Correlations in entanglement-assisted prepare-and-measure scenarios

TL;DR

This paper explores the set of correlations achievable in entanglement-assisted prepare-and-measure scenarios, providing new methods to characterize these correlations and applications for device-independent dimension testing.

Contribution

It introduces a hierarchy of semidefinite programming relaxations and an alternative relaxation scheme to analyze entanglement-assisted correlations, extending previous frameworks.

Findings

Correlations can be characterized by a hierarchy of SDP relaxations.

An alternative relaxation scheme based on informationally-restricted quantum correlations is proposed.

Device-independent tests for system dimension that include entanglement are developed.

Abstract

We investigate the correlations that can arise between Alice and Bob in prepare-and-measure communication scenarios where the source (Alice) and the measurement device (Bob) can share prior entanglement. The paradigmatic example of such a scenario is the quantum dense coding protocol, where the communication capacity of a qudit can be doubled if a two-qudit entangled state is shared between Alice and Bob. We provide examples of correlations that actually require more general protocols based on higher-dimensional entangled states. This motivates us to investigate the set of correlations that can be obtained from communicating either a classical or a quantum $d$-dimensional system in the presence of an unlimited amount of entanglement. We show how such correlations can be characterized by a hierarchy of semidefinite programming relaxations by reducing the problem to a non-commutative polynomial optimization problem. We also introduce an alternative relaxation hierarchy based on the notion of informationally-restricted quantum correlations, which, though it represents a strict (non-converging) relaxation scheme, is less computationally demanding. As an application, we introduce device-independent tests of the dimension of classical and quantum systems that, in contrast to previous results, do not make the implicit assumption that Alice and Bob share no entanglement. We also establish several relations between communication with and without entanglement as resources for creating correlations.