with generality,

[**Result:**]{} States ${\mbox{$| \psi \rangle$}}$ and ${\mbox{$| \phi \rangle$}}$ are equivalent under stochastic local operations and classical communication —SLOCC— iff an invertible local operator —ILO— relating them \[as in, for instance, equation (\[phipsi\])\] exists.

Bipartite entanglement under SLOCC
----------------------------------

What does this classification implies in the well-known case [@Lo97; @Vi99; @Ni99] of bipartite systems? Since LU are included in SLOCC, we can take the Schmidt decomposition of a pure state ${\mbox{$| \psi \rangle$}}\in\C^n\otimes\C^m$, $n\leq m$, as the starting point for our analysis. Thus, \_[i=1]{}\^[n\_]{} = U\_AU\_B;     \_i &gt;0,  n\_ n,\[Schmidt1\] where $U_A$ and $U_B$ are some proper local unitaries, the coefficients $\lambda_i$ decrease with $i$, and $n_{\psi}$ is the number of non-vanishing terms in the Schmidt decomposition. Clearly, the ILO (\_[i=1]{}\^[n\_]{} + \_[i=n\_+1]{}\^n )1\_B transforms (\[Schmidt1\]) into a maximally entangled state \_i\^[n\_]{} , \[maxi\] which depends only on the Schmidt number $n_{\psi}$. Since SLOCC cannot modify the rank of the reduced density matrices $\rho_A$ and $\rho_B$, which is given by $n_{\psi}$, we conclude that in $\C^n\otimes\C^m$, $n\leq m$, there are $n$ different kinds of entangled states, corresponding to $n$ different classes under SLOCC. Each of these classes is characterized by a given Schmidt number, and we can choose 