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We show that the majorization lattice provides the appropriate framework inorder to characterize the allowed transformations of pure entangled statesunder local operations and classical communication. The underlying notions ofmeet $\land$ 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 forthe (single-copy) conversion of incomparable bipartite pure states, which wename greedy and thrifty. Both protocols reduce to Vidal's protocol [G. Vidal,Phys. Rev. Lett. 83, 1046 (1999)] if the initial and final states arecomparable, but otherwise the thrifty protocol can be shown to be superior tothe greedy protocol as it yields a more entangled residual state when it fails(they both yield the same entangled state with the same optimal probabilitywhen they succeed). Finally, we consider the generalization of these protocolsto entanglement transformations involving multiple possible initial or finalstates.

Probabilistic pure state conversion on the majorization lattice