More nonlocality with less incompatibility in higher dimensions: Bell vs prepare-measure scenarios

TL;DR

This paper explores the complex relationship between measurement incompatibility and nonlocality in higher-dimensional quantum systems, revealing non-monotonic behaviors and dependencies not seen in two-dimensional cases, with implications for quantum information protocols.

Contribution

It demonstrates that in higher dimensions, measurement incompatibility does not always correlate with nonlocality, and the maximum violation states depend on incompatibility levels, extending understanding beyond qubits.

Findings

Violation of CGLMP inequality is non-monotonic with incompatibility in qutrits.

Maximally violating states depend on measurement incompatibility in higher dimensions.

Incompatibility can exist without Bell inequality violation in certain measurement scenarios.

Abstract

Connecting incompatibility in measurements with the violation of local realism is one of the fundamental avenues of research. For two qubits, any incompatible pair of projective measurements can violate Clauser-Horne-Shimony-Holt (CHSH) inequality for some states, and there is a monotonic relationship between the level of measurement incompatibility (projective) and the violation. However, in the case of two qutrits, we exhibit that the violation of the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality responds non-monotonically with the amount of incompatibility; we term this more nonlocality with less incompatibility. Furthermore, unlike in the CHSH case, the maximally violating state in higher dimensions depends on the amount of measurement incompatibility. We illustrate that similar patterns can also be observed in an experimentally viable interferometric measuring technique. In such a measurement scenario, we provide an explicit example of incompatible (not jointly measurable) measurements that do not violate the CGLMP inequality for any shared quantum state. We extend our study of incompatibility in the prepare and measure scenario, focusing on quantum random access codes (QRACs). Surprisingly, we show that the monotonicity of average success probability with measurement incompatibility does not hold for higher dimensions, as opposed to two dimensions, even though the maximum probability of QRAC behaves monotonically with incompatibility.