Simulating the photon statistics of multimode Gaussian states by automatic differentiation of generating functions

Abstract

Advances in photonics require photon-number resolved simulations of quantum optical experiments with Gaussian states. We demonstrate a simple and versatile method to simulate the photon statistics of general multimode Gaussian states. The derived generating functions enable simulations of the photon number distribution, cumulative probabilities, moments, and factorial moments of the photon statistics of Gaussian states as well as of multimode photon-added and photon-subtracted Gaussian states. Numerical results are obtained by automatic differentiation of these generating functions by employing the machine learning framework PyTorch. Our approach is particularly well suited for practical simulations of the photon statistics of quantum optical experiments in realistic scenarios with low photon numbers, in which various sources of imperfections have to be taken into account. As an example, we calculate the detection probabilities for a recent multipartite time-bin coding quantum key distribution setup and compare them with the corresponding experimental values.

Multi-mode Gaussian state analysis with total-photon counting

“…Examples of such expressions are in [33]. A review of related work and further expressions can be found in [34]. These expressions depend non-linearly on Γ and ⃗ d. Hamilton et al [35] notes that their evalution requires computing Hafnians, and that computing Hafnians is in general #P hard, suggesting that working directly with the joint photon-number probabilities could be difficult.…”