# Is entanglement nonincreasing on average by local operations for all possible ensemble decompositions?

We know for a pure state conversion $$|\psi \rangle \rightarrow_\textrm{LOCC} |\phi \rangle$$ via local operation and classical communication (LOCC), an entanglement monotone should not increase, that is, $$E(|\psi \rangle) \geq E(|\phi \rangle)$$.

Say for a pure to mixed state conversion $$|\psi \rangle \rightarrow_\textrm{LOCC} \rho$$ given by the unique ensemble $$l = \{ q_l, |\xi_l \rangle \} \in \mathcal{D}$$ fixed by pointer states (which are omitted), where $$\mathcal{D}$$ is the set of all decompositions for $$\rho$$.

Naturally, the convex roof extension $$E_\textrm{min}(\rho) = \min_\mathcal{D} \Sigma_i q_i |\xi_i \rangle \langle \xi_i|$$ should not increase, nor should that of the obtained ensemble $$l$$ on average, i.e., $$E(|\psi \rangle) \geq E_l(\rho) \geq E_\textrm{min}(\rho)$$.

So, is it for certain there is not some decomposition $$j = \{ q_j, |\xi_j \rangle \}$$ that violates the nonincreasing criterion for $$E$$ on average, i.e., $$E_j(|\rho \rangle) \geq E(\psi \rangle)$$?

You don't even define $$E_j(\rho)$$. This makes your question hard to answer.
• $|\psi \rangle \rightarrow_\textrm{LOCC} \mathbb{I}/d$ where $\mathbb{I}/d$ is given by maximally entangled states ensemble $\{ p_i, |\phi \rangle_i \}$ marked by pointer states. – 2ub Sep 13 '20 at 3:38