  We investigate manipulations of pure quantum states under incoherent or
strictly incoherent operations assisted by a coherence battery, that is, a
storage device whose degree of coherence is allowed to fluctuate in the
process. This leads to the derivation of fluctuation relations for quantum
coherence, analogous to Jarzynski's and Crooks' relations for work in
thermodynamics. Coherence is thus revealed as another instance of a physical
resource, in addition to athermality and entanglement, for which a connection
is established between the majorisation framework (regulating pure state
transformations under suitable free operations) and the emergence of
fluctuation theorems. Our study is hoped to provide further insight into the
general structure of battery assisted quantum resource theories, and more
specifically into the interplay between quantum coherence and quantum
thermodynamics.
