Coexistence of CHSH Nonlocality and KCBS Contextuality in a Single Quantum State

TL;DR

This paper explores the simultaneous presence of quantum contextuality and nonlocality within a single quantum state, analyzing their dependence on different physical parameters through analytical expressions and quantum circuit simulations.

Contribution

It provides the first analytical expressions for CHSH and KCBS inequalities in a hybrid qubit-qutrit system and identifies parameter regimes for their coexistence.

Findings

Contextuality depends solely on the population parameter p2.

Nonlocality depends on coherence parameters (X_i, Y_i).

Optimal regimes for KCBS violation do not overlap with those for CHSH violation.

Abstract

Contextuality and nonlocality are distinct manifestations at the foundation of quantum mechanics, yet their coexistence within a single quantum state remains subtle. In a hybrid CHSH--KCBS scenario involving the entanglment of a qubit and a qutrit, the qutrit supports the KCBS contextuality test, and the CHSH nonlocality arises from correlations between the qubit and qutrit. Here, we derive the analytical closed-form expressions for both inequalities and also simulate this physics on a quantum circuit. We show that contextuality is governed solely by a population parameter $p_2$, associated with the occupation of the qutrit subsystem in the $|2\rangle$ level, which plays a distinguished role in the KCBS structure. In contrast, nonlocality depends irreducibly on coherence, involving both amplitudes and phases encoded in parameters $(X_i, Y_i)$. This separation of physical resources reveals parameter regimes that optimize KCBS violation while suppress CHSH violation, and vice versa. As a result, the optimal regions do not overlap, and coexistence is restricted to a narrow intermediate regime in parameter space.