Advances in quantum learning theory with bosonic systems

TL;DR

This review summarizes recent progress in quantum learning theory for continuous-variable systems, focusing on sample complexity, Gaussian state learning, and bounds on state distinguishability.

Contribution

It provides a concise overview of developments in CV quantum state tomography, highlighting open problems and recent bounds relevant to the field.

Findings

Sample complexity for non-Gaussian state learning under energy constraints

Bounds on trace distance between CV states based on covariance matrices

Methods for testing Gaussianity of quantum states

Abstract

This paper reviews recent advances in quantum learning theory for continuous-variable (CV) systems. Quantum learning theory investigates how to extract classical information from quantum systems as efficiently as possible. CV systems are ubiquitous in nature and in quantum technologies, as they describe bosonic and quantum-optical systems. While quantum learning theory for finite-dimensional systems has been extensively studied, the corresponding theory for CV systems has only recently begun to develop; here we provide a concise review. We focus on the following questions: what is the minimum number of copies (the sample complexity) required to learn a non-Gaussian state, possibly under energy constraints? What is the sample complexity for learning Gaussian states? How does the performance of CV state learning depend on non-Gaussianity? How can one test whether a state is Gaussian or far from the set of Gaussian states? And how can Gaussian processes be learned efficiently? Central to these topics, we also review several bounds on the trace distance between CV states in terms of their covariance matrices, which may be of independent interest. Overall, this work summarises selected developments in tomography of CV systems and highlights a selection of open problems.