Optimal estimates of trace distance between bosonic Gaussian states and applications to learning

TL;DR

This paper establishes optimal bounds on how errors in estimated moments of bosonic Gaussian states affect the trace distance, providing insights for quantum state learning and tomography.

Contribution

It provides the first tight bounds linking moment estimation errors to trace distance, improving understanding of Gaussian state approximation in quantum information.

Findings

Trace distance error scales linearly with moment estimation error

Bounds are proven to be tight and optimal

Improves sample complexity bounds for Gaussian state tomography

Abstract

Gaussian states of bosonic quantum systems enjoy numerous technological applications and are ubiquitous in nature. Their significance lies in their simplicity, which in turn rests on the fact that they are uniquely determined by two experimentally accessible quantities, their first and second moments. But what if these moments are only known approximately, as is inevitable in any realistic experiment? What is the resulting error on the Gaussian state itself, as measured by the most operationally meaningful metric for distinguishing quantum states, namely, the trace distance? In this work, we fully resolve this question by demonstrating that if the first and second moments are known up to an error $\varepsilon$, the trace distance error on the state also scales as $\varepsilon$, and this functional dependence is optimal. To prove this, we establish tight bounds on the trace distance between two Gaussian states in terms of the norm distance of their first and second moments. As an application, we improve existing bounds on the sample complexity of tomography of Gaussian states.