Information metrics give lower bounds for the estimation of parameters. The Cencov-Morozova-Petz Theorem classifies the monotone quantum Fisher metrics. The optimum bound for the quantum estimation problem is offered by the metric which is obtained from the symmetric logarithmic derivative. To get a better bound, it means to go outside this family of metrics, and thus inevitably, to relax some general conditions. In the paper we defined logarithmic derivatives through a phase-space correspondence. This introduces a function which quantifies the deviation from the symmetric derivative. Using this function we have proved that there exist POVMs for which the new metric gives a higher bound from that of the symmetric derivative. The analysis was performed for the one qubit case.
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.
We examine important properties of the difference between the variance and the quantum Fisher information over four, i.e.,
(ΔA)2−FQ[ϱ,A]/4. We find that it is equal to a generalized variance defined in Petz [J. Phys. A 35, 929 (2002)] and Gibilisco, Hiai, and Petz [IEEE Trans. Inf. Theory 55, 439 (2009)]. We present an upper bound on this quantity that is proportional to the linear entropy. As expected, our relation shows that for states that are close to being pure, the quantum Fisher information over four is close to the variance. We also obtain the variance and the quantum Fisher information averaged over all Hermitian operators, and examine its relation to the von Neumann entropy. Apart from the usual quantum Fisher information, we also consider the Kubo-Mori-Bogoliubov quantum Fisher information.
Quantum generalizations of the Fisher information are important in quantum information science, with applications in high energy and condensed matter physics and in quantum estimation theory, machine learning, and optimization. One can derive a quantum generalization of the Fisher information matrix in a natural way as the Hessian matrix arising in a Taylor expansion of a smooth divergence. Such an approach is appealing for quantum information theorists, given the ubiquity of divergences in quantum information theory. In contrast to the classical case, there is not a unique quantum generalization of the Fisher information matrix, similar to how there is not a unique quantum generalization of the relative entropy or the Rényi relative entropy. In this paper, I derive information matrices arising from the log-Euclidean,
α-
z, and geometric Rényi relative entropies, with the main technical tool for doing so being the method of divided differences for calculating matrix derivatives. Interestingly, for all non-negative values of the Rényi parameter
α, the log-Euclidean Rényi relative entropy leads to the Kubo-Mori information matrix, and the geometric Rényi relative entropy leads to the right-logarithmic derivative Fisher information matrix. Thus, the resulting information matrices obey the data-processing inequality for all non-negative values of the Rényi parameter
α even though the original quantities do not. Additionally, I derive and establish basic properties of
α-
z information matrices resulting from the
α-
z Rényi relative entropies. For parameterized thermal states and time-evolved states, I establish formulas for their
α-
z information matrices and hybrid quantum-classical algorithms for estimating them, with applications in quantum Boltzmann machine learning.