Noise-Robust Self-Testing: Detecting Non-Locality in Noisy Non-Local Inputs

TL;DR

This work addresses how to assess and compare the robustness of self-testing non-local games to noisy inputs. It introduces three metrics—noise-tolerance, convincingness (with a $p$-value interpretation), and a gapped analytic score $\kappa_G$—to enable fair comparisons across games with different input-output structures and dimensions. Analytically and computationally, the authors show that convincingness provides the most nuanced ranking of noise-robustness, with CHSH often being the most robust under equal resources, while certain optimized 2-CHSH variants can outperform CHSH given enough resources. The framework informs practical choices for DIQKD, randomness generation, and resource-efficient entanglement certification by enabling selection of games aligned with noise models and resource budgets, and it lays groundwork for future theoretical developments in the noise-robustness of self-testing protocols.

Abstract

Non-local games test for non-locality and entanglement in quantum systems and are used in self-tests for certifying quantum states in untrusted devices. However, these protocols are tailored to ideal states, so realistic noise prevents maximal violations and leaves many partially non-local states undetected. Selecting self-tests based on their 'robustness' to noise can tailor protocols to specific applications, but current literature lacks a standardized measure of noise-robustness. Creating such a measure is challenging as there is no operational measure for comparing tests of different dimensionalities and input-output settings. We propose and study three comparative measures: noise-tolerance, convincingness, and an analytic approximation of convincingness called the gapped score. Our computational experiments and analytic framework demonstrate that convincingness provides the most nuanced measure for noise-robustness. We then show that the CHSH game has the highest noise-robustness compared to more complex games (2-CHSH variants and the Magic Square Game) when given equal resources, while with unequal resources, some 2-CHSH variants can outperform CHSH at a high resource cost. This work provides the first systematic and operational framework for comparing noise-robustness in self-testing protocols, laying a foundation for theoretical advances in understanding noise-robustness of self-tests and practical improvements in quantum resource utilization.

Figures (9)

• Figure 1: The CHSH game has 2 inputs and 2 outputs while the 2-CHSH game has 4 inputs and 4 outputs. What is inside the box (quantum or classical) is unknown to the referee, as these are self-tests.
• Figure 2: Optimization axes for non-local games, illustrating the four adjustable components of a non-local game configuration. That is, we can optimize or evaluate over the following--I: The weights, $\pi(x,y)$, of the measurements in $\mathbf{S}$, II: Quantum state $\rho$, III: The coefficients derived from $V(a,b|x,y)$ for the operators in $\mathbf{S}$, and IV: The measurement operators $\mathcal{M}_1, \mathcal{M}_2$ used to construct $\mathbf{S}$.
• Figure 3: Experiments for evaluating noise-robustness with a comparative score. Illustrating two ways to look at how a game can be robust to noise: robust across a region, or robust for a particular noise type. Green axes are varied and Red axes are fixed. 3a (left): Fixing known (i.e., optimal) weights, coefficients, and measurements while varying the visibility $\eta$ of the input state $\rho_\eta$. 3b (right): Optimizing the coefficients for a pre-selected state with visibility $\eta'$, fixing the rest of the axes. The resulting configuration from 3b can be analyzed using the method in 3a.
• Figure 4: Convincingness Behaviour Analysis for Finite Resources (low $N_{res}$). The convincingness curves were computed for the CHSH, MSG, 2-CHSH, and 2-CHSH-OPT games using the methods proposed in Sec. \ref{['sec:comp-methods']}. Each point on a curve was computed assuming $N_{res}$$\eta$-noisy EPR pairs are available to that game. The $\eta$-level for which 2-CHSH-OPT games were optimized is noted in brackets in the legend. Games which have similar $\kappa_G$, computed according to the methods in Sec. \ref{['sec:gapped-score']}, are coloured similarly. The dotted red line denotes the significance threshold, $\alpha=0.05$. The left figure shows the $\mathfrak{C}_G(\eta)$ scores ($p$-values), assuming $N_{res}=1000$ are available for all games. The right figure shows the same, but assumes $N_{res}=10,000$. All plotted values were averaged over 10 runs, each with a random seed, where the $p$-value computed for each random seed is an average of three runs with this seed.
• Figure 5: Convincingness Behaviour Analysis for Near-Infinite Resources (high $N_{res}$). The convincingness was computed the same way as in Figure \ref{['fig:pvals-N=1000,N=10000']}, with the exception that left figure shows the convincingness scores for when $N_{res}=100,000$ for all games, and the right shows the same for when $N_{res}=1,000,000$.

Theorems & Definitions (59)

• Definition II.1: Hilbert Space
• Definition II.2: Bra-ket
• Definition II.3: Qubit and $n$-Qubit System
• Definition II.4: Quantum State
• Definition II.5: Separable & Entangled States
• Definition II.6: Bipartite System
• Definition II.7: Maximally Mixed State
• Definition II.8: Maximally Entangled State
• Definition II.9: POVM
• Definition II.10: Projective Measurement