  Pure state entanglement transformations have been thought of as irreversible,
with reversible transformations generally only possible in the limit of many
copies. Here, we show that reversible entanglement transformations do not
require processing on the many copy level, but can instead be undertaken on
individual systems, provided the amount of entanglement which is produced or
consumed is allowed to fluctuate. We derive necessary and sufficient conditions
for entanglement manipulations in this case. As a corollary, we derive an
equation which quantifies the fluctuations of entanglement, which is formally
identical to the Jarzynski fluctuation equality found in thermodynamics. One
can also relate a forward entanglement transformation to its reverse process in
terms of the entanglement cost of such a transformation, in a manner equivalent
to the Crooks relation. We show that a strong converse theorem for entanglement
transformations is formally related to the second law of thermodynamics, while
the fact that the Schmidt rank of an entangled state cannot increase is related
to the third law of thermodynamics. Achievability of the protocols is done by
introducing an entanglement battery, a device which stores entanglement and
uses an amount of entanglement that is allowed to fluctuate but with an average
cost which is still optimal. This allows us to also solve the problem of
partial entanglement recovery, and in fact, we show that entanglement is fully
recovered. Allowing the amount of consumed entanglement to fluctuate also leads
to improved and optimal entanglement dilution protocols.
