The development of quantum devices capable of outperforming classical counterparts is a significant goal. However, certifying the correct function of these devices presents a challenge, especially as complexity increases. Traditional characterization methods like tomography and classical simulation become infeasible for large systems due to exponential scaling of resources and the inherent difficulty of classically simulating quantum computations. Therefore, efficient characterization methods are crucial. Self-testing, which minimizes assumptions about the system, is a promising approach. Initially defined for quantum states, self-testing has been extended to various settings, including bipartite and multipartite systems, Bell state measurements, and prepare-and-measure scenarios. However, these existing self-testing schemes rely on quantum nonlocality and are unsuitable for single, complex quantum systems, a critical challenge in building large-scale quantum computers. This work addresses this gap by extending self-testing to the single-system setting.

Previous self-testing schemes primarily focused on entangled systems, utilizing Bell nonlocality. This paper builds upon recent work exploring self-testing using non-contextuality inequalities, which are applicable to single systems. However, these prior methods often required exact matches between experimental and ideal statistics. This research overcomes this limitation by providing a robustness curve, allowing for self-testing even when experimental statistics deviate slightly from ideal ones. The authors also compare their method to another single-system self-testing scheme based on cryptographic assumptions, noting that the latter requires a significantly larger number of qubits, making it currently impractical.

The proposed self-testing method leverages non-contextuality inequalities, specifically the KCBS inequality for qutrits (a three-level quantum system). The core assumption, termed KCBS orthogonality, requires projective measurements satisfying specific orthogonality relations. A self-test, I, is defined as a linear expression of input/output probabilities. A quantum device can make I = 0, while classical devices must yield I ≥ C > 0, where C is a constant. The KCBS configuration, characterized by a specific state and measurements, achieves I = 0. To quantify the closeness of a configuration to the KCBS configuration, total fidelity, F, is defined. A lower bound on F is established as a function of I using an SDP relaxation. This involves constructing an isometry, expressed in terms of the device's measurement operators, mapping the device's configuration to the KCBS configuration when I = 0. The calculation of F when I > 0 becomes an optimization problem, numerically solved via a hierarchy of semidefinite programs (SDPs), involving over 15,000 constraints. The experimental setup utilizes a single trapped ⁴⁰Ca⁺ ion, where the qutrit levels are encoded in the ion's energy levels. Measurements are performed using 729 nm lasers, fluorescence detection, and controlled rotations. Two types of experiments were conducted: (a) p-configuration, varying the initial state's purity, and (b) θ-configuration, modifying measurement settings. These experiments generate I values, enabling the application of the robustness curve to lower bound the total fidelity. Conventional tomography was also performed to determine the actual fidelity for comparison.

The research successfully demonstrates self-testing of a single quantum system. The theoretical framework provides a lower bound on the fidelity between an experimentally obtained configuration and the ideal KCBS configuration as a function of the KCBS value (a measure of deviation from classicality). Experiments were performed on a single ⁴⁰Ca⁺ ion using two configurations: the p-configuration (varying initial state purity) and the θ-configuration (modifying measurement angles). Both configurations showed consistency with the theoretical model; the experimental results lie above the computed lower bound on total fidelity (as expected since the experimental data only provides an estimate of the true values). The fidelity was calculated using conventional quantum tomography and compared to the theoretically derived fidelity lower bounds. This comparison confirmed the validity of the self-testing scheme and illustrated its robustness. The study also highlights the importance of quantitatively accounting for experimental deviations from the idealized assumptions (perfect detection efficiency, random measurement selection) in self-testing. The authors acknowledge the need for future work to address the effects of measurement errors on the fidelity lower bounds. The experiment is notably comprehensive in simultaneously quantifying and minimizing deviations from these basic assumptions, exceeding the scope of previous contextuality experiments.

The successful experimental self-testing of a single quantum system addresses a significant challenge in quantum technology. The method, based on contextuality rather than nonlocality, opens new avenues for characterizing complex quantum systems for which traditional methods are impractical. The robustness curve allows for realistic experimental scenarios with some deviations from idealized conditions, enhancing the applicability of self-testing. The agreement between theoretical predictions and experimental results validates the approach and underscores the potential for extending self-testing to more complex quantum systems. The comprehensive nature of the experiment, with quantitative assessment of experimental imperfections, sets a high standard for future contextuality experiments.

This research presents a significant advancement in quantum characterization. The development of a robust self-testing method for single quantum systems based on contextuality overcomes limitations of previous nonlocality-based methods. The experimental demonstration using a single trapped ion validates the theoretical framework and highlights the method's practicality. Future work should focus on extending the analysis to larger systems, exploring different non-contextuality scenarios, and incorporating a more comprehensive treatment of experimental imperfections into the robustness analysis.

The current analysis assumes nearly perfect detection efficiency and random measurement settings. Relaxing these assumptions to include measurement errors and deviations from perfect randomness is an area for future research. While the experimental setup minimizes these deviations, their full impact on the fidelity lower bound requires further investigation. The SDP involved in calculating fidelity lower bounds is computationally intensive; future work could aim to improve efficiency for scaling to larger systems.