Machine learning for quantum state tomography: robust covariance matrix estimation for squeezed vacuum states with thermal noise

We present a supervised machine learning-based method using convolutional neural networks to estimate the covariance matrix of Gaussian quantum states in the presence of thermal noise. Unlike computationally intensive density matrix reconstructions, our machine learning-based method allows for the reconstruction of impure squeezed vacuum states using sparse measurements of quadrature sequences based on a model employing a two-component state mixed together from thermal and squeezed thermal states. The method achieves high fidelity and precision, notably also at high squeezing levels, while offering an effective characterization of physical quantities and accurately estimating the covariance matrix. We benchmark our machine against experimental data of single-mode squeezed vacuum states, demonstrating its accuracy and capability to quantify experimental degradation to squeezing and purity. We experimentally verify that our covariance matrix estimation exhibits robustness to state degradation induced by thermal state admixtures. We provide a method for lightweight, compact, and complete representation of lab-generated Gaussian states and lay the foundation for extending real-time quantum state tomography for thermal multi-component Gaussian states to multi-mode systems.