques also provide an interpretation of the subentropy of E\mathcal{E}E.
Any given density matrix can be represented as an infinite number of ensembles of pure states. This leads to the natural question of how to uniquely select one out of the many, apparently equally-suitable, possibilities. Following Jaynes' information-theoretic perspective, this can be framed as an inference problem. We propose the Maximum Geometric Quantum Entropy Principle to exploit the notions of Quantum Information Dimension and Geometric Quantum Entropy. These allow us to quantify the entropy of fully arbitrary ensembles and select the one that maximizes it. After formulating the principle mathematically, we give the analytical solution to the maximization problem in a number of cases and discuss the physical mechanism behind the emergence of such maximum entropy ensembles.
In this article we exploit the geometric structure of statistical manifolds to compare vis-a-vis quantum and classical dynamics. We show that the fixed points of geodesic dynamics in a classical setting coincide with the distributions that minimise the Shannon's entropy. Whereas in a quantum setting this happens only for particular initial conditions, which correspond to classical submanifolds. This result can be interpreted as a manifestation of the uncertainty relation. Other differences put forward concern the (in)stability of geodesic dynamics. The classical scenario results in an isotropic manifold where the geodesic flow is stable. Conversely, on the quantum setting the instability of geodesic dynamics matches the spread out of geodesic ball volumes described by negative constant scalar curvature. spread out of geodesic ball volumes described by negative constant scalar curvature.
Any given density matrix can be represented as an infinite number of ensembles of pure states. This leads to the natural question of how to uniquely select one out of the many, apparently equally suitable, possibilities. Following Jaynes' information-theoretic perspective, this can be framed as an inference problem. We propose the Maximum Geometric Quantum Entropy Principle to exploit the notions of Quantum Information Dimension and Geometric Quantum Entropy. These allow us to quantify the entropy of fully arbitrary ensembles and select the one that maximizes it. After formulating the principle mathematically, we give the analytical solution to the maximization problem in a number of cases and discuss the physical mechanism behind the emergence of such maximum entropy ensembles.
We analyze the geometry of a joint distribution over a set of discrete random variables. We briefly review Shannon's entropy, conditional entropy, mutual information and conditional mutual information. We review the entropic information distance formula of Rokhlin and Rajski. We then define an analogous information area. We motivate this definition and discuss its properties. We extend this definition to higher-dimensional volumes. We briefly discuss the potential utility for these geometric measures in quantum information processing.