[Paper Review] Continuous-variable quantum compressed sensing

Matthias Ohliger, Vincent Nesme+3 more|arXiv (Cornell University)|Nov 3, 2011

Sparse and Compressive Sensing Techniques5 references3 citations

TL;DR

This paper extends quantum compressed sensing to continuous-variable systems using tight frames—overcomplete, non-orthogonal measurement families—enabling efficient reconstruction of low-rank quantum states from few measurements. It introduces a certification method that guarantees reconstruction success without prior assumptions, and demonstrates robust universal reconstruction via restricted isometry properties, significantly improving efficiency and noise resilience in optical and continuous-variable tomography.

ABSTRACT

We significantly extend recently developed methods to faithfully reconstruct unknown quantum states that are approximately low-rank, using only a few measurement settings. Our new method is general enough to allow for measurements from a continuous family, and is also applicable to continuous-variable states. As a technical result, this work generalizes quantum compressed sensing to the situation where the measured observables are taken from a so-called tight frame (rather than an orthonormal basis) --- hence covering most realistic measurement scenarios. As an application, we discuss the reconstruction of quantum states of light from homodyne detection and other types of measurements, and we present simulations that show the advantage of the proposed compressed sensing technique over present methods. Finally, we introduce a method to construct a certificate which guarantees the success of the reconstruction with no assumption on the state, and we show how slightly more measurements give rise to "universal" state reconstruction that is highly robust to noise.

Motivation & Objective

• Address the scalability problem in quantum state tomography for large systems, especially in continuous-variable quantum optics.
• Develop a compressed sensing framework applicable to continuous families of measurements, which are common in optical experiments.
• Provide a certification mechanism that guarantees successful reconstruction without assuming the state is low-rank or pure.
• Achieve universal state reconstruction with strong error bounds using slightly more measurements, enabling robustness to noise and full-rank states with decaying spectra.
• Demonstrate practical applicability to real-world scenarios such as homodyne detection and Wigner function pointwise measurements.

Proposed method

• Generalize quantum compressed sensing to tight frames—overcomplete, non-orthogonal operator families—extending beyond orthonormal bases.
• Introduce new incoherence properties for tight frames that ensure efficient compressed sensing for low-rank states, using an extended 'golfing' proof technique.
• Develop a convex duality-based certification method that verifies reconstruction success a posteriori, without assumptions on the unknown state.
• Apply the restricted isometry property (RIP) to achieve universal reconstruction with strong error bounds, even under statistical noise and decoherence.
• Use random measurement selection from unitary-invariant distributions to enhance information gain per observable, leveraging concentration of measure.
• Implement numerical simulations using displacement operators and homodyne detection to validate performance on single-mode and multi-mode optical states.

Experimental results

Research questions

• RQ1 Can quantum compressed sensing be generalized to continuous families of measurements, such as those in continuous-variable quantum optics?
• RQ2 What incoherence conditions on tight frames ensure efficient reconstruction of low-rank quantum states?
• RQ3 How can one certify the success of quantum state reconstruction without prior assumptions about the state?
• RQ4 Can a single fixed set of measurements achieve universal reconstruction of all low-rank states with robust error bounds?
• RQ5 How does the proposed method perform in realistic experimental settings like homodyne detection or Wigner function tomography?

Key findings

• The method achieves compressed sensing reconstruction using only $ O(rn ext{ polylog}(n)) $ measurements for a rank-$ r $ state in a Hilbert space of dimension $ n $, significantly reducing the number of settings compared to standard tomography.
• For typical low-rank states, $ O(n ext{ polylog}(n)) $ measurements suffice when the tight frame satisfies $ \|w_\alpha\|_1 = O(\text{polylog}(n)) $, enabling efficient reconstruction even with generic frames.
• A certification protocol based on convex duality guarantees successful reconstruction a posteriori, with only a moderate overhead in measurement count, and remains valid under statistical noise and decoherence.
• Using a Fourier-type tight frame and slightly more measurements, the method achieves universal reconstruction with strong error bounds, ensuring robustness to noise and accurate low-rank approximation even for full-rank states.
• Simulations show successful reconstruction of single-mode optical states via homodyne detection using displacement operators, with high success probability and effective certification.
• Global random measurements (invariant under unitary group) outperform local random measurements due to better concentration of measure, yielding more information per observable.