  We show that the majorization lattice provides the appropriate framework in
order to characterize the allowed transformations of pure entangled states
under local operations and classical communication. The underlying notions of
meet $\land$ and join $\lor$ in the majorization lattice lead us to define,
respectively, the optimal common resource and optimal common product states.
Based on these two states, we introduce two optimal probabilistic protocols for
the (single-copy) conversion of incomparable bipartite pure states, which we
name greedy and thrifty. Both protocols reduce to Vidal's protocol [G. Vidal,
Phys. Rev. Lett. 83, 1046 (1999)] if the initial and final states are
comparable, but otherwise the thrifty protocol can be shown to be superior to
the greedy protocol as it yields a more entangled residual state when it fails
(they both yield the same entangled state with the same optimal probability
when they succeed). Finally, we consider the generalization of these protocols
to entanglement transformations involving multiple possible initial or final
states.

Quantum Physics