Wishart and random density matrices: Analytical results for the mean-square Hilbert-Schmidt distance

TL;DR

This paper derives exact formulas for the mean-square Hilbert-Schmidt distance between random and fixed density matrices, with applications in quantum information and validation through simulations.

Contribution

It provides novel analytical results for distances involving Wishart and density matrices, enhancing understanding in quantum information theory.

Findings

Exact formulas for mean-square Hilbert-Schmidt distances derived

Verification of results through Monte Carlo simulations

Application to reduced density matrices in coupled kicked tops

Abstract

Hilbert-Schmidt distance is one of the prominent distance measures in quantum information theory which finds applications in diverse problems, such as construction of entanglement witnesses, quantum algorithms in machine learning, and quantum state tomography. In this work, we calculate exact and compact results for the mean square Hilbert-Schmidt distance between a random density matrix and a fixed density matrix, and also between two random density matrices. In the course of derivation, we also obtain corresponding exact results for the distance between a Wishart matrix and a fixed Hermitian matrix, and two Wishart matrices. We verify all our analytical results using Monte Carlo simulations. Finally, we apply our results to investigate the Hilbert-Schmidt distance between reduced density matrices generated using coupled kicked tops.