On estimates of trace-norm distance between quantum Gaussian states

TL;DR

This paper introduces an alternative method for estimating the trace-norm distance between quantum Gaussian states using a fidelity-like measure called states overlap, providing potentially more stringent bounds especially for pure or gauge-invariant states.

Contribution

It presents a new, more straightforward approach to estimate trace-norm distances between quantum Gaussian states, extending to infinite modes and including fermionic states, differing from previous methods.

Findings

New estimates based on states overlap are sometimes more stringent.

Estimates do not depend on the number of modes, applicable to infinite modes.

Provides an alternative approach to existing inequalities for quantum Gaussian states.

Abstract

In the paper of F.A. Mele, A.A. Mele, L. Bittel, J. Eisert, V. Giovannetti, L. Lami, L. Leone, S.F.E. Oliviero, ArXiv:2405.01431, estimates for the trace-norm distance between two quantum Gaussian states in terms of the mean vectors and covariance matrices were derived and used to evaluate the sample complexity of learning quantum energy-constrained Gaussian states. In the present paper we obtain different estimates; our proof is based on a fidelity-like quantity which we call states overlap, and is more straightforward leading to estimates which are sometimes even more stringent, especially in the cases of pure or gauge-invariant states. They do not depend on number of modes and hence can be extended to the case of bosonic field with infinite number of modes. These derivations are not aimed to replace the useful inequalities from ArXiv:2405.01431; they just show an alternative approach to the problem leading to different results. In the Appendix we briefly recall our results concerning estimates of the overlap for general fermionic Gaussian states of CAR. The problem studied in this paper can be considered as a noncommutative analog of estimation of the total variance distance between Gaussian probability distributions in the classical probability theory.