Truncated moment sequences and a solution to the channel separability problem

We consider the problem of separability of quantum channels via the Choi matrix representation given by the Choi-Jamiołkowski isomorphism. We explore three classes of separability across different cuts between systems and ancillae, and we provide a solution based on the mapping of the coordinates of the Choi state (in a fixed basis) to a truncated moment sequence (tms) $y$. This results in an algorithm which gives a separability certificate using semidefinite programming. The computational complexity and the performance of it depend on the number of variables $n$ in the tms and on the size of the moment matrix $M_t\left(y\right)$ of order $t$. We exploit the algorithm to numerically investigate separability of families of two-qubit and single-qutrit channels; in the latter case we can provide an answer for examples explored earlier through the criterion based on the negativity $N$, a criterion which remains inconclusive for Choi matrices with $N=0$.

Figure 1

Difference between separable (left) and entanglement-breaking (right) channels for a bipartite system $AB$ with ancillae $A^{\prime }{B}^{\prime }$. The chains represent entanglement. A separable channel preserves separability between ($A-A^{\prime }$) and ($B-B^{\prime }$), whereas an entanglement-breaking channel destroys entanglement between $A$ and all the ancillae and $B$ and all the ancillae, giving separability between ($A-B$) and ($A^{\prime }-B^{\prime }$).

Figure 2

Region of $x$ and $y$ parameters for which $C_{{\mathrm{\Phi }}_D}$ of the damping qutrit channel is positive semidefinite; the color function corresponds to the negativity values in the range of [0,1] with steps for the contour lines of 0.02. The central plateau corresponds to the region of zero negativity where the PPT criterion remains inconclusive. Gray points correspond to states found separable by our algorithm, signifying entanglement breaking channels; red points correspond to states where the algorithm needs to go to a higher extension order and remains inconclusive with our numerical resources. The white point marks the maximally mixed state.