Entanglement catalysis ($\lambda \otimes c\prec \mu \otimes c\text{ despite }\lambda \nprec \mu$)

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.^[ref]