Calculating trace distances of bosonic states in Krylov subspace

TL;DR

This paper presents an efficient numerical method for calculating the trace distance between Gaussian states in continuous-variable quantum systems, facilitating state discrimination and certification without exponential computational costs.

Contribution

It introduces a generalized Lanczos algorithm that computes trace distances using only moments, extending to non-Gaussian states and providing practical bounds for quantum state analysis.

Findings

Efficient numerical computation of trace distances between Gaussian states.

Extension of the method to non-Gaussian states as linear combinations of Gaussian states.

Provision of lower bounds on trace distances for mixed Gaussian states.

Abstract

Continuous-variable quantum systems are central to quantum technologies, with Gaussian states playing a key role due to their broad applicability and simple description via first and second moments. Distinguishing Gaussian states requires computing their trace distance, but no analytical formula exists for general states, and numerical evaluation is difficult due to the exponential cost of representing infinite-dimensional operators. We introduce an efficient numerical method to compute the trace distance between a pure and a mixed Gaussian state, based on a generalized Lanczos algorithm that avoids explicit matrix representations and uses only moment information. The technique extends to non-Gaussian states expressible as linear combinations of Gaussian states. We also show how it can yield lower bounds on the trace distance between mixed Gaussian states, offering a practical tool for state certification and learning in continuous-variable quantum systems.