Self-testing of a single quantum system from theory to experiment

The paper addresses whether self-testing—typically established in multipartite Bell scenarios—can be extended to a single quantum system. Conventional device characterization methods like tomography and classical simulation are infeasible for large systems. Prior self-testing relies on nonlocality and no-communication assumptions, unsuitable for single devices. The authors propose a self-testing framework for single systems using contextuality, focusing on the KCBS qutrit scenario. The goal is to construct a robust, experimentally applicable method that certifies a state and measurements up to a global isometry from observed statistics, with minimal assumptions and a quantified robustness curve linking witness value to configuration fidelity.

Self-testing originated with Mayers and Yao and has expanded to bipartite and multipartite entangled states, entangled measurements, prepare-and-measure, and steering scenarios. Existing contextuality-based self-testing proposals required exact ideal statistics and lacked practical robustness functions. A recent single-system self-testing approach under computational (cryptographic) assumptions needs thousands of qubits and is impractical experimentally. Compared to prior contextuality experiments, this work quantifies deviations from assumptions such as detection, randomness of basis choice, compatibility, and sharpness, providing a more comprehensive treatment.

Theory: The scheme assumes KCBS orthogonality (Assumption 1): an odd cycle (n=5) of repeatable, Hermitian binary projective measurements with exclusivity Π1|i Π1|j = 0 on adjacent vertices. Define the KCBS value as the sum of single-click probabilities Σi p_i with p_i = tr(Π_i ρ), and the self-test operator I := Q_n − Σi p_i, where Q_n = cos(π/n)/(1 + cos(π/n)). I=0 uniquely identifies the KCBS configuration (qutrit |ψ⟩ and projectors Π_i) up to a global isometry. For nonzero ε = I, define the total fidelity F as the sum of state and measurement fidelities after an isometry V, and derive a lower bound on F as a function of I via an SDP hierarchy. A swap-isometry construction maps an arbitrary realization to the KCBS reference using operators built from the device’s Π_i; the ensuing fidelity objective is relaxed into an NPA-like hierarchy with localizing matrix constraints and solved numerically. Numerics required symbolic derivation of an explicit SDP (≈192×192 with 16,859 constraints and 769 localizing constraints) and were solved with CVXPY/MOSEK. Experiment: A single trapped 40Ca+ ion encodes a qutrit in Zeeman sublevels: |0⟩=|2S1/2,m_j=−1/2⟩, |1⟩=|2D5/2,m_j=−3/2⟩, |2⟩=|2D5/2,m_j=−1/2⟩. Arbitrary qutrit rotations are implemented via 729 nm laser-driven transitions R1(θ1,φ1), R2(θ2,φ2). Projective measurements Π_i=U_i†|0⟩⟨0|U_i are realized by rotate-detect-unrotate sequences with fluorescence detection (PMT), near-unit detection efficiency, and random edge and order selection using a quantum RNG. Two deviation families were tested while maintaining KCBS orthogonality: (a) p-configuration: depolarizing the initial state toward maximally mixed with parameter p∈{0,0.1,0.2}; (b) θ-configuration: varying measurement directions parameterized by θ with a pure state chosen as the top-eigenvector of Σ Π_i. Each configuration measured sequential outcomes for 10,000 trials per setting to estimate p_i and Σi p_i and then applied the robustness SDP to bound F. Conventional tomography provided independent fidelity estimates for comparison.

- The authors provide the first robustness curve for contextuality-based self-testing of a single system, giving a lower bound on total fidelity F (state + five measurements) as a function of the KCBS value Σi p_i. - Experimental demonstration on a single 40Ca+ ion with random measurement selection and near-perfect detection shows self-testing in practice. - p-configuration (normal order): Σp_i ≈ 2.233 (p=0), 2.186 (p=0.1), 2.118 (p=0.2). SDP lower bounds on total fidelity: 5.296, 3.002, 2.343, respectively. Tomography yields total fidelities: 5.965, 5.901, 5.812. - p-configuration (reverse order): Σp_i ≈ 2.236 (p=0), 2.182 (p=0.1), 2.124 (p=0.2). SDP lower bounds: 5.892, 2.933, 2.386. Tomography: 5.965, 5.901, 5.812. - θ-configuration: For θ≈22.04°, Σp_i≈2.058 (normal) and 2.057 (reverse). SDP bounds: 2.561 (normal), 2.579 (reverse). Tomography: 3.956. For θ≈150.61°, Σp_i≈2.043 (normal) and 2.048 (reverse). SDP bounds: 2.222 (normal), 2.579 (reverse). Tomography: 4.050. - Experimental KCBS sums slightly exceeded √5 in some cases due to small deviations from assumptions; error bars remained consistent with the quantum bound. Across all tested points, tomography results lie above the SDP lower bounds, validating the robustness curve and demonstrating single-system self-testing.

The work answers the central question of whether self-testing can be achieved for single quantum systems by leveraging contextuality instead of nonlocality. Under a pragmatic KCBS orthogonality assumption, the authors derive a robustness curve via an SDP hierarchy and validate it experimentally in a trapped-ion qutrit. The findings show that observed KCBS statistics suffice to certify, up to a global isometry, both the state and measurement projectors with quantitative fidelity guarantees. This advances device characterization for single-system platforms, relevant to scalable quantum processors where multipartite Bell assumptions are inapplicable. The analysis highlights the need to monitor and quantify deviations from assumptions (detection efficiency, randomness, compatibility, sharpness), which were measured and found negligible here, and frames how future full robustness analyses could incorporate such deviations explicitly.

This paper introduces and experimentally validates a robust, contextuality-based self-testing framework for single quantum systems using the KCBS inequality for a qutrit. Theoretical contributions include a swap-isometry construction and an SDP-based robustness curve that lower-bounds total configuration fidelity from observed KCBS values. The trapped 40Ca+ experiment demonstrates practical self-testing with randomly chosen measurements and near-perfect detection, and tomography confirms the SDP-derived bounds. Future work includes extending to larger odd cycles (n>5), analyzing nonideal repeatability and orthogonality within the robustness curve, relaxing IID assumptions, and seeking analytic bounds via convex optimization techniques.

- The robustness curve presently assumes ideal KCBS orthogonality with repeatable, projective, Hermitian measurements; incorporating quantified deviations—errors ε in repeatability and δ in orthogonality—into the fidelity bound remains open. - Requires near-perfect detection efficiency and random, independent basis selection to avoid loopholes; although quantified here, the current theory does not explicitly parameterize these imperfections. - Assumes IID devices/runs; extending to non-IID scenarios (e.g., via martingale techniques) is suggested. - Numerical approach is computationally intensive (large SDP with many constraints), limiting immediate generalization to larger n without further optimization. - Slight deviations from assumptions can affect observed KCBS values; full analysis incorporating these effects is future work.