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Physical Review A

Information and metrics in Hilbert space

M. Raviculé, M. Casas, Angel Ricardo Plastino

Published Mar 1, 1997• 35 citations
Publisher

Abstract

The concept of distance in Hilbert space is relevant in a variety of scenarios, in particular for investigating the quality of different approximations. In this work we study the relations between (i) statistical distances (SD) on a probability space, on the one hand, and (ii) different metrics on Hilbert space (MHS), on the other hand. As a result, we are able to establish some universal relations between SD and MHS and to apply them to one-dimensional problems.

DOI
10.1103/physreva.55.1695

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The concept of distance in Hilbert space is relevant in a variety of scenarios, in particular for investigating the quality of different approximations. In this work we study the relations between (i) statistical distances (SD) on a probability space, on the one hand, and (ii) different metrics on Hilbert space (MHS), on the other hand. As a result, we are able to establish some universal relations between SD and MHS and to apply them to one-dimensional problems.
A concept of "statistical distance" is defined between different preparations of the same quantum system, or in other words, between different rays in the same Hilbert space. Statistical distance is determined entirely by the size of statistical fluctuations occurring in measurements designed to distinguish one state from another. It is not related, a priori, to the usual distance (or angle) between rays. One finds, however, that these two kinds of distance are in fact the same, a result which depends on certain peculiarities of quantum mechanics.
Abstract Our purpose here is to draw attention to an interesting relationship that can be shown to hold between (a) information geometry, and (b) quantum geometry. By ‘information geometry’, we mean the natural geometry associated with families of probability distributions. This is a subject that goes back more than half a century to the pioneering work of Rao (1945), who showed that the Fisher information matrix associated with a parametrised family of probability dis tributions gives rise to a natural Riemannian metric on the parameter space. This then allows one to speak in a precise way of the ‘distance’ between two probability distributions, an idea that turns out to be useful in problems of statistical inference. An extensive literature has thus developed, following this line of enquiry, on applications of differential geometry to statistics (see, e.g., Amari (1985), Barndorff-Nielsen et al. (1986), Murray and Rice (1993)). By ‘quantum geometry’, on the other hand, we mean the geometry of the manifold of states associated with a given quantum mechanical system. In particular, we consider the manifold of ‘pure’ states associated with a given complex Hilbert space. This is the space of ‘rays’ through the origin in the Hilbert space, which has the structure of a complex projective space. It is known that the usual rules of quantum mechanics allow one to construct a natural metric on this space, namely the Fubini-Study metric, the specification of which is equivalent to all the familiar structure associated with ordinary quantum mechanics.
The statistical distance between pure quantum states is obtained by finding a measurement that is optimal in a sense defined by Wootters. As such, one may expect that the statistical distance will turn out to be different if the set of possible measurements is restricted in some way. It nonetheless turns out that if the restriction is to local operations and classical communication on any multipartite system, then the statistical distance is the same as it is without restriction, being equal to the angle between the states in Hilbert space.
The question of distinguishability of quantum states is at the heart of quantum information processing, an issue is here addressed with reference to different distances in probability space vis-a-vis metrics in Hilbert's one. We provide further reconfirmation of Wootters' hypothesis: the possibility that statistical fluctuations in the outcomes of measurements be regarded as responsible for the Hilbert-space structure of quantum mechanics, a view that becomes here considerably strengthened. We show that distances between neighboring states, whether of statistical or Hilbert's metric origin, have as a lower bound Fisher's measure, up to second-order approximation. As a consequence, the structure of the vicinitv of a given quantum state is to a large extent determined by the fluctuations of the pertinent observables. It is also shown that Tsallis' non-extensivity parameter q can be used as a tool for increasing discernibility between wave functions.
Combining gravity with quantum theory is still a work in progress. On the one hand, classical gravity is the geometry of space-time determined by the energy–momentum tensor of matter and the resulting nonlinear equations; on the other hand, the mathematical description of a quantum system is Hilbert space with linear equations describing evolution. In this paper, various measures in Hilbert space will be presented. In general, distance measures in Hilbert space can be divided into measures determined by energy and measures determined by entropy. Entropy measures determine quasi-distance because they do not satisfy all the axioms defining distance. Finding a general rule to determine such a measure unambiguously seems to be fundamental.