Distributed Inner Product Estimation of Continuous-Variable Quantum States

Continuous-variable distributed inner product estimation (CV-DIPE) compares unknown states prepared by distant bosonic and quantum optical systems. It supports benchmarking without efficient classical descriptions or joint measurements. We develop three CV-DIPE protocols for the natural class of energy-constrained continuous-variable quantum states, based on coordinated displaced parity, coordinated Haar measurements, and independent homodyne shadows. For arbitrary multimode inputs in this class, including entangled and non-Gaussian states, we establish truncation error bounds and worst-case sample complexity guarantees for the exact inner product. The leading terms of these bounds favor coordinated Haar measurements, whose global implementation is experimentally challenging. The three protocols provide certified inner product estimates for distributed continuous-variable systems from local measurement data.