  Many fruitful analogies have emerged between the theories of quantum
entanglement and thermodynamics, motivating the pursuit of an axiomatic
description of entanglement akin to the laws of thermodynamics. A long-standing
open problem has been to establish a true second law of entanglement, and in
particular a unique function which governs all transformations between
entangled systems, mirroring the role of entropy in thermodynamics. Contrary to
previous promising evidence, here we show that this is impossible, and no
direct counterpart to the second law of thermodynamics can be established. This
is accomplished by demonstrating the irreversibility of entanglement theory
from first principles -- assuming only the most general microscopic physical
constraints of entanglement manipulation, we show that entanglement theory is
irreversible under all non-entangling transformations. We furthermore rule out
reversibility without significant entanglement expenditure, showing that
reversible entanglement transformations require the generation of
macroscopically large amounts of entanglement according to certain measures.
Our results not only reveal fundamental differences between quantum
entanglement transformations and thermodynamic processes, but also showcase a
unique property of entanglement which distinguishes it from other known quantum
resources.
