Experimental device-independent certified randomness generation with an instrumental causal structure

The paper addresses the problem of generating and certifying true randomness in a device-independent (DI) fashion, crucial for applications in cryptography, simulation, and privacy. Classical randomness is fundamentally epistemic and cannot be certified without assumptions, whereas quantum measurements can provide intrinsic randomness. Traditional DI approaches rely on Bell inequality violations (e.g., CHSH) under causal assumptions like locality and measurement independence. The authors ask whether a simpler causal structure—the instrumental scenario, central in causal inference and involving one-way communication from Alice’s outcome to Bob’s measurement choice—can support DI certified randomness. They propose and analyze a DI randomness generation protocol in this instrumental causal model, prove its security against general quantum adversaries, and experimentally demonstrate it with entangled photons, showing conditions under which it can outperform CHSH-based protocols.

Prior work established DI randomness generation via Bell inequality violations, where the degree of violation bounds extractable randomness (e.g., Pironio et al., Nature 2010). Advances include randomness amplification, expansion, and DI-QKD, with recent loophole-free tests and security against general attacks using entropy accumulation. From a causal perspective, Bell nonlocality challenges classical causal models. The instrumental causal structure, important in causal inference with latent variables, imposes instrumental inequalities (Pearl; Bonet) and was shown to admit quantum violations (Chaves et al., Nat. Phys. 2018). Techniques to quantify randomness with quantum side information use min-entropy bounds, NPA hierarchy relaxations for quantum correlations, and entropy accumulation (Dupuis et al.; Arnon-Friedman et al.). The authors build on these, adapting the EAT-based DI security framework from Bell to the instrumental scenario and leveraging classical extractors (Trevisan) for post-processing.

The authors certify randomness using violations of an instrumental inequality in a two-party temporal scenario. Alice chooses among I=3 measurements with d=2 outcomes using an instrument variable x, independent of shared resources, while Bob’s measurement choice y is set to Alice’s outcome a (y=a). The key causal assumption is that x has no direct influence on B beyond through a, i.e., p(b|x,a,λ)=p(b|a,λ). For the dichotomic-output, three-input case, they use Bonet’s instrumental inequality (1), which bounds classical correlations by 3 and admits quantum violations up to I=1+2√2. Given observed statistics p(a,b|x), they bound the conditional smooth min-entropy of outcomes against a quantum adversary with side information e. For each input x, they compute a lower bound f_x(I) on H_min via semidefinite programming that maximizes adversarial guessing probability subject to quantum constraints enforced by the NPA hierarchy, accounting for only observed terms compatible with the instrumental process. To avoid i.i.d. assumptions and handle memory attacks, they adapt the Entropy Accumulation Theorem (EAT) to the instrumental setting. Over n runs, the smooth min-entropy is bounded as H_min^ε(O^n|S,E) ≥ n t(I_exp) − v√n, where t is a convex tradeoff function determined by the expected violation I_exp in a honest (possibly noisy) implementation, v depends on the smoothing ε and protocol error ε_EAT, and S,E denote leaked and side information. The protocol uses a Bernoulli variable T per run with parameter γ: if T=0 (accumulation) they set x deterministically to 2 (chosen to give higher f(I)); if T=1 (test) they choose x uniformly from {1,2,3}. After m test runs they estimate the instrumental violation I; if I<I_exp−δ′ (δ′ experimental uncertainty) the protocol aborts; otherwise they apply the EAT bound to compute the total certified min-entropy. Implementation: They generate polarization-entangled photon pairs via SPDC in a 2-mm BBO crystal pumped at 392.5 nm; 785 nm photons are distributed to Alice and Bob. Alice chooses one of three observables using a liquid crystal and PBS, with choices seeded by NIST Randomness Beacon bits. Alice’s outcome triggers a fast micro Pockels cell (LiNbO3, <1 ns risetime) to realize active feed-forward: Bob’s photon is delayed by 600 ns using a 125 m single-mode fiber to await Alice’s outcome; the Pockels cell plus a fixed HWP (56.25°) selects Bob’s measurement O_a accordingly. Detection events are recorded to estimate I. Certified randomness is extracted using a Trevisan extractor with a public random seed (also from NIST Beacon), using a strong quantum-proof implementation; extractor parameters (seed length polylogarithmic in input size) and error ε_ext are set per security requirements. Figures provide theoretical parameter examples (e.g., n=10^12, ε=10^-6, etc.), and experimental settings use thresholds I_exp and δ′ tailored to the achieved visibility.

• Theoretically, device-independent certified randomness can be generated in the instrumental causal scenario by violating an instrumental inequality, with security against general quantum adversaries via EAT. The min-entropy per run increases monotonically with the observed violation/visibility; functions f_x(I) are obtained via SDP with NPA constraints.
• Comparative advantage: For high state visibility (≈0.98) and large invested input randomness, the total certified random bits from the instrumental protocol can exceed those from a CHSH-based protocol when both are fed the same number of input random bits. This advantage arises because each instrumental test run uses log2(3) input bits versus 2 for CHSH, and min-entropy per run is comparable in the high-visibility regime. Figure 4 shows the gain ratio H_min^Instr/H_min^CHSH > 1 for n of invested bits 10^9–10^10 at high visibilities.
• Experimental demonstration: With an expected visibility of 0.915 and threshold I_exp=3.5 (δ′=0.011), they performed 172,095 runs. The observed instrumental violation was I=3.516±0.011 (consistent with visibility v=0.9186±0.030). Using the EAT bound, they certified a smooth min-entropy of 0.031125 (per run), enabling extraction of 5,270 random bits with extractor error ε_ext=10^-6. Each run lasted ~1 s; the rate was limited by the ~700 ms response time of the liquid crystal in Alice’s station (replaceable by faster electro-optic devices for ~100 ns response).

The findings show that DI randomness generation need not rely exclusively on spatially separated Bell tests; instrumental processes with one-way communication suffice under suitable causal shielding, enabling a new platform within causal inference frameworks. The adaptation of EAT provides robustness against non-i.i.d. behavior and quantum side information. In regimes of high visibility and sufficient input randomness, the instrumental protocol can be more resource-efficient than CHSH-based schemes due to lower input entropy per test and comparable per-run min-entropy at high visibilities. The experimental proof-of-principle validates feasibility with current photonics, active feed-forward, and standard classical extractors, and identifies practical bottlenecks and clear routes to higher rates via faster modulators. The approach suggests broader applicability of instrumental causal structures to DI tasks traditionally tied to Bell scenarios.

The paper introduces and experimentally demonstrates a device-independent randomness generation protocol based on the instrumental causal scenario, proving security against general quantum adversaries via entropy accumulation and semidefinite relaxations. It shows that, for sufficiently high visibilities and large run numbers, the instrumental approach can surpass CHSH-based protocols in net certified random bits for a fixed input randomness budget. Experimentally, the authors certify and extract 5,270 random bits from 172,095 runs with an observed instrumental inequality violation. Future work includes improving hardware to boost rates (e.g., fast electro-optics for Alice), operating in regimes with higher visibility and larger datasets to realize the predicted advantage, and extending instrumental-scenario techniques to other DI tasks such as self-testing and communication complexity.

• The experimental rate is limited by Alice’s liquid crystal response (~700 ms per setting), leading to ~1 s per run; replacing with fast electro-optic modulators could dramatically improve throughput.
• The demonstrated advantage over CHSH appears only at high visibilities (~0.98) and with large invested input randomness; the current experiment operated at ~0.919 visibility and did not target that regime.
• The protocol’s causal assumption requires shielding Alice’s station so that only the outcome bit is communicated to Bob, which may impose engineering constraints.
• Min-entropy bounds rely on SDP relaxations (NPA) and finite-size EAT terms, leading to conservative estimates; only a subset of terms is tested in the instrumental process, potentially lowering certifiable randomness compared with fully tested scenarios.
• Proof-of-principle scale (172,095 runs) yields modest total random bits; scaling up is needed for practical deployment.