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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Statistical distance and Hilbert space

William K. Wootters

Published Jan 15, 1981• 789 citations
Publisher

Abstract

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.

DOI
10.1103/physrevd.23.357

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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.
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.
In this paper with two equivalent representations of the information contained by a SAT formula, the reason why string generated by succinct SAT formula can be greatly compressed is firstly presented based on Kolmogorov complexity theory. Then what strings can be greatly compressed were classified and discussed. In this way we discovered the SAT problem was composed of a basic distinguish problem: distinguish two different distributions induced under the computer with certain SAT formula ensemble. We then tried to map this problem into quantum mechanics, or the quantum version basic distinguish problem: this time two different distributions are induced under quantum mechanics. Based on the equivalence of statistical distance between probability space and Hilbert space, in the same time this distance is invariant under all unitary transformations. The quantum version basic problem cannot be efficiently solved by any quantum computer. In the worst case, any quantum computer must perform exponential times measurement in order to solve it. In the end we proposed the main theorem : The statistical distance in program space and probability space are identical. We tried to prove it using the relationship of Kolmogorov complexity and entropy. It showed there is no difference to solve the basic problem in SAT formula space or probability space. In the worst case, exponential trials must be performed to solve it. NP!=P.