Optimal Discrimination of Gaussian States by Gaussian Measurements

TL;DR

This paper investigates whether Gaussian measurements can optimally distinguish between Gaussian states using max-relative entropy, deriving conditions for optimality and revealing instances of Gaussian data hiding with significant measurement gaps.

Contribution

It provides a complete covariance matrix-based condition for optimal Gaussian measurements and explicitly constructs these measurements when possible.

Findings

Derived a covariance matrix condition for optimal Gaussian measurements.

Identified cases where Gaussian measurements are suboptimal, creating a gap in distinguishability.

Presented examples of Gaussian data hiding with arbitrarily large measurement gaps.

Abstract

Are Gaussian measurements enough to distinguish between Gaussian states? Here, we tackle this question by focusing on the max-relative entropy as an operational distinguishability metric. Given two general multimode Gaussian states, we derive a condition, based on their covariance matrices, that completely determines whether or not there exists an optimal Gaussian measurement achieving the max-relative entropy. When the condition is satisfied, we find this optimal measurement explicitly. When the condition is not met, there is a strict gap between the distinguishability achievable by Gaussian measurements and the unconstrained max-relative entropy in which all measurements are allowed. We illustrate our results in the single-mode setting, and show examples of states for which this gap can be made arbitrarily large, revealing novel instances of Gaussian data hiding.