An approximate quantum Cram\'{e}r--Rao bound based on skew information

TL;DR

This paper derives a closed-form expression for Wigner-Yanase skew information in mixed quantum states, showing it can approximate the Helstrom information, thus providing a computationally simpler lower bound for quantum parameter estimation.

Contribution

It introduces a new approximate quantum Cramér-Rao bound based on Wigner-Yanase skew information, simplifying calculations compared to Helstrom information.

Findings

Wigner-Yanase information equals Helstrom information at certain mixing coefficients.

Wigner-Yanase information serves as an upper bound for classical Fisher information.

Inverse Wigner-Yanase information offers an approximate lower bound for estimator variance.

Abstract

A closed-form expression for Wigner-Yanase skew information in mixed-state quantum systems is derived. It is shown that limit values of the mixing coefficients exist such that Wigner-Yanase information is equal to Helstrom information. The latter constitutes an upper bound for the classical expected Fisher information, hence the inverse Wigner-Yanase information provides an approximate lower bound to the variance of an unbiased estimator of the parameter of interest. The advantage of approximating Helstrom's sharp bound lies in the fact that Wigner-Yanase information is straightforward to compute, while it is often very difficult to obtain a feasible expression for Helstrom information. In fact, the latter requires the solution of an implicit second order matrix differential equation, while the former requires just scalar differentiation.