On quantum large sieve inequalities and operator recovery from incomplete information

TL;DR

The paper develops a deterministic, operator-level analogue of the large sieve in the time–frequency plane by introducing quantum large sieve inequalities for the operator STFT of density operators. It provides abstract bounds that connect phase-space sparsity, via the Nyquist-type measure $\nu(\Omega,R)$ or geometric kernels, to concentration and recoverability results, including an operator Logan-type recovery from incomplete data. By analyzing both Gaussian/thermal-state windows and polyradial windows within the modulation-operator framework, it derives explicit bounds for Husimi and Cohen's class distributions and proves a local reproducing formula for higher-rank operator windows. The results yield concrete recovery guarantees for quantum state tomography and operator reconstruction from incomplete phase-space measurements, highlighting a trade-off between sparsity and concentration and offering new deterministic tools for continuous low-rank recovery in infinite dimensions.

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 $Ω$, 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 $Ω$, obtained by large sieve methods, introduce a trade-off between sparsity and concentration properties of $Ω$: If $Ω$ is sparse, the large sieve bound may significantly improve known operator norm bounds, while if $Ω$ is concentrated, it produces worse bounds.

Theorems & Definitions (49)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Lemma 2.1: Laguerre connection
• Proposition 2.2
• Lemma 2.3
• Definition 2.4
• Proposition 2.5: Special case of Proposition 5.13 in OpSTFT
• Proposition 2.6
• proof