Theorem 13: entangled inputs reach LOCC-distillable states

For all positive natural numbers $a,b$ and every positive semidefinite trace-one matrix $\rho$ on $\mathbb C^a\otimes\mathbb C^b$ that is not separable positive, there exists a channel $L$ to operators on $\mathbb C^2\otimes\mathbb C^2$ such that both $L$ and its trace-bilinear dual are non-entangling and $L(\rho)$ is LOCC-distillable in the following precise sense: $L(\rho)$ is positive semidefinite of trace $1$, and there is a sequence $\Lambda_n$, each both a channel and a finite LOCC map from $n+1$ copies of its two local systems to two qubits, such that $\|\Lambda_n(L(\rho)^{\otimes(n+1)})-\Phi_2\|_1\to0$ as $n\to\infty$, where $\Phi_2$ has entries $1/2$ when both row and column pairs have equal components and $0$ otherwise. A finite LOCC map is generated recursively from identity maps by finite complete local Kraus measurements on either party and an outcome-dependent finite LOCC continuation, summing the resulting branches. A channel has a finite Kraus representation with $\sum_jE_j^\dagger E_j=I$; a non-entangling map has a finite Kraus representation and preserves all finite nonnegative sums of tensor products of positive semidefinite trace-one matrices, including $0$. The dual is $(L^\vee(Y))_{ij}=\operatorname{Tr}(L(E_{ji})Y)$, and $\|M\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{M^\dagger M}$. The hypotheses allow one-dimensional factors, although no density matrix with such a factor satisfies the stated nonseparability assumption.