Non-Additivity of the Entanglement of Purification (Beyond Reasonable Doubt)

We demonstrate the convexity of the difference between the regularized
entanglement of purification and the entropy, as a function of the state. This
is proved by means of a new asymptotic protocol to prepare a state from
pre-shared entanglement and by local operations only. We go on to employ this
convexity property in an investigation of the additivity of the (single-copy)
entanglement of purification: using numerical results for two-qubit Werner
states we find strong evidence that the entanglement of purification is
different from its regularization, hence that entanglement of purification is
not additive.