Random density matrices: Closed form expressions for the variance of squared Hilbert-Schmidt distance

Abstract

One of the fundamental aspects of quantum information theory is the study of distance measures between quantum states. Hilbert-Schmidt distance serves as a convenient choice for exploring a wide range of quantum information-related phenomena. This study contributes by deriving precise closed-form expressions for the variance of the squared Hilbert-Schmidt distance, both between one fixed and one random density matrix and between two independent random density matrices. Notably, our findings go beyond recently acquired results concerning the mean-square Hilbert-Schmidt distance. By incorporating both mean and variance, we extend our analysis to provide a gamma-distribution-based approximation for the probability density function of the squared Hilbert-Schmidt distance. The validity of these analytical results is confirmed through Monte Carlo simulations. Additionally, we perform a comparative analysis, aligning our analytical predictions with the mean and variance of the squared Hilbert-Schmidt distance calculated in correlated spin chain simulations, revealing a satisfactory agreement.