Generic aspects of the resource theory of quantum coherence

TL;DR

This paper investigates the resource theory of quantum coherence, showing that in high dimensions, randomly chosen pure states are almost never comparable under incoherent operations, and derives asymptotic formulas for conversion success probabilities.

Contribution

It provides a probabilistic analysis of state convertibility in high-dimensional quantum coherence, revealing that comparability becomes negligible and offering explicit asymptotic formulas.

Findings

Probability of two random states being comparable vanishes as dimension increases

Explicit asymptotic formula for maximal success probability of incoherent conversions

Distribution of extreme components of random states approximated by Markov chains

Abstract

The class of incoherent operations induces a pre-order on the set of quantum pure states, defined by the possibility of converting one state into the other by transformations within the class. We prove that if two $n$-dimensional pure states are chosen independently according to the natural uniform distribution, then the probability that they are comparable vanishes as $n\rightarrow\infty$. We also study the maximal success probability of incoherent conversions and find an explicit formula for its large-$n$ asymptotic distribution. Our analysis is based on the observation that the extreme values (largest and smallest components) of a random point uniformly sampled from the unit simplex are distributed asymptotically as certain explicit homogeneous Markov chains.