= Entanglement catalysis
{title2=$\lambda\otimes c\prec\mu\otimes c\text{ despite }\lambda\not\prec\mu$}
An auxiliary entangled <pure state> can enable an otherwise forbidden deterministic bipartite <LOCC> transformation while being returned exactly and uncorrelated with the output. <Nielsen's pure-state conversion theorem> reduces this to the displayed <majorization> condition on sorted squared <Schmidt coefficients>. It does not follow from a scalar comparison of <entanglement entropies>: tensoring changes which products occupy the largest cumulative-weight sums. An exact returned catalyst can be reused; merely restoring its marginal while allowing correlations is a different resource task.<Entanglement-assisted local manipulation of pure quantum states by Daniel Jonathan and Martin B. Plenio>[Introduction and explicit two-qubit catalyst example]{ref}
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