f transforming a bipartite entangled pure state into another one allowing only local operations on the separate subsystems respectively and classical communication between them (or LOCC for short). The asymptotic case when arbitrarily large number of copies are provided is considered by Bennett and his collaborators [@BBPS96]. While in deterministic and finite manner, the first and significant step was made by Nielsen [@NI99] who discovered the connection between the theory of majorization in linear algebra [@MO79] and entanglement transformation. Nielsen proved that a bipartite entangled pure state $|\psi_1\rangle$ can be transformed into another bipartite entangled pure state $|\psi_2\rangle$ by LOCC if and only if $\lambda_{\psi_1}\prec\lambda_{\psi_2}$, where the probability vectors $\lambda_{\psi_1}$ and $\lambda_{\psi_2}$ denote the Schmidt coefficient vectors of $|\psi_1\rangle$ and $|\psi_2\rangle$, respectively. Here the symbol $\prec$ stands for ‘majorization relation’. Generally, an $n$-dimensional real vector $x$ is said to be majorized by another $n$-dimensional real vector $y$, denoted by $x\prec y$, if the following relations hold: $$\sum_{i=1}^l x^\downarrow_i \leq \sum_{i=1}^l y^\downarrow_i {\rm \
\ \ for\ any\ \ \ }1\leq l\leq n,$$ with equality holding when $l=n$, where $x^\downarrow$ denotes the vector obtained by rearranging the components of $x$ in nonincreasing order.

Nielsen’s theorem gives a necessary and sufficient condition when two entangled pure states are comparable in the sense that one can be transformed into another by LOCC. There exist, however, incomparable states such that any one cannot be transformed into another only using LOCC. To treat the case of transformations between incomparable states, Vidal [@Vi99] generalized Nielsen’s work by allowing probabilistic transformations. He found that the maximal probability of transforming $|\psi_1\rangle$ into $|\psi_2\rangle$ by LOCC can be calculated by $$\label{eq:Vidal}
P(|\psi_1\rangle \rightarrow |\psi_2\rangle)=\min_{