Perturbation theory for quantum information quantities of Gaussian states

Abstract

Many quantum-information tasks can benefit from Gaussian quantum resources, while their fundamental performance limits are characterized by quantum information quantities. However, a unified perturbation theory for these quantities directly within the Gaussian-state formalism has been lacking. We develop such a theory for the von Neumann entropy, quantum relative entropy, quantum Chernoff quantity, and quantum fidelity, using perturbations of the moments of a reference Gaussian state. Using general Fréchet-derivative identities, we derive compact second-order expressions for reference-state covariance matrices whose symplectic eigenvalues are all greater than $1/2$. We also show that the perturbative quantum Chernoff quantity is maximized at its symmetric point, yielding the quantum Chernoff exponent (QCE). We further show that symplectic eigenvalues equal to 1/2 can lead to slower asymptotic vanishing, potentially improving the corresponding fundamental performance limits. For reference-state covariance matrices containing symplectic eigenvalues both equal to and greater than 1/2, we identify sufficient conditions under which the former determine the leading covariance-matrix contributions, allowing the remaining modes to be traced out without changing the leading-order asymptotic scaling or the coefficient. Applying the framework to quantum illumination and incoherent-light sub-Rayleigh object discrimination, we recover known asymptotic QCEs directly within the Gaussian-state formalism. In the latter problem, the slower asymptotic vanishing explains the known quadratic scaling of the QCE with the source separation.