Hence, the capability to simulate the photon statistics of GSs is of great practical relevance. Transformations by optical setups described by such Hamiltonians, introduced, e.g., by phase shifters or beam splitters, map GSs to other GSs. States with such Hamiltonians are called Gaussian states (GSs) and include, e.g., vacuum, coherent states, squeezed states, thermal states, or states generated by spontaneous parametric down-conversion (SPDC). Therefore, many common photonic quantum states are described by Hamiltonians that are at most quadratic in the creation and annihilation operators. The strength of nonlinear optical interactions between light and matter typically decreases rapidly with the order of the nonlinear effect. Photon-number resolved (PNR) detection of quantum states opens new pathways to experiments and applications requiring the simulation of such experiments. Padgett, “ Imaging with quantum states of light,” Nat. Pan, “ Secure quantum key distribution with realistic devices,” Rev. Other applications such as quantum key distribution (QKD) or quantum imaging have gained considerable attention over the last few years and become commercially relevant, requiring the development of sophisticated optical setups working at the single-photon level. Milburn, “ A scheme for efficient quantum computation with linear optics,” Nature 409, 46– 52 (2001). An example is photonic quantum computing, which can be realized by using only single-photon sources, beam splitters, phase shifters, and photon detectors. Recent progress in the generation, manipulation, and detection of photonic quantum states has led to new applications in the field of photonic quantum information processing and sensing. 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. 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. Numerical results are obtained by the automatic differentiation of these generating functions by employing the software framework PyTorch. 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. We demonstrate a simple and versatile method to simulate the photon statistics of general multimode Gaussian states. Advances in photonics require photon-number resolved simulations of quantum optical experiments with Gaussian states.
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