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arXiv · 2607.11759

Higher-order covariance matrices for non-Gaussian quantum states

Vojtěch Kala·Petr Marek·Nicolas J. Cerf
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Abstract

Covariance matrices lie at the heart of the powerful and well-established symplectic framework for describing continuous-variable Gaussian quantum states. However, since this framework only relies on first- and second-order moments, it is not sufficient for the analysis of non-Gaussian states because their higher-order moments are essential to capture some of their key properties. Here, we define higher-order covariance matrices -- more precisely, covariance matrices built from higher-order quadrature monomials -- which provide a simple way to evaluate the effect of Gaussian transformations on non-Gaussian states and can be used, for example, to address nonlinear squeezing or non-Gaussian nullifiers. Higher-order covariance matrices can be estimated from homodyne measurement data using only a limited number of quadrature angles, which involves matrices of moderate dimension compared with a full simulation in the Fock basis. The dimension of a higher-order covariance matrix does not depend on the span of the quantum states in Fock basis and, furthermore, scales only polynomially with the number of modes.

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Vojtěch Kala, Petr Marek, Nicolas J. Cerf. 2026-07-13. Higher-order covariance matrices for non-Gaussian quantum states. https://arxiv.org/abs/2607.11759

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