Truncated moment sequences and a solution to the channel separability problem

Verifying that quantum devices work in a properly quantum way has become of high relevance since the availability of the first small-scale quantum processors; an important requirement for such devices is the ability to create entanglement. To understand how entanglement evolves under physical operations acting on quantum states, we consider the problem of separability of quantum channels via the Choi matrix representation. 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 a necessary and sufficient condition for a channel to be separable or entanglement breaking. The algorithm we provide gives definiteness in the answer to the channel separability problem using semidefinite programming. The computational complexity and the performance depend on the number of variables n in the tms and on the size of the matrices involved in the semidefinite program. We investigate separability of 2-qubit and single-qutrit channels; in the latter case we can provide an answer in cases where more straight-forward separability criteria based on positive but not completely positive maps fail.