= Schmidt projection entanglement concentration
{c}
{title2=$d_k=\binom nk$}
For $n$ copies of a normalized <pure state> $\alpha|00\rangle+\beta|11\rangle$, regroup Alice's and Bob's tensor factors. Alice performs a <projective measurement> of the total number $k$ of zeroes, preserving all coherence among strings with that count. The outcome has <binomial distribution>
$$
p_k=\binom nk|\alpha|^{2k}|\beta|^{2(n-k)}.
$$
Each retained string has the same complex amplitude $\alpha^k\beta^{n-k}$, so, up to a common phase, the conditional <quantum state> is
$$
|T_k\rangle=\binom nk^{-1/2}\sum_{z(x)=k}|x\rangle_A|x\rangle_B.
$$
Its equal <Schmidt coefficients> make it a <maximally entangled state> of <Schmidt rank> $\binom nk$. Alice sends the count to Bob. They can relabel the corresponding local <computational basis> strings with local unitary operations, and convert the resulting resource to <Bell pairs>. Measuring the individual bits instead would reveal the entire string and leave a <product state>.
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