Violation of Bell Inequalities in Qubit-Qutrit Systems: The Hidden Role of Local Polarization

A new theoretical research has expanded the understanding of quantum entanglement and nonlocality by deriving the necessary and sufficient conditions for the violation of Bell inequalities in asymmetric $2\times3$ dimensional systems, known as qubit-qutrit systems. Focusing on the Clauser-Horne (CH) inequality, physicists demonstrated that, unlike traditional two-qubit systems, local polarization plays a critical and indispensable role in the violation of local realism for mixed states.

The Context of High-Energy Physics and Asymmetric Systems

The application of quantum information theory methods in high-energy physics has grown significantly, especially in the analysis of particle collision processes. Recently, collaborations at the Large Hadron Collider (LHC) reported the measurement of spin entanglement between top quarks, motivating the investigation of quantum correlations in systems composed of particles with different spins. A practical example is the qubit-qutrit system, which can emerge in collision processes involving a fermion and a vector boson, such as the system formed by a top quark and a W boson. Historically, verifying nonlocality depended on models for symmetric matrices, but the asymmetric nature of the $2\times3$ model prevents the direct use of established metrics for pure qutrits, such as the CGLMP inequality, requiring the rigorous adaptation of the CH inequality.

Mathematical Solution and the Impact of Local Polarization

Mathematical analysis revealed a fundamental divergence from two-qubit ($2\times2$ dimension) models. In the classic Bell-CHSH scenario for two qubits, the violation of the inequality depends exclusively on the bipartite correlation tensor. However, for the qubit-qutrit system, the mathematical formulation proved that the local vector polarization directly affects the evaluation outcome. The authors mathematically demonstrated that there are families of mixed states where the bipartite correlation part alone completely obeys the Bell condition, and the breach of locality only occurs due to the presence of local polarization parameters. Furthermore, the results proved to remain valid even when Positive Operator-Valued Measures (POVMs) are used instead of standard projective measurements.

“In other words, there are some families of mixed states violating the inequality for which the correlation part alone is useless in the CH test.”

Impact and Next Steps in Cryptography and Quantum Theory

The discoveries open new research avenues in both fundamental physics and applied quantum technology. From a technological standpoint, the asymmetry in the local parameter dependence suggests possible applications in cryptography and secure random-number generation. The authors suggest that a directionality phenomenon may occur, where the amount of local randomness secure against a quantum adversary varies depending on which part of the asymmetric system performs the generation. As next steps, researchers plan to investigate the behavior and security against no-signaling adversaries in this context and explore whether this strong dependence on local polarization extends to other mixed-dimensional states ($n\times k$, where $n\neq k$).