Almost all quantum channels are equidistant

TL;DR

This paper demonstrates that in large quantum systems, the distance between most random quantum channels converges to a constant, indicating they become nearly indistinguishable as system size increases.

Contribution

The study applies random matrix theory and free probability to show that quantum channels become equidistant in high dimensions, providing new insights into quantum channel distinguishability.

Findings

Distance between two random channels converges to a constant as system size increases.

For flat Hilbert-Schmidt distribution, the distance converges to approximately 0.886.

Maximum success probability for distinguishing channels approaches a specific limit.

Abstract

In this work we analyze properties of generic quantum channels in the case of large system size. We use random matrix theory and free probability to show that the distance between two independent random channels converges to a constant value as the dimension of the system grows larger. As a measure of the distance we use the diamond norm. In the case of a flat Hilbert-Schmidt distribution on quantum channels, we obtain that the distance converges to $\frac12 + \frac{2}{\pi}$, giving also an estimate for the maximum success probability for distinguishing the channels. We also consider the problem of distinguishing two random unitary rotations.