Violation of Bell inequalities in $2\times3$ dimensional systems

Abstract: We consider the Clauser-Horn (CH) inequality for a qubit-qutrit system. We derive the necessary and sufficient conditions for the violation of the inequality as well as some sufficient conditions. Remarkably, we demonstrate the importance of local parameters in violation of the inequality. In other words, there are some families of mixed states violating the inequality for which the correlation part alone is useless in the Bell-CH test.

• The paper derives necessary and sufficient conditions for Bell-CH violations by analyzing both local polarization vectors and correlation matrices in 2×3 systems.
• It presents an explicit analytical bound based on singular values of the correlation tensor and local Bloch vectors that distinguishes local from nonlocal mixed states.
• Practical examples illustrate how certain states violate the inequality only due to local contributions, reshaping Bell nonlocality criteria in asymmetric quantum systems.

Violation of Bell Inequalities in $2\times3$ Dimensional Systems

Overview and Motivation

This paper presents a rigorous analysis of Bell nonlocality in the asymmetric bipartite scenario of qubit-qutrit ($2\times3$) systems, focusing on the Clauser-Horne (CH) variant of Bell inequalities. While for two-qubit systems the violation of Bell inequalities (such as CHSH or CH) is well-understood, especially in terms of the correlation tensor, the qubit-qutrit case presents distinctive mathematical and physical challenges. The authors derive necessary and sufficient criteria for Bell-CH violation in mixed states of $2\times3$ systems, highlighting the interplay between local parameters and the structure of quantum correlations.

Mathematical Formulation

The general state of a qubit-qutrit system is parametrized as:

$$\rho= \frac{1}{6} (I_2 \otimes I_3 + \sum_i r_i \sigma_i \otimes I_3 + \sum_j R_j I_2 \otimes \lambda_j + \sum_{ij} T_{ij} \sigma_i \otimes \lambda_j)$$

where $\sigma_i$ are Pauli matrices for the qubit subsystem, $\lambda_j$ are Gell-Mann matrices for the qutrit, $r_i$ and $R_j$ are local Bloch and polarization vectors, and $T_{ij}$ is the $3\times8$ real correlation matrix.

A central result is the derivation of the maximal quantum value of the CH expression for such states:

$2\times3$0

$2\times3$1 and $2\times3$2 are measurement directions for the qubit, $2\times3$3 encode qutrit measurement choices and characterize effective projectors (via SU(3) adjoint action).

The crucial theorem states that local hidden variable (LHV) correlations are ensured if:

$2\times3$4

for all admissible measurement directions and effective qutrit projectors.

Key Findings

Role of Local Parameters

A principal finding is that the local vector $2\times3$5, representing qubit polarization, directly influences the Bell-CH violation in $2\times3$6 systems. This stands in contradistinction to the two-qubit scenario, where only the correlation tensor governs nonlocality. Consequently, mixed states may display nonlocality exclusively due to the local polarization — even when their correlation part, $2\times3$7, alone cannot induce violation.

Analytical Bound

The authors derive an explicit upper bound for the maximal Bell-CH expression for any qubit-qutrit state:

$2\times3$8

where $2\times3$9 are the two largest singular values of the correlation matrix $2\times3$0. If this quantity does not exceed $2\times3$1, the state is guaranteed to be local. Importantly, the set $2\times3$2 of admissible qutrit projector parameters is a constrained subset of the $2\times3$3 sphere, not reachable by arbitrary $2\times3$4 rotations, further limiting the possible Bell violations.

Examples

1. Family from [Bernal et al.]: For certain $2\times3$5 parameters, states are entangled but do not violate the CH inequality unless the combined local and correlation contributions cross the threshold.
2. States with Nonzero Local Vector: Some states exhibit regions in parameter space where only the local vector’s magnitude enables violation, despite the correlation tensor alone failing to do so.
3. Maximally Entangled Mixed States (TGX states): The authors analyze states from [MMH2017-2x3MES]. They identify domains where Bell locality is ensured due to the derived bound, and demonstrate the dependence on both purity and negativity.

Theoretical Implications

The results fundamentally alter the understanding of Bell nonlocality in higher-dimensional, asymmetric bipartite systems. The necessity to consider local parameters broadens the scope for constructing and characterizing nonlocal resources in quantum networks and hybrid quantum systems.

From a mathematical standpoint, the constrained structure of admissible measurements for the qutrit is tightly linked to SU(3) symmetry and Gell-Mann algebra. The adjoint representation’s action on the seven-dimensional sphere, with the star-product constraint, is critical for precise analysis of quantum nonlocality.

Practical Implications and Future Directions

This work provides analytical tools to certify quantum correlations in platforms comprising hybrid systems (such as trapped ions, photonic qutrits, and solid-state qubits). For device-independent quantum tasks, these bounds clarify which physical states and measurements can exhibit nonlocality. Furthermore, the explicit role of local polarization should influence quantum state engineering and detection strategies.

Future research might extend these results to $2\times3$6 systems (qubit-qudit, $2\times3$7), analyze multipartite generalizations, and incorporate it into quantum information protocols such as entanglement certification in noisy environments. The authors also point towards potential applications in high-energy physics — e.g., quantum correlations in particle decays and collider events, referencing recent works on Bell tests for relativistic vector bosons and Higgs decays.

Conclusion

This paper establishes rigorous, tight conditions for Bell nonlocality in $2\times3$8 dimensional quantum systems. The necessity of including local polarization vectors in violation criteria marks a significant departure from the traditional two-qubit paradigm. The analytical bounds and constructive examples clarify the boundaries of nonlocality and locality, and the results provide a foundation for quantum information protocols in hybrid systems and inform future theoretical and experimental investigations of quantum correlations in systems of arbitrary local dimensions (2605.09474).