# Degradable states and one-way entanglement distillation

We derive an upper bound on the one-way distillable entanglement of bipartite quantum states. To this end, we revisit the notion of degradable, conjugate degradable, and antidegradable bipartite quantum states [Smith et al., IEEE Trans. on Inf. Th. 54.9 (2008), pp. 4208-4217], and prove that for each of these classes of states the one-way distillable entanglement is equal to the coherent information, and thus given by a single-letter formula. We use these results to derive an upper bound for arbitrary bipartite quantum states, which is based on a convex decomposition of a bipartite state into degradable and antidegradable states. This upper bound is always at least as good an upper bound as the entanglement of formation. Applying our bound to the qubit depolarizing channel, we obtain an upper bound on its quantum capacity that is strictly better than previously known bounds in the high noise regime. We also transfer the concept of approximate degradability [Sutter et al., arXiv:1412.0980 [quant-ph]] to quantum states and show that this yields another easily computable upper bound on the one-way distillable entanglement. Moreover, both methods of obtaining upper bounds on the one-way distillable entanglement can be combined into a generalized one. Finally, we prove a "pretty strong" converse for the one-way distillable entanglement of a degradable state, in analogy to the corresponding result for degradable quantum channels [Morgan and Winter, IEEE Trans. on Inf. Th. 60.1 (2014), pp. 317-333].

**Comments:**21 pages, 1 figure, comments welcome

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