Self-testing quantum systems of arbitrary local dimension with minimal number of measurements

Quantum nonlocality, revealed by violations of Bell inequalities, enables device-independent (DI) certification tasks such as quantum cryptography, randomness generation, and certification of entanglement and system dimension based solely on observed correlations. The strongest DI method is self-testing, which infers (up to local unitaries and ancillary degrees of freedom) the underlying quantum state and measurements from maximal violation of a Bell inequality. While self-testing is well understood for qubit systems (e.g., all two-qubit entangled states via tilted CHSH), higher-dimensional (qudit) systems remain far less explored. Existing approaches to arbitrary dimensions often combine multiple two-outcome inequalities and require more than the minimal number of measurements per party, leaving open whether a single, truly d-outcome inequality with only two measurements per party can self-test qudit maximally entangled states. This work addresses these gaps by providing a self-testing statement for the d-dimensional maximally entangled state |ψ⟩_d = (1/√d) Σ_{i=0}^{d−1} |ii⟩ based on maximal violation of a single d-outcome Bell inequality, using the minimal number of two measurements per party. The approach leverages the SATWAP inequality (a d-outcome generalization of CHSH) whose Tsirelson bound is achieved by |ψ⟩_d and optimal CGLMP measurements. Consequences include a simpler scheme for parallel self-testing of N copies of |ψ⟩_2 when d=2^n, and perfect local randomness in outcomes enabling unbounded randomness expansion with only one random input bit.

Prior self-testing research has largely focused on qubit systems, with robust self-tests of the singlet and all two-qubit entangled states based on variants of CHSH (including tilted CHSH). Extension to higher local dimensions has seen fewer results; some protocols self-test arbitrary pure bipartite entangled states using many two-outcome inequalities or larger measurement sets, which are not optimal in measurement count and do not use a single truly d-outcome inequality. The SATWAP family of inequalities was introduced as Bell inequalities tailored to maximally entangled states in any local dimension, achieving a quantum bound B_Q=2(d−1) with optimal CGLMP measurements. Experimental violation of SATWAP has been demonstrated in integrated photonics platforms. Remaining open issues in the literature include analytical robustness guarantees beyond qubits, self-testing of partially entangled qudit states with minimal measurements, and optimal randomness certification using local POVMs or pairs of projective measurements.

- Scenario and correlator formalism: Consider a bipartite Bell setup with two parties (Alice, Bob), each choosing one of two d-outcome measurements per run (x,y∈{1,2}), yielding outcomes a,b∈{0,…,d−1}. Correlations p(a,b|x,y) are represented both as probabilities and via their two-dimensional Fourier-transformed correlators (A^k B^l)_{xy} using unitary d-outcome observables with eigenvalues being d-th roots of unity. Any quantum point can be realized with projective measurements and (possibly higher-dimensional) pure states, exploiting the convexity of the quantum set Q. - Bell inequality used: Employ the SATWAP Bell inequality, a truly d-outcome inequality involving only two measurements per party, written in the correlator picture as I_d = Σ_{k=1}^{d−1} [ a_k (A^k B^k)_{xy} + a_k^* (A^k B^k)_{xy}^† + a_k^* (A^k B^{l−k})_{xy} + a_k (A^k B^{l−k})_{xy}^† ], with specific complex coefficients a_k = (1−i) ω^{k/4}/2 and local bound B_L = (1/2)(3 cot(π/4d) − cot(3π/4d) − 2). Its maximal quantum value is B_Q = 2(d−1), achieved by the maximally entangled state |ψ⟩_d with optimal CGLMP measurements. - Target observables: Define Z = diag(1, ω, …, ω^{d−1}) and a unitary T_d with eigenvalues ω^j (j=0,…,d−1). These form d-outcome observables unitarily equivalent to the CGLMP optimal measurements. - Self-testing strategy: Assume maximal violation of SATWAP is observed. Using a sum-of-squares (SOS) decomposition of the Bell operator at the Tsirelson bound, derive operator constraints on Alice’s and Bob’s measurement unitaries (e.g., tracelessness and equal multiplicities of eigenvalues), implying they act on a Hilbert space that factorizes as system ⊗ auxiliary with the system dimension a multiple of d. Construct local unitaries (isometries) U_A, U_B mapping the unknown observables to canonical forms: on Bob’s side U_B B_1 U_B^† = Z_a ⊗ I_B and U_B B_2 U_B^† = T_d ⊗ I_B; on Alice’s side U_A A_1 U_A^† = (α_1 Z_a + 2 α_1^3 T_d) ⊗ I_A and U_A A_2 U_A^† = (α_1 Z_a − α_1 T_d) ⊗ I_A with α_1 = (1+i) ω^{1/4}/2. The SOS constraints further impose stabilizer-like conditions (Z_a ⊗ Z_a) |φ⟩ = |φ⟩ and (T_d ⊗ T_d) |φ⟩ = |φ⟩ on the mapped state |φ⟩. - State extraction: Solve the imposed relations to conclude that, up to local isometries and tensoring an auxiliary state, the global state equals the maximally entangled state: U_A ⊗ U_B |Ψ_AB⟩ = |ψ⟩_d ⊗ |aux_AB⟩. - Randomness certification: With Bob’s observables traceless and forming mutually unbiased-type structure in the correlator picture at maximal violation, evaluate Eve’s optimal guessing probability G(y,p) over all strategies consistent with p to be 1/d, certifying −log2 G = log2 d bits of perfect randomness per outcome. Because only two inputs are used, one random bit suffices for input selection, enabling unbounded randomness expansion as d grows.

- Main self-testing result: Maximal violation of the d-outcome SATWAP Bell inequality self-tests the d-dimensional maximally entangled state |ψ⟩_d and a specific pair of d-outcome measurements per party (unitarily equivalent to optimal CGLMP measurements), using only two measurements per subsystem. The extracted observables satisfy U_B B_1 U_B^† = Z_a ⊗ I, U_B B_2 U_B^† = T_d ⊗ I, and U_A A_1 U_A^† = (α_1 Z_a + 2 α_1^3 T_d) ⊗ I, U_A A_2 U_A^† = (α_1 Z_a − α_1 T_d) ⊗ I; the state obeys U_A ⊗ U_B |Ψ⟩ = |ψ⟩_d ⊗ |aux⟩. - Bell inequality performance: For SATWAP, the classical bound is B_L = (1/2)(3 cot(π/4d) − cot(3π/4d) − 2) and the Tsirelson bound is B_Q = 2(d−1), achieved by |ψ⟩_d with optimal measurements. - Minimal measurement setting: The protocol uses the minimal number (two) of measurements per party necessary to witness nonlocality while achieving full self-testing for arbitrary local dimension. - Randomness certification: At maximal violation, the local guessing probability G = 1/d, certifying log2 d bits of perfect randomness in the outcomes. Only one random bit is required to select the measurement input, enabling unbounded randomness expansion as d can be arbitrarily large. - Parallel self-testing implication: For d = 2^n, the construction yields a simpler parallel self-testing of N copies of the two-qubit maximally entangled state |ψ⟩_2 compared to previous methods.

The work resolves whether a single, truly d-outcome Bell inequality with only two measurements per party can self-test maximally entangled states of arbitrary local dimension. By leveraging the SATWAP inequality and an SOS-based analysis, the authors show that maximal violation uniquely determines (up to local isometries and auxiliary systems) both the target state |ψ⟩_d and the associated d-outcome measurements. This addresses practical constraints for experiments by minimizing the number of required measurements while operating in higher-dimensional systems. The result directly supports device-independent tasks: it enables simpler parallel self-testing of multiple singlets when d is a power of two and certifies the maximum possible per-outcome randomness (log2 d bits), which, combined with only one random input bit, leads to unbounded randomness expansion. The findings also suggest broader implications for understanding the geometry of the quantum correlation set at its extremal boundary and for applications in delegated quantum computation and DI cryptographic primitives.

This paper introduces a minimal-measurement, single-inequality self-testing protocol for maximally entangled states in any local dimension d. Using the SATWAP inequality and an SOS decomposition, it certifies both the state |ψ⟩_d and specific d-outcome measurements from maximal violation with only two measurements per party. Applications include simplified parallel self-testing for powers-of-two dimensions and device-independent certification of log2 d bits of perfect randomness per outcome, enabling unbounded randomness expansion. Future research directions include: deriving analytical robustness bounds for arbitrary d; adapting the inequality or measurements to self-test partially entangled qudit states with minimal measurements; and extending the framework to certify optimal local or global randomness (up to 2 log2 d bits) using local POVMs or pairs of projective measurements.

- The self-testing guarantees are exact at maximal violation; the paper does not provide general analytical robustness bounds for noise or near-maximal violations in arbitrary dimensions (beyond numerical studies for d=3). - The protocol targets maximally entangled states; self-testing of partially entangled qudit states with the same minimal measurement resources remains open. - Practical implementation relies on achieving or closely approaching the Tsirelson bound of the SATWAP inequality, which can be experimentally demanding in higher dimensions. - The work assumes finite-dimensional Hilbert spaces and does not address potential complications from dimension mismatch or imperfect projectivity of measurements in realistic devices.