Proposition 21: rank-three detector channels are not 3-non-entangling

For every natural number $d\ge3$ and every real number $\beta$ satisfying $\frac{d-3}{3}<\beta\le\frac{d-2}{2}$, the following assertions hold. Use basis indices $0,\ldots,s-1$ for $\mathbb C^s$, write $E_{ij}$ for a matrix unit, and write $|v\rangle\langle v|$ for the matrix with entries $v_i\overline{v_j}$. For a bipartite matrix, define its partial transpose on the first factor by $T_1(X)_{(i,j),(i',j')}=X_{(i',j),(i,j')}$. Let $F_d$ exchange the two factors, so its $((i,j),(i',j'))$-entry is $1$ when $i=j'$ and $j=i'$, and $0$ otherwise, and put

For any matrix $W$ on $\mathbb C^d\otimes\mathbb C^d$, define the map to matrices on $\mathbb C^2\otimes\mathbb C^2$ by

Here a density matrix means a positive semidefinite matrix of trace $1$; a separable positive matrix means any finite sum $\sum_{j=0}^{q-1}w_j A_j\otimes B_j$, where $q$ is a natural number, every $w_j\ge0$, and every $A_j,B_j$ is a density matrix, with no requirement that the weights sum to $1$; the empty sum is allowed and equals zero. A map is a channel when it has a finite Kraus representation $L(X)=\sum_jK_jXK_j^\dagger$ with $\sum_jK_j^\dagger K_j=I$. Say that a map has the separability-preserving property when it has a finite Kraus representation, without a completeness requirement, and sends every separable positive matrix to a separable positive matrix. Its trace-bilinear dual is the map defined by $(L^\vee(Y))_{ij}=\operatorname{tr}(L(E_{ji})Y)$, with no complex conjugation in this pairing. For a map from the bipartite system $A\otimes B$ to $C\otimes D$, define its party-grouped Choi matrix by

The Choi condition used here means that $L$ has a finite Kraus representation and that both $C_L$ and its partial transpose on the grouped factor $C\otimes A$ are positive semidefinite. The property with index $3$ that is excluded below means that $L$ has a finite Kraus representation and that its extension by a three-dimensional identity system on each party has the separability-preserving property across $(A\otimes\mathbb C^3)\mid(B\otimes\mathbb C^3)$; explicitly, this extension satisfies

where $u,u',v,v'\in\{0,1,2\}$. Finally, call a bipartite matrix $\sigma$ distillable here when it is a density matrix and there exists a sequence of channels $D_n$, each realizable by the following finite recursive local construction, such that $\|D_n(\sigma^{\otimes(n+1)})-\Omega_2\|_1\to0$ as $n\to\infty$: the construction starts with identity maps and permits either party to apply a finite complete local Kraus family and then, for each Kraus outcome, continue with another map in the construction, summing the resulting branches. More explicitly, an Alice step has the form $X\mapsto\sum_iD_i((K_i\otimes I)X(K_i\otimes I)^\dagger)$, with $\sum_iK_i^\dagger K_i=I$, and a Bob step uses $I\otimes K_i$ instead; local dimensions and the finite number of outcomes are natural numbers, including zero where the required equations permit it. The copies are grouped by party and indexed using the fixed bijections between strings of basis indices and their finite index sets, and $\|X\|_1=\operatorname{Re}\operatorname{tr}\sqrt{X^\dagger X}$. With these meanings, $L_{T_1(|r_d\rangle\langle r_d|)}$ is a channel, both it and its trace-bilinear dual have the separability-preserving property, it satisfies the Choi condition, its output on $\rho_{d,\beta}$ is distillable, and it does not have the property with index $3$. Moreover, for every unit vector $\eta\in\mathbb C^d\otimes\mathbb C^d$, let $\rho_\eta=|\eta\rangle\langle\eta|$. If exactly three eigenvalues of the reduced matrix $(\sum_b\eta_{i,b}\overline{\eta_{j,b}})_{ij}$, counted with multiplicity, are strictly positive, and

then $L_{T_1(\rho_\eta)}$ is a channel, both it and its trace-bilinear dual have the separability-preserving property, it satisfies the Choi condition, its output on $\rho_{d,\beta}$ is distillable, and it does not have the property with index $3$.