  Entanglement is among the most fundamental-and at the same time
puzzling-properties of quantum physics. Its modern description relies on a
resource-theoretical approach, which treats entangled systems as a means to
enable or accelerate certain informational tasks. Hence, it is of crucial
importance to determine whether-and how-different entangled states can be
converted into each other under free operations (those which do not create
entanglement from nothing). Here, we show that the majorization lattice
provides an efficient 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 initial or final states.
