It was recently conjectured by Vivo, Pato, and Oshanin [ Phys. Rev. E 93, 052106 (2016)] that for a quantum system of Hilbert dimension $mn$ in a pure state, the variance of the von Neumann entropy of a subsystem of dimension $m\le n$ is given by $$-{\psi }_1(mn+1)+\frac{m+n}{mn+1}{\psi }_1\left(n\right)-\frac{(m+1)(m+2n+1)}{4n^2(mn+1)},$$ where ${\psi }_1(·)$ is the trigamma function. We give a proof of this formula.

We study quantum bipartite systems in a random pure state, where von Neumann entropy is considered as a measure of the entanglement. Expressions of the first and second exact cumulants of von Neumann entropy, relevant respectively to the average and fluctuation behavior, are known in the literature. The focus of this paper is on its skewness that specifies the degree of asymmetry of the distribution. Computing the skewness requires additionally the third cumulant, an exact formula of which is the main result of this work. In proving the main result, we obtain as a byproduct various summation identities involving polygamma and related functions. The derived third cumulant also leads to an improved approximation to the distribution of von Neumann entropy.

The Bures-Hall distance metric between quantum states is a unique measure that satisfies various useful properties for quantum information processing. In this work, we study the statistical behavior of quantum entanglement over the Bures-Hall ensemble as measured by von Neumann entropy. The average von Neumann entropy over such an ensemble has been recently obtained, whereas the main result of this work is an explicit expression of the corresponding variance that specifies the fluctuation around its average. The starting point of the calculations is the connection between correlation functions of the Bures-Hall ensemble and those of the Cauchy-Laguerre ensemble. The derived variance formula, together with the known mean formula, leads to a simple but accurate Gaussian approximation of the distribution of von Neumann entropy of finite-size systems. This Gaussian approximation is also conjectured to be the limiting distribution for large dimensional systems.

Sarkar and Kumar recently conjectured (2019 J. Phys. A: Math. Theor. 52 295203) 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 ⩽ n over the Bures–Hall measure are given by 2 n ( 2 n + m ) − m 2 + 1 2 n ( 2 m n − m 2 + 2 ) and ψ 0 m n − m 2 2 + 1 − ψ 0 n + 1 2 , respectively, where ψ ₀ (⋅) 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 et al .

The Rayleigh product channel model is useful in capturing the performance degradation due to rank deficiency of MIMO channels. In this paper, such a performance degradation is investigated via the distribution of mutual information assuming the block fading channels and the uniform power transmission scheme. Using techniques of free probability theory, the asymptotic variance of mutual information is derived when the dimensions of the channel matrices approach infinity. In this asymptotic regime, the mutual information is rigorously proven to be Gaussian distributed. Using the obtained results, a fundamental tradeoff between multiplexing gain and diversity gain of Rayleigh product channels under the uniform power transmission can be characterized by the closed-form expression at any finite signal-to-noise ratio. Numerical results are provided to compare the outage performance between the Rayleigh product channels and the conventional Rayleigh MIMO channels.

The Tsallis entropy is a useful one-parameter generalization to the standard von Neumann entropy in quantum information theory. In this work, we study the variance of the Tsallis entropy of bipartite quantum systems in a random pure state. The main result is an exact variance formula of the Tsallis entropy that involves finite sums of some terminating hypergeometric functions. In the special cases of quadratic entropy and small subsystem dimensions, the main result is further simplified to explicit variance expressions. As a byproduct, we find an independent proof of the recently proven variance formula of the von Neumann entropy based on the derived moment relation to the Tsallis entropy.