Theorem 14: NPT conversion by PPT and dually non-entangling channels

For all positive natural numbers $a,b$ and every positive semidefinite trace-one matrix $\rho$ on $\mathbb C^a\otimes\mathbb C^b$ whose first partial transpose is not positive semidefinite, 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, its party-grouped Choi matrix has positive partial transpose, and $L(\rho)$ is LOCC-distillable. Here $(X^{T_A})_{(i,j),(i',j')}=X_{(i',j),(i,j')}$, and the grouped Choi matrix is

The Choi condition requires both $J_L$ and its partial transpose on the $(u,i)$ factor to be positive semidefinite, as well as a finite Kraus representation for $L$. A channel has such a representation $L(X)=\sum_jE_jXE_j^\dagger$ with $\sum_jE_j^\dagger E_j=I$. Each non-entangling condition means having a finite Kraus representation and preserving finite nonnegative sums of tensor products of positive semidefinite trace-one matrices, including the empty sum; the dual is $(L^\vee(Y))_{ij}=\operatorname{Tr}(L(E_{ji})Y)$. Distillability means that $L(\rho)$ is a density matrix and there are channels $\Lambda_n$, each a finite LOCC map, with $\|\Lambda_n(L(\rho)^{\otimes(n+1)})-\Phi_2\|_1\to0$, where $\Phi_2=|\Omega_2\rangle\langle\Omega_2|$, $|\Omega_2\rangle=(|00\rangle+|11\rangle)/\sqrt2$, and $\|M\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{M^\dagger M}$. Finite LOCC maps are generated by identity maps and finite complete local Kraus measurements with branch-dependent finite LOCC continuations. One-dimensional input factors are permitted by the dimension hypotheses, but cannot satisfy the assumed failure of positivity of the partial transpose.