The difference between two random mixed quantum states: exact and asymptotic spectral analysis

TL;DR

This paper provides exact and asymptotic spectral analyses of the difference between two random mixed quantum states, offering insights into their typical distances using free probability and spectral density functions.

Contribution

It introduces a closed-form for the joint eigenvalue distribution and derives the asymptotic eigenvalue density for the difference of two random quantum states.

Findings

Exact joint eigenvalue density derived for arbitrary dimensions.

Asymptotic eigenvalue density obtained using free probability.

Asymptotic behavior of operator norm and trace distance characterized.

Abstract

We investigate the spectral statistics of the difference of two density matrices, each of which is independently obtained by partially tracing a random bipartite pure quantum state. We first show how a closed-form expression for the exact joint eigenvalue probability density function for arbitrary dimensions can be obtained from the joint probability density function of the diagonal elements of the difference matrix, which is straightforward to compute. Subsequently, we use standard results from free probability theory to derive a relatively simple analytic expression for the asymptotic eigenvalue density (AED) of the difference matrix ensemble, and using Carlson's theorem, we obtain an expression for its absolute moments. These results allow us to quantify the typical asymptotic distance between the two random mixed states using various distance measures; in particular, we obtain the almost sure asymptotic behavior of the operator norm distance and the trace distance.