Duistermaat–Heckman measure and the mixture of quantum states

Abstract

In this paper, we present a general framework to solve a fundamental problem in Random Matrix Theory (RMT), i.e., the problem of describing the joint distribution of eigenvalues of the sum A + B of two independent random Hermitian matrices A and B. Some considerations about the mixture of quantum states are basically subsumed into the above mathematical problem. Instead, we focus on deriving the spectral density of the mixture of adjoint orbits of quantum states in terms of Duistermaat-Heckman measure, originated from the theory of symplectic geometry. Based on this method, we can obtain the spectral density of the mixture of independent random states. In particular, we obtain explicit formulas for the mixture of random qubits. We also find that, in the two-level quantum system, the average entropy of the equiprobable mixture of n random density matrices chosen from a random state ensemble (specified in the text) increases with the number n. Hence, as a physical application, our results quantitatively explain that the quantum coherence of the mixture monotonously decreases statistically as the number of components n in the mixture. Besides, our method may be used to investigate some statistical properties of a special subclass of unital qubit channels.Mathematics Subject Classification. 22E70, 81Q10, 46L30, 15A90, 81R05

Harmonic analysis for rank-1 Randomised Horn Problems

“…This set of inequalities for the general case was proved to be necessary and sufficient by Knutson and Tao [38], using a combinatorial method. Horn's problem can be encountered in various fields, such as in representation theory [35,21,8], combinatorics [34], algebraic geometry [40], quantum information [37,48,47] and, indeed, linear algebra [5]. In recent years a randomised version of Horn's problem has been considered where a and b are fixed, while the diagonalizing unitary matrices U, V , i.e.…”

Abelianization and the Duistermaat–Heckman theorem

“…on 𝔱 * + is another non-abelian variant of the Duistermaat-Heckman measure, and has received some attention in the literature [1,5,6,15].…”

“…In particular, the measures discussed in Remark 7 are defined on different spaces in the case of our arbitrary compact connected Lie group 𝐺. One may therefore regard DH 𝔱 * The measure DH 𝔱 * + and variants thereof have been studied quite extensively [1,3,5,6,11,12,14,15].…”

Probability density functions of quantum mechanical observable uncertainties

“…Recently, the non-additivity of quantum channel capacity [23] has been cracked via probabilistic tools. The Duistermaat-Heckman measure on moment polytope has been used to derive the probability distribution density of one-body quantum marginal states of multipartite random quantum states [24,25] and that of classical probability mixture of random quantum states [26,27].…”