Certifying Randomness or its Lack Thereof for General Network Scenarios

TL;DR

The paper tackles certifying intrinsic randomness in general quantum networks, extending device-independent approaches beyond standard Bell scenarios. It deploys the nonfanout inflation technique to upper-bound an adversary's guessing probability $p_{ ext{guess}}^{ar{A}_{ar{x}}}$ given network constraints and a fixed observed distribution, yielding concrete randomness certificates in bilocality and triangle networks under a beyond-quantum adversary. To address the converse problem, it develops inner-approximation methods that certify lack of randomness by constructing causal models with classical parents or by embedding Bell-like structures within networks, and applies these to Fritz-inspired, entanglement-swapping, and RGB3 distributions with explicit bilinear/program-based feasibility proofs. The work highlights conceptual subtleties in randomness certification in networks, including the distinction between single-party and multipartite randomness and the limitations of current methods, and points to open problems such as non-exogenous network scenarios and quantification of the remaining randomness resource.

Abstract

The certification of intrinsic randomness is foundational to quantum information theory and central in many practical applications thereof, such as in the generation of unquestionably random numbers and in cryptographic protocols. Device-independent randomness certification based on violations of Bell inequalities has been thoroughly investigated within the standard Bell scenario. In this work, we aim to extend this line of research by exploring randomness certification in more general causal structures, namely, network scenarios. To address this task, we demonstrate how the computational tool known as the inflation technique can be adapted. As proof of concept, we use inflation to certify randomness relative to a beyond-quantum adversary for sample probability distributions obtained in the bilocality and triangle scenarios. Complementarily, we also provide computational methods for the problem of certifying an absence of randomness, which should not be conflated with certifying the classicality of a given probability distribution. We conclude with a discussion of conceptual subtleties regarding randomness certification in networks, highlighting important open problems in this nascent research field.

Figures (14)

• Figure 1: Represenation of the standard Bell scenario (a) and the standard Bell scenario with an eavesdropper, Bell+E (b).
• Figure 2: Unpacked bilocality scenario for Charlie's settings. The blue triangles represent the classical sources, while the orange ones represent nonclassical sources.
• Figure 3: Triangle scenario with two classical sources and bilocality scenario with one classical source.
• Figure 4: Different extensions of the standard Bell scenario to include an eavesdropper. In (a) Eve is "listening to" the source, whereas in (b) she is "controlling" the source.
• Figure 5: Representation of the standard Bell scenario where the settings are produced by classical latent sources.

Theorems & Definitions (2)

• Proposition 3.1
• Proposition B.1