Cross-Platform Verification of Continuous-Variable Quantum Systems

Abstract

Cross-platform verification compares output states of independently operated bosonic and quantum optical systems when classical predictions and joint measurements are unavailable. Continuous-variable distributed inner product estimation (CV-DIPE) provides a local measurement approach, but infinite-dimensional state spaces require rigorous control of truncation error. We develop three CV-DIPE protocols for the natural class of energy-constrained continuous-variable quantum states. One uses coordinated displaced parity, another coordinated Haar measurements, and the third 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. These protocols provide certified strategies for cross-platform verification of continuous-variable systems using only local measurement data.

The paper studies distributed estimation of $\operatorname{tr}(\rho\sigma)$ for unknown bosonic states held in separate laboratories, using local measurements and classical communication without exchanging quantum systems or sharing entanglement. It treats arbitrary entangled and non-Gaussian $m$-mode states satisfying $\operatorname{tr}[(\hat n_{\mathrm{tot}}+m/2)\tau]\le mE$, and develops three protocols with certified truncation bias and worst-case copy bounds. Coordinated displaced-parity measurements sample common points inside a phase-space ball. An energy-based Wigner second-moment bound gives overlap bias at most $mE/R^2$; choosing $R^2=2mE/\varepsilon$ and applying Hoeffding’s inequality yields $N\ge 8\varepsilon^{-2}(8eE/\varepsilon)^{2m}\log(2/\delta)$ copies per laboratory. Coordinated Haar measurements instead truncate each mode at $L=\lceil4mE/\varepsilon\rceil$, encode overflow into distinct orthogonal flags, and apply an existing finite-dimensional DIPE protocol. The sufficient count is $O(\max\{\varepsilon^{-2},\sqrt{L^m+2}/\varepsilon\}\log(2/\delta))$. Independent homodyne shadows use the same cutoff in classical postprocessing; snapshot-norm and variance bounds give $N=O([1+C_1^mL^{7m/3}/\varepsilon^2+C_2^mL^{3m}/\varepsilon]/\delta)$. Independent batching can improve the confidence dependence to logarithmic. Both Fock-based protocols use an overlap-bias bound of $2mE/L$. The paper also shows that no fixed finite-volume phase-space region uniformly controls truncation bias over unrestricted states. These are sufficient bounds, not matching lower bounds or established separations between measurement resources. Haar measurements require challenging coherent operations beyond passive interferometers on the original modes, whose costs are excluded. The energy promise requires external justification; finite data alone cannot certify an unrestricted mean. Optimal energy-constrained copy complexity and extensions to loss and detector noise remain open.