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.
Highlights
We investigate the correlations that can be generated in prepare-and-measure experiments in which parties share entanglement and communicate either classical or quantum systems of a given dimension
We show that the strongest forms of quantum correlations require protocols that go beyond the paradigmatic quantum densecoding protocol and we develop general methods for bounding the correlations that can be obtained in such experiments when an unlimited amount of entanglement is allowed
How can one overcome this limitation in both a conceptual and practical manner? Second, we show that the strongest correlations possible from EA qubit communication in general require high-dimensional entanglement [70]
Summary
Much research has been directed at studying the correlations p(b|x, y) that arise from the communication of a classical or quantum d-dimensional system This covers a wide range of topics, including foundations of quantum theory [1,2], dimension witnessing [3,4,5,6], random access coding [7,8,9], quantum random-number generation [10,11], quantum key distribution [12,13], self-testing [14,15,16], and various protocols for characterizing and certifying quantum devices [17,18,19]. While such questions have been the topic of previous research efforts [26,32,39,40,41,42], our analysis requires no additional assumptions and is tolerant to noise
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