= Average monotonicity of pure-state entanglement entropy
{title2=$\sum_r p_r S(\rho_{A,r})\leq S(\rho_A)$}
For a bipartite <pure state>, <entanglement entropy> is the <Von Neumann entropy> of either <reduced density matrix>. When Alice makes a local measurement with pure conditional branches, Bob's reductions average to his original reduction. <Concavity of Von Neumann entropy> therefore gives $\sum_r p_rS(\rho_{B,r})\leq S(\rho_B)$. When Bob measures, apply concavity to Alice's reductions. In each pure branch the two reduced entropies coincide by the <Schmidt decomposition>. Refining measurement outcomes to individual <Kraus operators> and iterating proves the inequality for arbitrary finite-round <LOCC>. Discarded local systems can be measured and their outcomes refined as well, so discarding does not evade the bound.
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