Proof of Sarkar-Kumar's Conjectures on Average Entanglement Entropies over the Bures-Hall Ensemble

Abstract

Sarkar and Kumar recently conjectured [J. Phys. A: Math. Theor. $\text{52}$, 295203 (2019)] that for a bipartite system of Hilbert dimension $mn$, the mean values of quantum purity and von Neumann entropy of a subsystem of dimension $m\le n$ over the Bures-Hall measure are given by \begin{equation*} \frac{2n(2n+m)-m^{2}+1}{2n(2mn-m^2+2)} \end{equation*} and \begin{equation*} ψ_{0}\left(mn-\frac{m^2}{2}+1\right)-ψ_{0}\left(n+\frac{1}{2}\right), \end{equation*} respectively, where ${\psi }_0(\cdot )$ is the digamma function. We prove the above conjectured formulas in this work. A key ingredient of the proofs is Forrester and Kieburg's discovery on the connection between the Bures-Hall ensemble and the Cauchy-Laguerre biorthogonal ensemble studied by Bertola, Gekhtman, and Szmigielski.

References in corpus (11)

• Average Entropy of a Subsystem
• Induced measures in the space of mixed quantum states
• Statistical properties of random density matrices
• Random quantum correlations and density operator distributions
• Random Bures mixed states and the distribution of their purity
• Random pure states: quantifying bipartite entanglement beyond the linear statistics
• Distribution of bipartite entanglement for random pure states
• Cauchy-Laguerre two-matrix model and the Meijer-G random point field
• Relating the Bures measure to the Cauchy two-matrix model
• A Proof of Vivo-Pato-Oshanin's Conjecture on the Fluctuation of von Neumann Entropy
• Bures-Hall Ensemble: Spectral Densities and Average Entropies

Cited by in corpus (10)

• Exact variance of von Neumann entanglement entropy over the Bures-Hall measure
• Second-order statistics of fermionic Gaussian states
• Average capacity of quantum entanglement
• Random density matrices: Analytical results for mean root fidelity and mean square Bures distance
• Quantum interpolating ensemble: Biorthogonal polynomials and average entropies
• Random density matrices: Analytical results for mean fidelity and variance of squared Bures distance
• Average relative entropy of random states
• Entanglement capacity of fermionic Gaussian states
• Average mutual information for random fermionic Gaussian quantum states
• Moments of quantum purity and biorthogonal polynomial recurrence