General Expressions for the Quantum Fisher Information Matrix with Applications to Discrete Quantum Imaging

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 calculationmethods rely on the diagonalization of the density matrix. In this paper, we derive general expressions for the quantum Fisher information matrix that 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 nonorthogonal states, such as coherent states. Our derivation relies on two matrix inverses that, 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.

Automated summary

The paper presents general expressions for the quantum Fisher information matrix that bypass matrix diagonalization and do not require the expansion of operators on an orthonormal set of states. The approach simplifies analytical calculations considerably, especially when dealing with density matrices expressed in terms of nonorthogonal states like coherent states. Demonstrating the power of this method in discrete quantum imaging, the paper derives novel results in estimating positions and intensities of incoherent point sources, expecting that this method will become standard in quantum metrology.