Spectral moments of Bures-Hall ensemble and applications to entanglement entropy

Abstract

We study spectral moments of the Bures-Hall random matrices ensemble. The main result establishes a recurrence relation for the $k$-th spectral moment valid for a real-valued $k$, in contrast to prevailing results in the literature of different ensembles of assuming an integer $k$. The key to establish the recurrence relation is the obtained Christoffel-Darboux formulas of correlation kernels of the ensemble that avoid tedious summations. As an application of our spectral moment results, we re-derive the formulas of average von Neumann entropy and quantum purity of Bures-Hall ensemble conjectured by Ayana Sarkar and Santosh Kumar. This work is dedicated to the memory of Santosh Kumar.

The paper studies spectral moments of the Bures–Hall ensemble, motivated by calculations of entanglement statistics for random reduced density matrices. Its main question is how to obtain a recurrence in a real-valued moment order, allowing differentiation with respect to that order to access logarithmic linear statistics. For the unconstrained ensemble with eigenvalue weight $\prod_i x_i^\alpha e^{-x_i}\prod_{i<j}(x_i-x_j)^2/(x_i+x_j)$, it derives a three-term relation $g_1(k)\mathbb E[R_{k+2}]=g_2(k)\mathbb E[R_k]+g_3(k)\mathbb E[R_{k-2}]$, where $R_k=\sum_i x_i^k$ and the coefficients are explicit polynomials. The derivation applies where the required integrals exist. It uses elementary biorthogonality arguments to obtain summation-free Christoffel–Darboux representations of Cauchy–Laguerre kernels, followed by kernel differentiation, integration by parts, and elimination of auxiliary integrals. Differentiating the recurrence yields one for $T_k=\sum_i x_i^k\ln x_i$. Together with initial conditions and trace normalization, this reproduces previously proved formulas for mean entropy and purity. For subsystem dimensions $m\le n$ and $\alpha=n-m-\tfrac12$, these are $\mathbb E[S]=\psi_0(mn-m^2/2+1)-\psi_0(n+1/2)$ and $\mathbb E[\operatorname{Tr}\rho_A^2]=(m^2-2mn-4n^2-1)/[2n(m^2-2mn-2)]$. The applications concern first moments rather than new entropy-distribution results. Extending the approach to higher-order entropy cumulants through higher-point spectral correlators is identified as future work.