Distributed Inner-Product Estimation for Continuous-Variable Quantum States

Distributed inner product estimation (DIPE) is a fundamental task in quantum information, aiming to estimate the inner product between two unknown quantum states prepared on distributed quantum platforms. For multimode continuous-variable (CV) systems, the infinite-dimensional Hilbert space poses an additional challenge. In this work, we develop a CV-DIPE protocol using only local homodyne measurements at shared phase settings and classical communication. Under a known bound on the mean energy per mode, we derive explicit finite-sample guarantees and sufficient copy counts for estimating the full inner product to prescribed accuracy and confidence. The energy constraint yields an inverse-square truncation-bias bound, while Gaussian structure strengthens the tail control to exponential decay and improves the resulting accuracy scaling. Numerical simulations with Gaussian and non-Gaussian states illustrate the balance between truncation bias and statistical error and its dependence on mode number and energy. Our results enable independent CV platforms to benchmark their state preparations against one another, even when neither state admits a tractable classical description.