A Preparation Nonstationarity Loophole in Superconducting-Qubit Bell Tests

TL;DR

This work identifies a preparation nonstationarity loophole in superconducting-qubit Bell tests, showing that slow drift in state preparation can cause different correlators to sample distinct, time-varying ensembles and relax the CHSH bound to $|S| \le 2 + 6\,\delta_{\mathrm{ens}}$. The authors formalize ensemble divergence $\delta_{\mathrm{ens}}$ and operational drift $\delta_{\mathrm{op}}$, derive the relaxed bound, and introduce Pauli-axis diagnostics as a drift witness, complemented by Qiskit-Aer simulations and IBM hardware experiments with schedule-aware analysis. Hardware results reveal statistically significant drift along fixed measurement axes even after SPAM mitigation, while CHSH-optimal measurements are confounded by basis drift and do not reliably diagnose nonstationarity. A schedule-aware lower bound shows that modest preparation divergence can explain observed Bell violations within a local-hidden-variable framework, underscoring the need for drift-aware protocols in quantum certification on noisy devices. Collectively, these findings reveal a preparation-dependent relaxation of Bell bounds on NISQ hardware and motivate drift-aware methodologies for reliable quantum certification.

Abstract

Bell or Clauser-Horne-Shimony-Holt (CHSH) tests on superconducting quantum processors are commonly interpreted under the assumption that repeated circuit executions sample a single, stationary preparation ensemble. Here we show that this assumption can be violated on contemporary hardware, with direct implications for the interpretation of observed Bell violations. We introduce an ensemble-divergence framework in which slow temporal drift of the preparation process induces context-dependent effective ensembles, even when measurement independence and locality are preserved. This leads to a relaxed Bell bound $|S| \le 2 + 6δ_{\mathrm{ens}}$, where $δ_{\mathrm{ens}}$ quantifies preparation nonstationarity. Because $δ_{\mathrm{ens}}$ is not directly observable, we develop an operational witness $δ_{\mathrm{op}}$ based on bin-resolved outcome statistics for fixed measurement channels. Using Pauli-axis measurements on IBM superconducting processors, we observe statistically significant operational drift that persists after full two-qubit readout mitigation, ruling out measurement artifacts. In contrast, drift extracted from CHSH-optimal measurements is eliminated by mitigation, demonstrating that such settings are unsuitable for diagnosing preparation nonstationarity. We further show that the observed Bell violations imply only modest ensemble divergences, comparable in scale to those required in Hall-type measurement-dependence models, but arising here solely from preparation drift combined with experimental scheduling. Our results identify a preparation-dependent loophole relevant to Bell tests on noisy intermediate-scale quantum devices and highlight the necessity of drift-aware protocols for reliable quantum certification.

Figures (4)

• Figure 1: Operational ensemble drift $\delta_{op}^{global}$ extracted from Monte-Carlo–simulated CHSH experiments under controlled preparation drift. Panels (a)–(c) correspond to the ibm_fez backend using qubit pair (0,1), while panels (d)–(f) correspond to the ibm_torino backend using qubit pair (40,41). Vertically aligned panels share the same drift amplitude: (a,d) $\theta_{\max}=0$, (b,e) $\theta_{\max}=0.01$, and (c,f) $\theta_{\max}=0.1$. For each panel, results are shown as a function of the number of temporal bins $B$ and scheduling protocol (RR: round-robin; UB: unbalanced). Blue hatched bars denote the observed $\delta_{op}^{global}$$\pm$ one standard deviation, while orange hatched bars indicate the Monte-Carlo null mean $\pm$ one standard deviation. Red asterisks indicate statistically significant deviations from the null distribution (* $p<0.05$, ** $p<0.01$, *** $p<10^{-3}$).
• Figure 2: Experimentally extracted operational ensemble drift $\delta_{\mathrm{op}}$ on (a) the ibm_fez backend using qubit pair (0,1) and (b) the ibm_torino backend using qubit pair (40,41). Results are shown for round-robin and unbalanced scheduling protocols. White and hatched blue bars denote the observed raw and readout-mitigated $\delta_{\mathrm{op}}$, respectively, while dotted and cross-hatched bars indicate the corresponding Monte-Carlo null expectations constructed using identical shot budgets and scheduling structure. Error bars denote one standard deviation. Red asterisks indicate statistically significant deviations from the null (* $p<0.05$, ** $p<0.01$).
• Figure 3: Operational drift $\delta_{\mathrm{op}}$ extracted from outcome statistics measured along CHSH-optimal axes. Panels (a,b) show raw experimental values for ibm_fez (qubit pair (0,1)) and ibm_torino (qubit pair (40,41)), respectively, compared with matched Monte Carlo (MC) null distributions for round-robin (RR) and unbalanced (UB) scheduling. While the raw data exhibit apparent deviations from the null, these differences are eliminated by mitigation (data not shown), rendering the experimental $\delta_{\mathrm{op}}$ statistically consistent with the null. This behavior indicates that $\delta_{\mathrm{op}}$ extracted from CHSH-optimal settings is dominated by basis-dependent measurement instabilities rather than genuine preparation nonstationarity.
• Figure 4: Absolute Bell parameter $|S|$ measured using CHSH-optimal settings on ibm_fez (qubit pair (0,1)) and ibm_torino (qubit pair (40,41)). Blue bars are computed from raw counts and hatched orange bars from the corresponding readout-mitigated counts, shown for round-robin (RR) and unbalanced (UB) schedules; error bars denote one standard deviation. This figure reports the standard CHSH estimator of $S$ but does not test the preparation-stationarity assumption required for the ideal bound $|S|\le 2$; stationarity is assessed independently via the bin-resolved drift diagnostics in Fig. 2. Consequently, $|S|$ should be interpreted jointly with Fig. 2 when assessing whether the operational premises of the standard CHSH derivation are satisfied on hardware.