General expressions for the quantum Fisher information matrix with applications to discrete quantum imaging

TL;DR

This paper introduces general, non-diagonalization-based formulas for the quantum Fisher information matrix, simplifying calculations especially for non-orthogonal states, and applies these to improve discrete quantum imaging techniques.

Contribution

It provides new analytical expressions for the quantum Fisher information matrix that work for arbitrary rank density matrices without diagonalization, advancing quantum estimation methods.

Findings

Derived formulas for quantum Fisher information matrix without diagonalization.

Applied methods to estimate positions and intensities of incoherent point sources.

Obtained analytical solutions for two and three point source estimation problems.

Abstract

The quantum Fisher information matrix is a central object in multiparameter quantum estimation theory. It is usually challenging to obtain analytical expressions for it because most calculation methods rely on the diagonalization of the density matrix. In this paper, we derive general expressions for the quantum Fisher information matrix which bypass matrix diagonalization and do not require the expansion of operators on an orthonormal set of states. Additionally, we can tackle density matrices of arbitrary rank. The methods presented here simplify analytical calculations considerably when, for example, the density matrix is more naturally expressed in terms of non-orthogonal states, such as coherent states. Our derivation relies on two matrix inverses which, in principle, can be evaluated analytically even when the density matrix is not diagonalizable in closed form. We demonstrate the power of our approach by deriving novel results in the timely field of discrete quantum imaging: the estimation of positions and intensities of incoherent point sources. We find analytical expressions for the full estimation problem of two point sources with different intensities, and for specific examples with three point sources. We expect that our method will become standard in quantum metrology.