= Asymptotic pure-state entanglement conversion rate
{title2=$R(\psi\to\phi)=S(\rho_A^\psi)/S(\rho_A^\phi)$}
For bipartite pure source and target states with positive target <entanglement entropy>, the optimal rate of <LOCC> conversion with vanishing output error is the ratio of their entanglement entropies. Here vanishing error means that the <trace distance> between the output and $|\phi\rangle\langle\phi|^{\otimes m}$ tends to zero; a recorded failure probability must also tend to zero. <Entanglement concentration> first provides $nS(\rho_A^\psi)-o(n)$ <Bell pairs>, and <entanglement dilution> needs $mS(\rho_A^\phi)+o(m)$ such pairs. This achieves the ratio. Conversely, refine the output into pure branches. <Average monotonicity of pure-state entanglement entropy> bounds their average entropy by the input entropy; <continuity bound for quantum conditional entropy>, with trivial conditioning, makes the output entropy loss per target tend to zero. This forbids a larger rate. The convention permits approximations for finite blocks: exact deterministic conversion of every finite block can obey stricter <majorization> constraints. The reversible concentration construction is given in the https://arxiv.org/abs/quant-ph/9511030[original concentration paper].
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