Towards sample-optimal learning of bosonic Gaussian quantum states
🤖 AI Summary
This work addresses the problem of efficiently learning an unknown $n$-mode Gaussian quantum state under energy constraints with minimal sample complexity. By integrating tools from quantum information theory, Wigner function analysis, and distance estimation, it establishes the first sample complexity lower bounds: $\Omega(n^3/\varepsilon^2)$ for Gaussian measurements and $\Omega(n^2/\varepsilon^2)$ for arbitrary measurements. The study demonstrates the necessity of non-Gaussian measurements for optimally learning passive Gaussian states, shows that adaptive strategies in the single-mode setting achieve nearly energy-independent sample complexity, and provides tight sample complexity bounds for multi-mode Gaussian state learning as well as nearly tight bounds for learning the associated Wigner distribution. These results lay a theoretical foundation for quantum sensing and benchmarking protocols.
Application Category
📝 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 $Ω(n^3/\varepsilon^2)$ for Gaussian measurements, matching the best known upper bound up to doubly-log energy dependence, and $Ω(n^2/\varepsilon^2)$ 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 $\widetildeΘ(E/\varepsilon^2)$ 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.
Problem
Research questions and friction points this paper is trying to address.
bosonic Gaussian states
sample complexity
quantum state learning
trace distance
continuous-variable systems
Innovation
Methods, ideas, or system contributions that make the work stand out.
sample complexity
bosonic Gaussian states
quantum state learning
Gaussian measurements
adaptive strategies
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