How to quantify coherence: Distinguishing speakable and unspeakable notions

TL;DR

This paper distinguishes between speakable and unspeakable quantum coherence, proposing resource-theoretic frameworks for quantifying each, with applications in quantum metrology, thermodynamics, and related fields.

Contribution

It introduces a formal distinction between speakable and unspeakable coherence and applies the resource theory of asymmetry to quantify unspeakable coherence, offering new insights and approaches.

Findings

Resource theory of asymmetry effectively quantifies unspeakable coherence.

Different approaches to speakable coherence are reviewed and contrasted.

Mathematical connections among coherence quantification methods are identified.

Abstract

Quantum coherence is a critical resource for many operational tasks. Understanding how to quantify and manipulate it also promises to have applications for a diverse set of problems in theoretical physics. For certain applications, however, one requires coherence between the eigenspaces of specific physical observables, such as energy, angular momentum, or photon number, and it makes a difference which eigenspaces appear in the superposition. For others, there is a preferred set of subspaces relative to which coherence is deemed a resource, but it is irrelevant which of the subspaces appear in the superposition. We term these two types of coherence unspeakable and speakable respectively. We argue that a useful approach to quantifying and characterizing unspeakable coherence is provided by the resource theory of asymmetry when the symmetry group is a group of translations, and we translate a number of prior results on asymmetry into the language of coherence. We also highlight some of the applications of this approach, for instance, in the context of quantum metrology, quantum speed limits, quantum thermodynamics, and NMR. The question of how best to treat speakable coherence as a resource is also considered. We review a popular approach in terms of operations that preserve the set of incoherent states, propose an alternative approach in terms of operations that are covariant under dephasing, and we outline the challenge of providing a physical justification for either approach. Finally, we note some mathematical connections that hold among the different approaches to quantifying coherence.