Towards sample-optimal learning of bosonic Gaussian quantum states

TL;DR

This paper establishes fundamental limits and efficient methods for learning unknown bosonic Gaussian quantum states with minimal samples, revealing the roles of measurement types, state purity, and adaptivity in optimizing quantum state estimation.

Contribution

It provides tight bounds on the sample complexity for learning bosonic Gaussian states, highlighting the necessity of non-Gaussian measurements and adaptivity in certain scenarios.

Findings

Lower bounds of (n^3/^2) for Gaussian measurements and (n^2/^2) for arbitrary measurements.

An upper bound of (n^2/^2) for pure or passive Gaussian states.

Nearly tight bounds for learning single-mode Gaussian states with non-adaptive schemes.

Abstract

Continuous-variable systems enable key quantum technologies in computation, communication, and sensing. Bosonic Gaussian states emerge naturally in various such applications, including gravitational-wave and dark-matter detection. A fundamental question is how to characterize an unknown bosonic Gaussian state from as few samples as possible. Despite decades-long exploration, the ultimate efficiency limit remains unclear. In this work, we study the necessary and sufficient number of copies to learn an $n$-mode Gaussian state, with energy less than $E$, to $\varepsilon$ trace distance with high probability. We prove a lower bound of $\Omega(n^3/\varepsilon^2)$ for Gaussian measurements, matching the best known upper bound up to doubly-log energy dependence, and ${\Omega}(n^2/\varepsilon^2)$ for arbitrary measurements. We further show an upper bound of $\widetilde{O}(n^2/\varepsilon^2)$ given that the Gaussian state is promised to be either pure or passive. Interestingly, while Gaussian measurements suffice for nearly optimal learning of pure Gaussian states, non-Gaussian measurements are provably required for optimal learning of passive Gaussian states. Finally, focusing on learning single-mode Gaussian states via non-entangling Gaussian measurements, we provide a nearly tight bound of $\widetilde\Theta(E/\varepsilon^2)$ for any non-adaptive schemes, showing adaptivity is indispensable for nearly energy-independent scaling. As a byproduct, we establish sharp bounds on the trace distance between Gaussian states in terms of the total variation distance between their Wigner distributions, and obtain a nearly tight sample complexity bound for learning the Wigner distribution of any Gaussian state to $\varepsilon$ total variation distance. Our results greatly advance quantum learning theory in the bosonic regimes and have practical impact in quantum sensing and benchmarking applications.