Direct approach to quantum extensions of Fisher information

Abstract

By manipulating classical Fisher information and employing various derivatives of density operators, and using entirely intuitive and direct methods, we introduce two families of quantum extensions of Fisher information that include those defined via the symmetric logarithmic derivative, via the right logarithmic derivative, via the Bogoliubov-Kubo-Mori derivative, as well as via the derivative in terms of commutators, as special cases. Some fundamental properties of these quantum extensions of Fisher information are investigated, a multi-parameter quantum Cramér-Rao inequality is established, and applications to characterizing quantum uncertainty are illustrated.

Clocks and fisher information

“…Unlike the classical scenario in which the Fisher information is unique, there is an infinite variety of quantum Fisher information [12], [18], mainly because of the noncommutative features of observables and states in quantum theory. For instance, another distinguished variant of quantum Fisher information is the skew information or, more generally, the Wigner-Yanase-Dyson information which was first introduced in [13] and has been extensively studied with deep and interesting applications in quantum information theory [13]- [25].…”

Generalized Wigner-Yanase-Dyson skew information and uncertainty relation

“…It is famous that the convexity of I ρ,α (H) with respect to ρ was successfully proven by E.H.Lieb in [10]. And also this quantity was generalized by Chen and Luo in [4] to…”

Probing the qudit depolarizing channel

“…where ∂ θ signifies differentiation. Quantum Fisher information originated with Helstrom [16]; its relationship to classical Fisher information is brought out in [17]; and recent presentations are [18,19]. The Cramér-Rao bound, Eq.…”

“…Realizing the precision promised by the quantum Fisher information in the Cramér-Rao inequality depends on optimal measurement of the channel output. The quantum measurement that saturates the inequality is not always obvious, and in some cases-certain vectors of parameters, for example-no such measurement exists [19]. For the qudit depolarizing channel, is is possible to describe a measurement which yields a (classical) Fisher information which equals the quantum Fisher information.…”