We prove that the theory of entanglement manipulation is asymptotically
irreversible under all non-entangling operations, showing from first principles
that reversible entanglement transformations require the generation of
entanglement in the process. Entanglement is thus shown to be the first example
of a quantum resource that does not become reversible under the maximal set of
free operations, that is, under all resource non-generating maps. Our result
stands in stark contrast with the reversibility of quantum and classical
thermodynamics, and implies that no direct counterpart to the second law of
thermodynamics can be established for entanglement -- in other words, there
exists no unique measure of entanglement governing all axiomatically possible
state-to-state transformations. This completes the solution of a long-standing
open problem [Problem 20 in arXiv:quant-ph/0504166]. We strengthen the result
further to show that reversible entanglement manipulation requires the creation
of exponentially large amounts of entanglement according to monotones such as
the negativity. Our findings can also be extended to the setting of
point-to-point quantum communication, where we show that there exist channels
whose parallel simulation entanglement cost exceeds their quantum capacity,
even under the most general quantum processes that preserve
entanglement-breaking channels. The main technical tool we introduce is the
tempered logarithmic negativity, a single-letter lower bound on the
entanglement cost that can be efficiently computed via a semi-definite program.Comment: 16+30 pages, 3 figures. v2: minor clarification