Distinguishability of generic quantum states

TL;DR

This paper investigates the distinguishability of large random quantum states, showing that their trace distance converges to a fixed value, and derives related bounds using free random calculus and numerical simulations.

Contribution

It introduces a novel analysis of the trace distance between large random quantum states, applying free random calculus and deriving explicit asymptotic distributions.

Findings

Trace distance between two large random states tends to 1/4+1/π

Derived the symmetrized Marchenko–Pastur distribution for the model

Provided bounds on distinguishability using quantum relative entropy and Chernoff quantity

Abstract

Properties of random mixed states of order $N$ distributed uniformly with respect to the Hilbert-Schmidt measure are investigated. We show that for large $N$, due to the concentration of measure, the trace distance between two random states tends to a fixed number ${\tilde D}=1/4+1/\pi$, which yields the Helstrom bound on their distinguishability. To arrive at this result we apply free random calculus and derive the symmetrized Marchenko--Pastur distribution, which is shown to describe numerical data for the model of two coupled quantum kicked tops. Asymptotic values for the fidelity, Bures and transmission distances between two random states are obtained. Analogous results for quantum relative entropy and Chernoff quantity provide other bounds on the distinguishablity of both states in a multiple measurement setup due to the quantum Sanov theorem.