  A central question since the beginning of quantum information science is how
two distant parties can convert one entangled state into another. Answers to
these questions enable us to optimize the performance of tasks such as quantum
key distribution and quantum teleportation, since certain entangled states are
more useful than others for these applications. It has been conjectured that
entangled state transformations could be executed reversibly in an asymptotic
regime, mirroring the reversible nature of Carnot cycles in classical
thermodynamics. While a conclusive proof of this conjecture has been missing so
far, earlier studies excluded reversible entanglement manipulation in various
settings. In this work, we investigate the concept of an entanglement battery,
an auxiliary quantum system that facilitates quantum state transformations
without a net loss of entanglement. We establish that reversible manipulation
of entangled states is achievable through local operations when augmented with
an entanglement battery. In this setting, two distant parties can convert any
entangled state into another of equivalent entanglement. The rate of asymptotic
transformation is quantitatively expressed as a ratio of the entanglement
present within the quantum states involved. Different entanglement quantifiers
give rise to unique principles governing state transformations, effectively
constituting diverse manifestations of a "second law" of entanglement
manipulation. Our methods provide a solution to the long-standing open question
regarding the reversible manipulation of entangled states and are also
applicable to entangled systems involving more than two parties, and to other
quantum resource theories, including quantum thermodynamics.
