= Deterministic reduction of maximally entangled Schmidt rank
{title2=$|\Phi_d\rangle\xrightarrow{\mathrm{LOCC}}|\Phi_m\rangle\quad(m\leq d)$}
Let $|\Phi_d\rangle=d^{-1/2}\sum_{j=0}^{d-1}|j,j\rangle$ and $1\leq m\leq d$. It can be converted deterministically by <LOCC> to $|\Phi_m\rangle$. For each $m$-element subset $S$ of the local basis labels, let $\Pi_S$ be its projector and let Alice's <Kraus operator> be $K_S=\Pi_S/\sqrt{\binom{d-1}{m-1}}$. Every label occurs in $\binom{d-1}{m-1}$ subsets, so $\sum_SK_S^\dagger K_S=I$. Each outcome has <probability> $m/[d\binom{d-1}{m-1}]=1/\binom dm$ and leaves the <maximally entangled state> $m^{-1/2}\sum_{j\in S}|j,j\rangle$. Alice reports $S$, and both parties relabel these $m$ basis states to $0,\ldots,m-1$ by <local unitary operations>.
For $d=6$, $m=4$, only three outcomes are needed: partition the six labels into disjoint pairs $T_0,T_1,T_2$ and use $K_r=(I-\Pi_{T_r})/\sqrt2$. Each label is retained twice, proving completeness, and each outcome has <probability> $1/3$. Every branch leaves four equal <Schmidt coefficients>, hence two independent <Bell pairs> after binary relabelling.
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