Optimal Discrimination of Gaussian States by Gaussian Measurements

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.

Figures (1)

• Figure 1: Contour plot of the Gaussian measured max-relative entropy $D^{\cal G}_{\max}(\rho\|\sigma)$ (shades of blue, increasing from darker to lighter) between two undisplaced single-mode Gaussian states, a squeezed thermal state $\rho$ with covariance matrix $V_\rho=(2m+1) {\rm diag}({\rm e}^{2r}, {\rm e}^{-2r})$ and a thermal state $\sigma$ with covariance matrix $V_\sigma=(2n+1) \mathds{1}_2$, for $n=0.5$ (left) and $n=2$ (right). In the gray background region, $D^{\cal G}_{\max}(\rho\|\sigma)=\infty$. In the reddish shaded central region, $D^{\cal G}_{\max}(\rho\|\sigma)=D_{\max}(\rho\|\sigma)$, that is, Gaussian measurements achieve the max-relative entropy. Along the orange dashed boundary, the optimal Gaussian measurement is a homodyne and attains the max-relative entropy as a limiting case.

Theorems & Definitions (16)

• Definition 1
• Proposition 2
• proof
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof