  Computing the exact rate at which entanglement can be distilled from noisy
quantum states is one of the longest-standing questions in quantum information.
We give an exact solution for entanglement distillation under the set of dually
non-entangling (DNE) operations -- a relaxation of the typically considered
local operations and classical communication, comprising all channels which
preserve the sets of separable states and measurements. We show that the DNE
distillable entanglement coincides with a modified version of the regularised
relative entropy of entanglement in which the arguments are measured with a
separable measurement. Ours is only the second known regularised formula for
the distillable entanglement under any class of free operations in entanglement
theory, after that given by Devetak and Winter for one-way LOCCs. An immediate
consequence of our finding is that, under DNE, entanglement can be distilled
from any entangled state. As our second main result, we construct a general
upper bound on the DNE distillable entanglement, using which we prove that the
separably measured relative entropy of entanglement can be strictly smaller
than the regularisation of the standard relative entropy of entanglement. This
solves an open problem in [Li/Winter, CMP 326, 63 (2014)].
