Coherence generating power of quantum unitary maps and beyond

TL;DR

This paper introduces a measure for the coherence generating power of quantum unitaries relative to a basis, providing analytical formulas, detection protocols, and statistical properties, including behavior of random unitaries.

Contribution

It defines and analyzes a new measure of coherence generating power for quantum unitaries, including explicit formulas, detection methods, and properties of random unitaries.

Findings

Random unitaries typically have nearly maximal coherence generating power in large dimensions.

Explicit analytical form of the measure is provided for any dimension.

Operational protocol for direct detection of the coherence generating power.

Abstract

Given a preferred orthonormal basis $B$ in the Hilbert space of a quantum system we define a measure of the coherence generating power of a unitary operation with respect to $B$. This measure is the average coherence generated by the operation acting on a uniform ensemble of incoherent states. We give its explicit analytical form in any dimension and provide an operational protocol to directly detect it. We characterize the set of unitaries with maximal coherence generating power and study the properties of our measure when the unitary is drawn at random from the Haar distribution. For large state-space dimension a random unitary has, with overwhelming probability, nearly maximal coherence generating power with respect to any basis. Finally, extensions to general unital quantum operations and the relation to the concept of asymmetry are discussed.