Characterization of quantum correlations with local dimension constraints and its device-independent applications

TL;DR

This paper develops a hierarchy of semidefinite programming relaxations to characterize quantum nonlocality under local dimension constraints, enabling device-independent certification of entanglement and non-separable measurements.

Contribution

It introduces a complete hierarchy for bounding quantum nonlocality with finite dimensions and applies it to certify tripartite entanglement and detect non-separable measurements.

Findings

First hierarchy level provides non-trivial bounds for bipartite scenarios.

Derived a Bell inequality certifying three-dimensional tripartite entanglement.

Method can detect non-separable measurements in two-qubit systems.

Abstract

The future progress of semi-device independent quantum information science depends crucially on our ability to bound the strength of the nonlocal correlations achievable with finite dimensional quantum resources. In this work, we characterize quantum nonlocality under local dimension constraints via a complete hierarchy of semidefinite programming relaxations. In the bipartite case, we find that the first level of the hierarchy returns non-trivial bounds in all cases considered, allowing to study nonlocality scenarios with four measurement settings on one side and twelve (12) on the other in a normal desktop. In the tripartite case, we apply the hierarchy to derive a Bell-type inequality that can only be violated when each of the three parties has local dimension greater than two, hence certifying three-dimensional tripartite entanglement in a device independent way. Finally, we show how the new method can be trivially modified to detect non-separable measurements in two-qubit scenarios.