Proposition 17: complete preservation and Choi classes

For all positive natural numbers $d,a,b$ and every map $L$ from operators on $\mathbb C^d\otimes\mathbb C^d$ to operators on $\mathbb C^a\otimes\mathbb C^b$ having a finite Kraus representation $L(X)=\sum_jE_jXE_j^\dagger$, without any trace-preservation requirement, the following equivalences hold: $L$ has a separable positive party-grouped Choi matrix if and only if its amplification by a $d$-dimensional auxiliary system on each party is non-entangling, and this is equivalent to its amplification being non-entangling for every positive auxiliary dimension $k$; likewise, its party-grouped Choi matrix and that matrix’s first partial transpose are both positive semidefinite if and only if its amplification by $d$-dimensional auxiliaries preserves positive-partial-transpose matrices, and this is equivalent to that preservation property for every positive auxiliary dimension $k$. Precisely, the grouped Choi entries are $J_L{}_{(u,i),(v,j);\,(u',i'),(v',j')}=L(E_{(i,j),(i',j')})_{(u,v),(u',v')}$, and the amplification acts by

A non-entangling map has a finite Kraus representation and preserves every separable positive matrix, meaning every finite nonnegative sum of tensor products of positive semidefinite trace-one matrices, including $0$. Positive-partial-transpose preservation requires a finite Kraus representation and preservation of the set of matrices $X$ for which both $X$ and $X^{T_A}$ are positive semidefinite, where $(X^{T_A})_{(i,j),(i',j')}=X_{(i',j),(i,j')}$. Each fixed-$k$ condition also explicitly requires $k>0$ and a finite Kraus representation for $L$; the complete conditions quantify over all positive $k$, excluding $k=0$.