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Communications in Theoretical Physics

Logarithm Versus Square Root: Comparing Quantum Fisher Information

Luo Shun-Long

Published Apr 1, 2007• 12 citations
Publisher

Abstract

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.

DOI
10.1088/0253-6102/47/4/005

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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.
A parametric quantum mechanical wavefunction naturally induces parametric probability distributions by taking absolute square, and we can consider its classical Fisher information. On the other hand, it also induces parametric rank-one projections which may be viewed as density operators, and we can talk about its quantum Fisher information. Among many versions of quantum Fisher information, there are two prominent ones. The first, defined via a quantum score function, was introduced by Helstrom in 1967 and is well known. The second, defined via the square root of the density operator, has its origin in the skew information introduced by Wigner and Yanase in 1963 and remains relatively unnoticed. This study is devoted to investigating the relationships between the classical Fisher information and these two versions of quantum Fisher information for wavefunctions. It is shown that the two versions of quantum Fisher information differ by a factor 2 and that they dominate the classical Fisher information. The non-coincidence of these two versions of quantum Fisher information may be interpreted as a manifestation of quantum discord. We further calculate the difference between the Helstrom quantum Fisher information and the classical Fisher information, and show that it is precisely the instantaneous phase fluctuation of the wavefunctions.
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.
Skew information was introduced by Wigner and Yanase in the context of quantum measurements. It was later recognized that skew information in addition to its original significance as a measure of the information content of states admits several interpretations of a more physical and information theory nature. This quantity has now found many applications in quantum information. An intriguing and subtle feature of skew information is the involvement of the square root of quantum states (density operators). Comparing skew information with some of its natural modifications reveals mathematical and also physical reasons for using the square root. We further use skew information to quantify the asymmetry of quantum states with respect to group representations.