Equivalence between simulability of high-dimensional measurements and high-dimensional steering

TL;DR

This paper establishes a fundamental link between high-dimensional quantum steering and measurement incompatibility, showing how high-dimensional correlations can be simulated with lower-dimensional resources and proposing criteria for testing measurement incompatibility.

Contribution

It introduces a novel connection between high-dimensional steering and measurement incompatibility, and presents methods to simulate high-dimensional correlations with reduced resources.

Findings

High-dimensional steering relates directly to measurement incompatibility.

Certain high-dimensional entangled states can be simulated with lower-dimensional entanglement.

Criteria for testing measurement incompatibility are derived.

Abstract

The effect of quantum steering arises from the judicious combination of an entangled state with a set of incompatible measurements. Recently, it was shown that this form of quantum correlations can be quantified in terms of a dimension, leading to the notion of genuine high-dimensional steering. While this naturally connects to the dimensionality of entanglement (Schmidt number), we show that this effect also directly connects to a notion of dimension for measurement incompatibility. More generally, we present a general connection between the concepts of steering and measurement incompatibility, when quantified in terms of dimension. From this connection, we propose a novel twist on the problem of simulating quantum correlations. Specifically, we show how the correlations of certain high-dimensional entangled states can be exactly recovered using only shared randomness and lower-dimensional entanglement. Finally, we derive criteria for testing the dimension of measurement incompatibility, and discuss the extension of these ideas to quantum channels.