Optimal and Feasible Contextuality-based Randomness Generation

TL;DR

This paper advances semi-device-independent randomness generation using Kochen-Specker contextuality, introducing new feasible graphs, relaxing measurement assumptions, and analyzing qubit contextuality and security threats.

Contribution

It introduces a family of orthogonality graphs for optimal randomness certification, relaxes measurement compatibility assumptions, and explores qubit contextuality and security issues.

Findings

Maximum randomness of log2(d) bits certified for certain graphs

Quantum correlations certify randomness under relaxed compatibility conditions

Single qubits are nearly contextual, challenging classical explanations

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 of randomness from a qu$d$it system with projective measurements for $d \geq 3$. We analytically derive the Lov\'asz 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 $\epsilon$-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.