# Truncated moment sequences and a solution to the channel separability problem

Abstract : We consider the problem of separability of quantum channels via the Choi matrix representation given by the Choi-Jamio{\l}kowski 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 $M_t(y)$ of order $t$. We exploit the algorithm to numerically investigate separability of families of 2-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 $N=0$.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-03017063
Contributor : Claudine Le Vaou <>
Submitted on : Friday, November 20, 2020 - 4:49:05 PM
Last modification on : Friday, December 18, 2020 - 5:20:05 PM

### Citation

Nadia Milazzo, Daniel Braun, Olivier Giraud. Truncated moment sequences and a solution to the channel separability problem. Physical Review A, American Physical Society 2020, 102 (5), ⟨10.1103/PhysRevA.102.052406⟩. ⟨hal-03017063⟩

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