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Quantum statistics and measures of quantum information

Frank Hansen

Published Oct 22, 2009• 0 citations

Abstract

The geometrical formulation of quantum statistics and the theory of measures of quantum information are intimately connected by the introduction of the notion of metric adjusted skew information.We survey the area with a focus on the ‘representing operator monotone functions’. In particular, we exhibit a new order structure that renders the set of metric adjusted skew informations into a lattice with a maximal element. The Wigner–Yanase–Dyson skew informations (with parameter p) are increasing with respect to this order structure for 0 < p ≤ 1/2 and decreasing for 1/2 ≤ p < 1 with maximum in the Wigner–Yanase skew information. Introduction The geometrical formulation of quantum statistics and the theory of measures of quantum information are two distinct theories with separate motivations and histories, and for a long time they did not seem to be related in any way. Early contributions which view quantum information in statistical terms were given by Hasegawa and Petz (Hasegawa and Petz 1996), and by Luo (Luo 2003a, Luo 2003b, Luo 2005). Today the two theories are largely fused with the notion of metric (or metrically) adjusted skew information and a common set of tools and techniques involving certain operator monotone functions and their representations. The geometrical approach to statistics reveals its fundamental nature when we try to generalise classical statistics to the quantum setting. The key to obtaining quantisation of the Fisher information is to consider it as a Riemannian metric with a certain behaviour in the presence of noise.

DOI: 10.1017/cbo9780511642401.021

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The geometrical formulation of quantum statistics and the theory of measures of quantum information are intimately connected by the introduction of the notion of metric adjusted skew information.We survey the area with a focus on the ‘representing operator monotone functions’. In particular, we exhibit a new order structure that renders the set of metric adjusted skew informations into a lattice with a maximal element. The Wigner–Yanase–Dyson skew informations (with parameter p) are increasing with respect to this order structure for 0 < p ≤ 1/2 and decreasing for 1/2 ≤ p < 1 with maximum in the Wigner–Yanase skew information. Introduction The geometrical formulation of quantum statistics and the theory of measures of quantum information are two distinct theories with separate motivations and histories, and for a long time they did not seem to be related in any way. Early contributions which view quantum information in statistical terms were given by Hasegawa and Petz (Hasegawa and Petz 1996), and by Luo (Luo 2003a, Luo 2003b, Luo 2005). Today the two theories are largely fused with the notion of metric (or metrically) adjusted skew information and a common set of tools and techniques involving certain operator monotone functions and their representations. The geometrical approach to statistics reveals its fundamental nature when we try to generalise classical statistics to the quantum setting. The key to obtaining quantisation of the Fisher information is to consider it as a Riemannian metric with a certain behaviour in the presence of noise.

In the search of appropriate Riemannian metrics on quantum state space, the concept of statistical monotonicity, or contraction under coarse graining, has been proposed by Chentsov. The metrics with this property have been classified by Petz. All the elements of this family of geometries can be seen as quantum analogs of Fisher information. Although there exists a number of general theorems shedding light on this subject, many natural questions, also stemming from applications, are still open. In this paper we discuss a particular member of the family, the Wigner–Yanase information. Using a well-known approach that mimics the classical pull-back approach to Fisher information, we are able to give explicit formulas for the geodesic distance, the geodesic path, the sectional and scalar curvatures associated to Wigner–Yanase information. Moreover, we show that this is the only monotone metric for which such an approach is possible.

We extend the concept of Wigner-Yanase-Dyson skew information to something we call "metric adjusted skew information" (of a state with respect to a conserved observable). This "skew information" is intended to be a non-negative quantity bounded by the variance (of an observable in a state) that vanishes for observables commuting with the state. We show that the skew information is a convex function on the manifold of states. It also satisfies other requirements, proposed by Wigner and Yanase, for an effective measure-of-information content of a state relative to a conserved observable. We establish a connection between the geometrical formulation of quantum statistics as proposed by Chentsov and Morozova and measures of quantum information as introduced by Wigner and Yanase and extended in this article. We show that the set of normalized Morozova-Chentsov functions describing the possible quantum statistics is a Bauer simplex and determine its extreme points. We determine a particularly simple skew information, the "lambda-skew information," parametrized by a lambda in (0, 1], and show that the convex cone this family generates coincides with the set of all metric adjusted skew informations.

In classical statistics, the Fisher information is unique in the sense that it is essentially the only monotone Riemannian metric on the space of probability densities. In quantum theory, this uniqueness breaks down, and there are many natural quantum analogues of the Fisher information, among which two particular versions distinguish themselves by their intuitive and informational significance: The first has its origin in the skew information introduced by Wigner and Yanase in 1963 in the context of quantum measurement, and is defined via the square root of the density operator. The second arises from Helstrom's study of quantum detection in 1967, and is defined via the symmetric logarithmic derivative. The aim of this paper is to compare these two versions of quantum Fisher information, and to establish two informational inequalities relating them.

In classical statistics,the Fisher information is unique in the sense that it is essentially the only monotone Riemannian metric on the space of probability densities.In quantum theory,this uniqueness breaks down,and there are many natural quantum analogues of the Fisher information,among which two particular versions distinguish themselves by their intuitive and informational significance:The first has its origin in the skew information introduced by Wigner and Yanase in 1963 in the context of quantum measurement,and is defined via the square root of the density operator. The second arises from Helstrom's study of quantum detection in 1967,and is defined via the symmetric logarithmic derivative.The aim of this paper is to compare these two versions of quantum Fisher information,and to establish two informational inequalities relating them.

The Wigner-Araki-Yanase theorem puts a limitation on the measurement of observables in the presence of a conserved quantity, and the notion of Wigner-Yanase skew information quantifies the amount of information on the values of observables not commuting with the conserved quantity. We demonstrate that the statistical idea underlying the skew information is the Fisher information in the theory of statistical estimation. A quantum Cramér-Rao inequality and a new uncertainty relation in terms of the skew information are established, which shed considerable new light on the relationships between quantum measurement and statistical inference. The result is applied to estimating the evolution speed of quantum states.