Spectral density of mixtures of random density matrices for qubits

TL;DR

This paper derives spectral densities for mixtures of random qubit density matrices, analyzes their entropy behavior, and provides exact fidelity calculations, advancing understanding of quantum state mixtures and their properties.

Contribution

It introduces explicit spectral densities for mixtures of random qubits and explores entropy and fidelity characteristics, offering new analytical tools in quantum information theory.

Findings

Average entropy of the mixture increases with the number of qubits.

Spectral densities are explicitly derived for different mixture rules.

Exact average squared fidelity is computed.

Abstract

We derive the spectral density of the equiprobable mixture of two random density matrices of a two-level quantum system. We also work out the spectral density of mixture under the so-called quantum addition rule. We use the spectral densities to calculate the average entropy of mixtures of random density matrices, and show that the average entropy of the arithmetic-mean-state of $n$ qubit density matrices randomly chosen from the Hilbert-Schmidt ensemble is never decreasing with the number $n$. We also get the exact value of the average squared fidelity. Some conjectures and open problems related to von Neumann entropy are also proposed.