The quest to efficiently certify quantum states receives a significant boost from new research into Bell nonlocality, a phenomenon that demonstrates quantum mechanics’ departure from classical physics. Eloïc Vallée, Owidiusz Makuta, and Patrick Emonts, all from Universiteit Leiden, alongside Rhine Samajdar from Princeton University and Jordi Tura also from Universiteit Leiden, present a framework for constructing Bell inequalities specifically tailored to the toric code, a crucial structure in quantum error correction. This work establishes inequalities that are maximally violated by states within the toric code, and importantly, demonstrates the first-ever self-testing of a qutrit subspace, meaning the quantum state can be fully verified without needing prior knowledge of the device preparing it. By enhancing the distinction between quantum and classical behaviour, this research paves the way for device-independent certification of highly entangled quantum matter and provides powerful new tools for validating qudit states in both error-correcting codes and quantum simulation platforms.
Researchers demonstrated that these inequalities enable the self-testing of the Z3 toric code, a significant achievement in verifying quantum states without making strong assumptions about the devices used to create them. Notably, this represents the first demonstration of self-testing an entangled qutrit subspace, extending the capabilities of self-testing techniques to a new class of quantum states. The construction of these inequalities relies on carefully selecting stabilizer operators from the toric code’s ground state and transforming them into measurable quantities.
The resulting Bell expressions were analyzed using mathematical techniques to determine both the maximum quantum violation and the classical limits, confirming their ability to certify the presence of the desired quantum state. While acknowledging that experimental implementation presents challenges, the team highlights that recent advances in qudit-based topological state preparation, anyon braiding, and multi-qudit readout suggest that realizing this self-testing protocol is within reach of current quantum technologies. The authors also note that increasing the number of special sites in the system requires measurements in a larger set of bases, potentially increasing experimental complexity. Nevertheless, this work establishes a pathway toward device-independent certification of highly entangled matter and provides new tools for validating qudit states in error-correcting codes and simulation platforms.
Entanglement Certification via Toric Code Bell Inequalities
This work presents a new method for certifying entanglement using Bell inequalities tailored to the toric code, a quantum error-correcting code with topological properties. Researchers have developed inequalities that enable the verification of quantum states without relying on strong assumptions about the underlying quantum devices, a crucial step towards building trustworthy and reliable quantum technologies. The team’s approach focuses on the Z3 toric code, a specific type of quantum state that offers advantages for fault-tolerant computation. The researchers achieve this by carefully selecting a subset of stabilizer operators, which describe the symmetries of the toric code, and mapping them to measurable quantities. They then compute mathematical expressions, known as Bell inequalities, and demonstrate that these inequalities are maximally violated by all states within the toric code’s ground state, confirming that the inequalities effectively detect and certify the presence of entanglement. The results demonstrate a powerful new tool for validating quantum states and assessing the performance of quantum devices.
👉 More information
🗞 Tailoring Bell inequalities to the qudit toric code and self testing
🧠 ArXiv: https://arxiv.org/abs/2512.00146
