Optimal and Feasible Contextuality-based Randomness Generation

• URL: http://arxiv.org/abs/2412.20126v1
• Date: Sat, 28 Dec 2024 12:11:07 GMT
• Title: Optimal and Feasible Contextuality-based Randomness Generation
• Authors: Yuan Liu, Ravishankar Ramanathan,
• Abstract summary: Semi-measurement-independent randomness generation protocols based on Kochen-Specker contextuality offer attractive features. We show that a single qubit is non-contextual, i.e., the qubit correlations can be explained by a non-contextual hidden variable (NCHV) model. We point out possible attacks by quantum and general consistent (non-signalling) adversaries for certain classes of contextuality tests.
• Score: 4.2126604059714685
• License:
• Abstract: Semi-device-independent (SDI) randomness generation protocols based on Kochen-Specker contextuality offer the attractive features of compact devices, high rates, and ease of experimental implementation over fully device-independent (DI) protocols. Here, we investigate this paradigm and derive four results to improve the state-of-art. Firstly, we introduce a family of simple, experimentally feasible orthogonality graphs (measurement compatibility structures) for which the maximum violation of the corresponding non-contextuality inequalities allows to certify the maximum amount of $\log_2 d$ bits from a qu$d$it system with projective measurements for $d \geq 3$. We analytically derive the Lovasz theta and fractional packing number for this graph family, and thereby prove their utility for optimal randomness generation in both randomness expansion and amplification tasks. Secondly, a central additional assumption in contextuality-based protocols over fully DI ones, is that the measurements are repeatable and satisfy an intended compatibility structure. We frame a relaxation of this condition in terms of $\epsilon$-orthogonality graphs for a parameter $\epsilon > 0$, and derive quantum correlations that allow to certify randomness for arbitrary relaxation $\epsilon \in [0,1)$. Thirdly, it is well known that a single qubit is non-contextual, i.e., the qubit correlations can be explained by a non-contextual hidden variable (NCHV) model. We show however that a single qubit is \textit{almost} contextual, in that there exist qubit correlations that cannot be explained by $\epsilon$-ontologically faithful NCHV models for small $\epsilon > 0$. Finally, we point out possible attacks by quantum and general consistent (non-signalling) adversaries for certain classes of contextuality tests over and above those considered in DI scenarios.

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