Qudit Clauser-Horne-Shimony-Holt Inequality and Nonlocality from Wigner Negativity

TL;DR

This paper introduces a new CHSH inequality for qudits linked to Wigner negativity, demonstrating maximal violation by certain stabilizer states and connecting Wigner negativity with nonlocality and contextuality in higher-dimensional quantum systems.

Contribution

It generalizes the CHSH inequality for qudits based on Wigner negativity and shows how Wigner negativity relates to nonlocality and contextuality in higher-dimensional quantum states.

Findings

Maximal violation of the inequality by stabilizer states with high Wigner negativity

Bell operator measures Wigner negativity volume and singlet fraction

Wigner negativity serves as a witness for contextuality in bipartite qudit states

Abstract

Nonlocality is an essential concept that distinguishes quantum from classical models and has been extensively studied in systems of qubits. For higher-dimensional systems, certain results for their two-level counterpart, like Bell violations with stabilizer states and Clifford operators, do not generalize. On the other hand, similar to continuous variable systems, Wigner negativity is necessary for nonlocality in qudit systems. We propose a new generalization of the CHSH inequality for qudits by inquiring correlations related to the Wigner negativity of stabilizer states under the adjoint action of a generalization of the qubit $\pi/8$-gate. A specified stabilizer state maximally violates the inequality among all qudit states based on its Wigner negativity. The Bell operator not only serves as a measure for the singlet fraction but also quantifies the volume of Wigner negativity. Additionally, we show how a bipartite entangled qudit state can serve as a witness for contextuality when it exhibits Wigner negativity. Furthermore, we identify rational-phase diagonal unitaries as the key resource that exactly reproduce the CGLMP violation with the maximally entangled state through simple phase-difference alignment.