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

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.

Figures (4)

• Figure 1: Observing nonlocality and contextuality in $\mathbb{C}^2 \otimes \mathbb{C}^3$. To observe nonlocality (CHSH), Alice performs measurements $R(\omega_0)$ and $R(\omega_2)$, while Bob measures observables $B_0$ and $B_m B_{m+1}$ and violate Eq.\ref{['CHSH_first']}. To observe contextuality (KCBS), Alice performs no measurement, and Bob measures compatible observables $B_j$ and $B_{j+1}$ to evaluate the correlators $\langle B_j B_{j+1} \rangle$. He then obtains Eq.\ref{['KCBS_ineq']}.
• Figure 2: Quantum circuit realization of the hybrid CHSH--KCBS protocol. (a) General architecture: the first stage prepares the quantum state, while the second stage implements the measurement via a qutrit Fourier test. The controlled-$U^a$ operation encodes the joint observable $U = A \otimes B$, where $A$ and $B$ denote Alice's and Bob's measurement operators. (b) Explicit implementation for the state in Eq. \ref{['stateI']}. All gate conventions (e.g., $F_3$, $R_{ij}(\theta)$, $D(\alpha,\beta)$, and controlled-$U^a$) follow Sec. \ref{['Gates']}.
• Figure 3: A state exhibiting both nonlocality and contextuality. (a) Analytical violation landscape for the hybrid CHSH-KCBS scenario. (b) Circuit-based simulation, showing excellent agreement with the analytical results. (c) One-dimensional slice at $\phi = 0$ (or $\phi = k\pi$), corresponding to the maximal CHSH violation. The intersection point identifies the optimal coexistence between nonlocality and contextuality.
• Figure 4: Scaling of the optimal coexistence point with the cycle size $n$ for State I. (a) CHSH expectation value as a function of $\theta$ for representative values of $n$ (with $\phi=0$). (b) Corresponding KCBS violation for the same set of $n$. (c) Intersection of the CHSH and KCBS violation margins $S - S_{\mathrm{classical}}$ for odd $n \in [5,55]$, defining the optimal coexistence point. The optimal angle $\theta_{\mathrm{opt}}$ decreases with $n$ (approximately $\sim n^{-1/2}$), while the maximal violation at the intersection diminishes, indicating a progressive weakening of the coexistence region.