On quantum large sieve inequalities and operator recovery from incomplete information

TL;DR

This paper introduces quantum large sieve inequalities for phase space representations, enabling operator recovery from incomplete information, and connects these results to low-rank matrix recovery and concentration estimates.

Contribution

It develops quantum large sieve inequalities for operators, providing a new method for operator recovery from sparse phase space data and linking to low-rank recovery techniques.

Findings

Guarantees operator recovery via $L^{1}$-minimization.

Establishes concentration estimates for Cohen's class distributions.

Demonstrates a trade-off between sparsity and concentration in phase space.

Abstract

We obtain large sieve type inequalities for the Rayleigh quotient of the restriction of phase space representations of higher rank operators, via an operator analogue of the short-time Fourier transform (STFT). The resulting bounds are referred to as `quantum large sieve inequalities'. On the shoulders of Donoho and Stark, we demonstrate that these inequalities guarantee the recovery of an operator whose phase-space information is missing or unobservable over a 'measure-sparse' region $\Omega $, by solving an $L^{1}$-minimization program. This is an operator version of what is commonly known as `Logan's phenomenon'. Moreover, our results can be viewed as a deterministic, continuous variable version, on phase space, of `low-rank' matrix recovery, which itself can be regarded as an operator version of (finite-rank) compressive sensing. Our results depend on an abstract large sieve principle for operators with integrable STFT and on a non-commutative analogue of the local reproducing formula in rotationally invariant domains (first stated by Seip for the Fock space of entire functions). As an application, we obtain concentration estimates for Cohen's class distributions and the Husimi function. We motivate the paper by comparing with Nicola and Tilli's Faber-Krahn inequality for the STFT, illustrating that norm bounds on a domain $\Omega $, obtained by large sieve methods, introduce a trade-off between sparsity and concentration properties of $\Omega $: If $\Omega $ is sparse, the large sieve bound may significantly improve known operator norm bounds, while if $\Omega $ is concentrated, it produces worse bounds.