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arXiv:2405.14367 · Quantum

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

Uta Isabella Meyer, Ivan Šupić, Damian Markham, Frédéric Grosshans

Published Jun 15, 2026• 0 citations
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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 π / 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 and SATWAP violation with the maximally entangled state through simple phase-difference alignment.

DOI
10.22331/q-2026-06-15-2139

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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 π / 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 and SATWAP violation with the maximally entangled state through simple phase-difference alignment.
We evaluate the maximal Clauser-Horne-Shimony-Holt (CHSH) violation for a generic (typically mixed) qubit-qudit state, obtaining easily computable expressions in arbitrary qudit dimension. This represents the optimal (2-2-2) Bell nonlocality for this kind of systems. The work generalizes the well-known Horodeckis result for a qubit-qubit setup. We also give simple lower and upper bounds on that violation. We apply our general results to address a number of issues, namely, we obtain a bound on the degree of purity required in a system to exhibit nonlocality and study the statistics of nonlocality in random density matrices. In addition, we show the impossibility of improving the amount of CHSH violation by embedding the qudit in a Hilbert space of larger dimension. Finally, the general result is illustrated with a family of density matrices in the context of a qubit-qutrit system.
Abstract Violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality by a bipartite quantum state is now used in many quantum applications. However, the explicit analytical expression for the maximal value of the CHSH expectation under local Alice and Bob spin-s measurements is still known only for s = 1 / 2 . In the present article, for an arbitrary state of two spin-s qudits, each of dimension d = 2 s + 1 ⩾ 2 , we introduce the notion of the spin-s correlation matrix, which has dimension 3 × 3 for all s ⩾ 1 2 ; establish its relation to the general correlation ( d 2 − 1 ) × ( d 2 − 1 ) matrix of this state within the generalized Pauli representation and derive in terms of the spin-s correlation matrix the explicit analytical expression for the maximal value of the CHSH expectation under local Alice and Bob spin-s measurements in this state. Specifying this general expression for the two-qudit Greenberger–Horne–Zeilinger state, the nonlocal two-qudit Werner state, and some nonseparable pure two-qudit states, we find that, under local Alice and Bob high-spin ( s ⩾ 1 ) measurements in each of these nonseparable states, including the maximally entangled one, the CHSH inequality is not violated. Moreover, unlike the case of spin- 1 / 2 measurements, where each pure nonseparable two-qubit state violates the CHSH inequality and the maximal value of its CHSH expectation increases monotonically with a growth of its entanglement, the situation under high-spin measurements is quite different—for a pure two-qudit state with a higher degree of entanglement, the maximal value of the CHSH expectation turns out to be less than for a pure two-qudit state with lower entanglement and even for a separable one.
The Clauser-Horne-Shimony-Holt inequality (CHSH) is one of the most popular and well-studied witnesses of Bell's theorem, separating classical from quantum correlations. In this work, for every $d\ensuremath{\ge}2$, we present a generalization of the CHSH inequality with the property that maximal violation is achieved uniquely by the maximally entangled state of local dimension ddd. This provides an avenue for device-independent certification of maximally entangled states of arbitrary local dimension.
Abstract Nonlocality, manifested by the violation of Bell inequalities, indicates entanglement within a joint quantum system. A natural question is how much entanglement is required for a given nonlocal behavior. Here, we explore this question by quantifying entanglement using a family of generalized Clauser–Horne–Shimony–Holt-type Bell inequalities. Given a Bell-inequality violation, we derive analytical lower bounds on the entanglement of formation, a measure related to entanglement dilution. The bounds also lead to an analytical estimation of the negativity of entanglement. In addition, we consider one-way distillable entanglement tied to entanglement distillation and derive tight numerical estimates. With the additional assumptions of qubit-qubit systems, we find that the relationship between entanglement and measurement incompatibility is not simply a trade-off under a fixed nonlocal behavior. Furthermore, we apply our results to two realistic scenarios—non-maximally entangled and Werner states. We show that one can utilize the nonlocal statistics by optimizing the Bell inequality for better entanglement estimation.
We construct logical Bell inequalities whose violations are determined by the Wigner function of a reduced quantum state. For odd-prime qudits, each phase-space point of any nonempty proper subsystem defines an inequality, violated precisely when the corresponding Gross Wigner value is negative. This correspondence allows us to quantify the contextuality that a reduced state guarantees for the composite system. The normalized violation gives a lower bound on the contextual fraction in the complete Weyl measurement scenario, valid for every joint state with that marginal, irrespective of its correlations. We identify a family of product states that saturates this bound. For these states, one marginal Wigner value fixes the contextual fraction of the entire system.
Nonlocality is one of the most important resources for quantum information protocols.The observation of nonlocal correlations in a Bell experiment is the result of appropriately chosen measurements and quantum states.We quantify the minimal purity to achieve a certain Bell value for any Bell operator.Since purity is the most fundamental resource of a quantum state, this enables us also to quantify the necessary coherence, discord, and entanglement for a given violation of two-qubit correlation inequalities.Our results shine a light on the Clauser-Horne-Shimony-Holt inequality by showing that for a fixed Bell violation an increase in the measurement resources does not always lead to a decrease of the minimal state resources.