= Solution
<Nielsen's pure-state conversion theorem> gives the exact deterministic <LOCC> criterion. Let the <Schmidt decompositions> be $|\psi\rangle=\sum_j\sqrt{\lambda_j}|a_jb_j\rangle$ and $|\phi\rangle=\sum_j\sqrt{\mu_j}|a'_jb'_j\rangle$. The vectors $\lambda,\mu$ consist of squared <Schmidt coefficients>, equivalently the eigenvalues of either reduced <density operator>. Order each in decreasing order and pad with zeros to a common length $d$.
Then deterministic conversion is possible if and only if
$$
\boxed{\lambda\prec\mu,\quad\text{meaning}\quad
\sum_{j=1}^k\lambda_j\leq\sum_{j=1}^k\mu_j\ (1\leq k<d),\qquad
\sum_{j=1}^d\lambda_j=\sum_{j=1}^d\mu_j=1.}
$$
This is <majorization>, with the input vector majorized by the output vector. The direction matters: a maximally entangled state has a uniform vector, which is majorized by a product state's vector $(1,0,\ldots)$, so entanglement can be discarded by <LOCC>. The criterion is for certain exact conversion, without catalysts or postselection on a successful branch.
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