Theorem 15: distillable output undetected by finitely many witnesses

For all natural numbers $a,b,c,d,n$ with $a,b>0$ and $c,d\ge2$, and every family $W_0,\ldots,W_{n-1}$ of Hermitian matrices on $(\mathbb C^c\otimes\mathbb C^a)\otimes(\mathbb C^d\otimes\mathbb C^b)$ satisfying $\operatorname{Re}\operatorname{Tr}(W_iX)\ge0$ for every separable positive $X$, there exists an LOCC-distillable matrix $\sigma$ on $\mathbb C^c\otimes\mathbb C^d$ such that the replacement map $R_\sigma(X)=\operatorname{Tr}(X)\sigma$ is a channel, has a finite Kraus representation, satisfies $\operatorname{Re}\operatorname{Tr}(W_iJ_{R_\sigma})\ge0$ for every $i<n$, and obeys $R_\sigma(\rho)=\sigma$ for every positive semidefinite trace-one input $\rho$. Here separable positive means a finite nonnegative sum of tensor products of positive semidefinite trace-one matrices, including the empty sum, and the party-grouped Choi matrix has entries $J_L{}_{(u,i),(v,j);\,(u',i'),(v',j')}=L(E_{(i,j),(i',j')})_{(u,v),(u',v')}$. A channel has a finite Kraus representation $L(X)=\sum_jE_jXE_j^\dagger$ with $\sum_jE_j^\dagger E_j=I$. LOCC-distillability means that $\sigma$ is positive semidefinite of trace $1$ and that a sequence of channels $\Lambda_m$, each a finite LOCC map acting on $m+1$ copies, satisfies $\|\Lambda_m(\sigma^{\otimes(m+1)})-\Phi_2\|_1\to0$; here $\Phi_2=|\Omega_2\rangle\langle\Omega_2|$, $|\Omega_2\rangle=(|00\rangle+|11\rangle)/\sqrt2$, and $\|M\|_1=\operatorname{Re}\operatorname{Tr}\sqrt{M^\dagger M}$. Finite LOCC maps are generated by identities and finite complete local Kraus measurements followed by branch-dependent finite LOCC continuations. The family may be empty when $n=0$, in which case its witness hypotheses and all its Choi inequalities are vacuous.