A quantum mechanical version of Price's theorem for Gaussian states
This work provides a theoretical extension of a classical signal processing result to quantum systems, relevant for researchers in quantum information and control.
The paper extends Price's theorem, an integro-differential identity for expectations of nonlinear functions of Gaussian random variables, to quantum Gaussian states using Weyl functional calculus. The quantum version relates Fréchet derivatives of generalized moments to other moments, with an example for quadratic-exponential moments relevant to risk-sensitive quantum control.
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.