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

Gaussian states of bosonic quantum systems enjoy several 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 ε, the trace distance error on the state also scales as ε, 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. In our analysis, we introduce the general notion of derivative of a Gaussian state and uncover its fundamental properties, enhancing our understanding of the structure of the set of Gaussian states.