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.Comment: A non-technical summary of the results and applications is provided
in the first section. V5 close to the published version. Typos correcte
