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We study the problem of deterministic transformations of an \textit{initial}pure entangled quantum state, $|\psi\rangle$, into a \textit{target} pureentangled quantum state, $|\phi\rangle$, by using \textit{local operations andclassical communication} (LOCC). A celebrated result of Nielsen [Phys. Rev.Lett. \textbf{83}, 436 (1999)] gives the necessary and sufficient conditionthat makes this entanglement transformation process possible. Indeed, thisprocess can be achieved if and only if the majorization relation $\psi \prec\phi$ holds, where $\psi$ and $\phi$ are probability vectors obtained by takingthe squares of the Schmidt coefficients of the initial and target states,respectively. In general, this condition is not fulfilled. However, one canlook for an \textit{approximate} entanglement transformation. Vidal \textit{et.al} [Phys. Rev. A \textbf{62}, 012304 (2000)] have proposed a deterministictransformation using LOCC in order to obtain a target state$|\chi^\mathrm{opt}\rangle$ most approximate to $|\phi\rangle$ in terms ofmaximal fidelity between them. Here, we show a strategy to deal withapproximate entanglement transformations based on the properties of the\textit{majorization lattice}. More precisely, we propose as approximate targetstate one whose Schmidt coefficients are given by the supremum between $\psi$and $\phi$. Our proposal is inspired on the observation that fidelity does notrespect the majorization relation in general. Remarkably enough, we find thatfor some particular interesting cases, like two-qubit pure states or theentanglement concentration protocol, both proposals are coincident.

Approximate transformations of bipartite pure-state entanglement from the majorization lattice