QIP2021 | The PPT-squared conjecture holds for all Choi type map (Satvik Singh)

Authors: Satvik Singh and Ion Nechita
Affiliations: Department of Physical Sciences, Indian Institute of Science Education and Research, Mohali | Université de Toulouse

Abstract:
In the rapidly developing field of Quantum technologies, the task of entanglement distribution between two parties occupies a central stage in many important protocols. However, as the distance between the two parties increases, the error probability in any transmission channel gets larger, resulting in degradation of the quality of the distributed entanglement. To overcome this problem, quantum repeater devices are used. The basic idea of such a device is to split up the long transmission channel into shorter manageable segments, each of which can be provided with high fidelity entangled states. Then, the well-known entanglement swapping technique can be used to transfer the entanglement from the intermediate segments to the ends of the long channel. A key conjecture in this regard was proposed by M. Christandl, which states that all PPT entangled states are useless from the perspective of repeater devices, since the swapping of entanglement in such states inevitably leads to a separable state. The conjecture admits an equivalent formulation in terms of linear maps, where it amounts to saying that the composition of any two PPT maps (these are the maps which are both completely positive and completely copositive) is entanglement-breaking. In the present work, we prove that this conjecture holds for all linear maps which are covariant under the diagonal unitary group’s action. Many salient examples like the Choi-type maps, Schur multipliers, Classical maps, etc. lie in this class. Our proof relies on a generalization of the matrix-theoretic notion of factor width for pairwise completely positive matrices, as well as on our previous characterization of the aforementioned class of maps. Hence, in a nutshell, our research proves the unsuitability of a large class of states from the perspective of repeater protocols and thus significantly contributes to the solution of a long-standing open problem in quantum information theory.

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