Distributed Inner-Product Estimation for Continuous-Variable Quantum States

Abstract

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.

The paper studies estimation of $\operatorname{Tr}(\rho\sigma)$ for unknown multimode continuous-variable states prepared on separate platforms, using local homodyne measurements and classical communication rather than joint measurements or state reconstruction. It assumes a common calibrated phase reference and a known bound $E$ on each state’s mean photon number per mode. The protocol samples shared random homodyne phases, takes $n$ shots per state at each setting, and combines all cross-platform outcome pairs through an explicit product kernel. The kernel analytically averages radial Fourier variables over a product of phase-space disks of radius $R$. The resulting estimator is unbiased for the truncated overlap. For $M$ modes, defining $c_R=(R^2/2)^M$, its variance is bounded by $[c_R^2/n^2+c_R(1-n^{-2})]/N_\theta$. An energy-controlled weighted second moment of the characteristic functions gives truncation bias at most $(4ME+2M)/R^2$. For Gaussian states, covariance bounds and Gaussian tail estimates improve this to $M\exp[-R^2/(2(4ME+2M))]$. Combining these certificates with Hoeffding concentration gives sufficient total-copy scaling, at fixed $M,E,n$, of $O(\epsilon^{-2M-2}\log(2/\delta))$ generally and $O(\epsilon^{-2}[\log(2M/\epsilon)]^{2M}\log(2/\delta))$ for Gaussian states. Bernstein bounds additionally expose the shots-versus-settings tradeoff. Simulations of coherent, squeezed, Fock, and cat-state benchmarks illustrate truncation bias versus statistical variance. The stated limitations include conservative uniform bounds, substantial mode-number dependence, and empirically optimized cutoffs that do not provide reliable data-driven selection for unknown states. Experimental noise and drift remain untreated.