A quantum mechanical version of Price's theorem for Gaussian states

TL;DR

This paper extends Price's theorem, originally for classical Gaussian variables, to quantum Gaussian states using the Weyl calculus, linking derivatives of quantum moments to covariance matrices with applications in quantum control.

Contribution

It introduces a quantum version of Price's theorem for Gaussian states, connecting derivatives of quantum moments to covariance matrices within the Weyl calculus framework.

Findings

Quantum Price's theorem established for Gaussian states.

Relations between derivatives of quantum moments and covariance matrices.

Application to risk-sensitive quantum control moments.

Abstract

This paper is concerned with integro-differential identities which are known in statistical signal processing as Price's theorem for expectations of nonlinear functions of jointly Gaussian random variables. We revisit these relations for classical variables by using the Frechet differentiation with respect to covariance matrices, and then show that Price's theorem carries over to a quantum mechanical setting. The quantum counterpart of the theorem is established for Gaussian quantum states in the framework of the Weyl functional calculus for quantum variables satisfying the Heisenberg canonical commutation relations. The quantum mechanical version of Price's theorem relates the Frechet derivative of the generalized moment of such variables with respect to the real part of their quantum covariance matrix with other moments. As an illustrative example, we consider these relations for quadratic-exponential moments which are relevant to risk-sensitive quantum control.