Practical Homodyne Shadow Estimation

Abstract: Shadow estimation provides an efficient framework for estimating observable expectation values using randomized measurements. While originally developed for discrete-variable systems, its recent extensions to continuous-variable (CV) quantum systems face practical limitations due to idealized assumptions of continuous phase modulation and infinite measurement resolution. In this work, we develop a practical shadow estimation protocol for CV systems using discretized homodyne detection with a finite number of phase settings and quadrature bins. We construct an unbiased estimator for the quantum state and establish both sufficient conditions and necessary conditions for informational completeness within a truncated Fock space up to $n_{\mathrm{max}}$ photons. We further provide a comprehensive variance analysis, showing that the shadow norm scales as $\mathcal{O}(n_{\mathrm{max}}<sup>4)$, improving upon previous $\mathcal{O}(n_{\mathrm{max}}<sup>{13/3})$ bounds. Our work bridges the gap between theoretical shadow estimation and experimental implementations, enabling robust and scalable quantum state characterization in realistic CV systems.