Exact mean and variance of the squared Hellinger distance for random density matrices

TL;DR

This work tackles the statistics of the squared Hellinger distance $D_H$ between quantum density matrices when states are random, analyzing both Hilbert-Schmidt and Bures-Hall ensembles. By decomposing density matrices as $\rho = U\Lambda U^{\dagger}$ and applying Weingarten calculus to perform unitary integrals, the authors derive exact expressions for the mean and second moment of the affinity $A(\rho_1,\rho_2) = \mathrm{tr}(\sqrt{\rho_1}\sqrt{\rho_2})$ in the cases of one random and one fixed state, and of two independent random states. These results yield the exact mean and variance of $D_H = 2-2A$ and enable a gamma-approximation of its PDF via cumulant matching. The analytical findings are validated by Monte Carlo simulations, showing excellent agreement and demonstrating that the gamma approximation captures the distribution well across parameter regimes. Overall, the work provides precise statistical tools for quantum state distinguishability in random-state ensembles and links random-matrix theory with practical distance measures in quantum information.

Abstract

The Hellinger distance between quantum states is a significant measure in quantum information theory, known for its Riemannian and monotonic properties. It is also easier to compute than the Bures distance, another measure that shares these properties. In this work, we derive the mean and variance of the Hellinger distance between pairs of density matrices, where one or both matrices are random. Along the way, we also obtain exact results for the mean affinity and mean square affinity. The first two cumulants of the Hellinger distance allow us to propose an approximation for the corresponding probability density function based on the gamma distribution. Our analytical results are corroborated through Monte Carlo simulations, showing excellent agreement.

Figures (5)

• Figure 1: Statistics of the squared Hellinger distance $D_\mathrm{H}$ between a fixed density matrix and a Hilbert-Schmidt distributed random density matrix, both of dimension $n=5$. The fixed density matrix has eigenvalues $(7/100,16/100,17/100,23/100,37/100)$. Panels (a) and (b) show a comparison between exact analytical results and simulations for the mean and variance of $D_\mathrm{H}$ as a function of ancilla dimension $m$ of the random density matrix. Panel (c) contrasts the distribution of $D_\mathrm{H}$ obtained from simulation, for $(n,m)=(5,10)$, with the analytical approximation based on gamma distribution.
• Figure 2: Statistics of the squared Hellinger distance $D_\mathrm{H}$ between two independent Hilbert-Schmidt distributed random density matrices of dimension $n=3$. In panels (a) and (b), compared between exact analytical and simulation results has been shown for the mean and variance of $D_\mathrm{H}$ for varying ancilla dimensions $m_1,m_2$ of the two random density matrices. In panel (c) the distribution of $D_\mathrm{H}$ obtained from simulation, for $(n,m_1,m_2)=(3,4,6)$, has been contrasted with the analytical gamma-distribution-based approximation.
• Figure 3: Statistics of the squared Hellinger distance $D_\mathrm{H}$ between a fixed density matrix and a Bures-Hall distributed random density matrix, both of dimension $n=5$. The considered fixed density matrix is same as in Fig. \ref{['HSF']}. In panels (a) and (b), comparison between exact analytical and simulation results has been shown for the mean and variance of $D_\mathrm{H}$ for varying ancilla dimension $m$ of the random density matrix. In panel (c) the distribution of $D_\mathrm{H}$ obtained from simulation, for $(n,m)=(5,10)$, has been contrasted with the analytical gamma-distribution-based approximation.
• Figure 4: Statistics of the squared Hellinger distance $D_\mathrm{H}$ between two independent Bures-Hall distributed random density matrices of dimension $n=3$. In panels (a) and (b), compared between exact analytical and simulation results has been shown for the mean and variance of $D_\mathrm{H}$ for varying ancilla dimensions $m_1,m_2$ of the two random density matrices. In panel (c) the distribution of $D_\mathrm{H}$ obtained from simulation, for $(n,m_1,m_2)=(3,4,6)$, has been contrasted with the analytical gamma-distribution-based approximation.
• Figure 5: Statistics of the squared Hellinger distance $D_\mathrm{H}$ between two independent random density matrices of dimension $n=3$, one taken from the Hilbert-Schmidt distribution and the other from the Bures-Hall distribution. In panels (a) and (b), compared between exact analytical and simulation results has been shown for the mean and variance of $D_\mathrm{H}$ for varying ancilla dimensions $m_1,m_2$ of the two random density matrices. In panel (c) the distribution of $D_\mathrm{H}$ obtained from simulation, for $(n,m_1,m_2)=(3,4,6)$, has been contrasted with the analytical gamma-distribution-based approximation.