= Tripartite Schmidt projection entanglement concentration
{title2=$R=h_2(p)$}
For $\sqrt p\,|000\rangle+\sqrt{1-p}\,|111\rangle$, a <projective measurement> of the number of ones in one party's $n$ local <qubits> gives a common type class of size $d_k=\binom nk$. The conditional <pure state> is $d_k^{-1/2}\sum_{|x|=k}|x,x,x\rangle$. It can be converted exactly and deterministically to a uniform common-label state of any dimension $m\leq d_k$: one party measures an $m$-element subset $S$ with <Kraus operator> $\Pi_S/\sqrt{\binom{d_k-1}{m-1}}$, and all parties relabel their common retained labels after receiving $S$. Choosing $m=2^f$ gives $f$ independent <GHZ states>. <Stirling's formula> and the <weak law of large numbers> give $f/n\to h_2(p)$ in probability. Measuring every local bit separately would instead destroy the coherence and leave a <product state>.
Back to article page
