Improved device-independent randomness expansion rates using two sided randomness

TL;DR

This work investigates how two-sided randomness can markedly improve device-independent randomness expansion rates by bounding extractable randomness through the entropy accumulation theorem. It develops both analytic and numerical bounds for six key entropies in CHSH-based DIRE protocols, and demonstrates that input randomness recycling and biased-input schemes can yield orders-of-magnitude rate gains under realistic experimental parameters. The authors provide upper and lower bounds for the relevant entropy curves, connect them via convex lower bounds, and propose two non-spot-checking protocols that close locality loopholes while substantially boosting practical expansion rates. The results suggest that two-sided randomness methods can make DIRE far more practical, while also highlighting open problems in analytic bounds, extended EAT applicability, and higher-dimensional Bell scenarios.

Abstract

A device-independent randomness expansion protocol aims to take an initial random string and generate a longer one, where the security of the protocol does not rely on knowing the inner workings of the devices used to run it. In order to do so, the protocol tests that the devices violate a Bell inequality and one then needs to bound the amount of extractable randomness in terms of the observed violation. The entropy accumulation theorem lower bounds the extractable randomness of a protocol with many rounds in terms of the single-round von Neumann entropy of any strategy achieving the observed score. Tight bounds on the von Neumann entropy are known for the one-sided randomness (i.e., where the randomness from only one party is used) when using the Clauser-Horne-Shimony-Holt (CHSH) game. Here we investigate the possible improvement that could be gained using the two-sided randomness. We generate upper bounds on this randomness by attempting to find the optimal eavesdropping strategy, providing analytic formulae in two cases. We additionally compute lower bounds that outperform previous ones and can be made arbitrarily tight (at the expense of more computation time). These bounds get close to our upper bounds, and hence we conjecture that our upper bounds are tight. We also consider a modified protocol in which the input randomness is recycled. This modified protocol shows the possibility of rate gains of several orders of magnitude based on recent experimental parameters, making device-independent randomness expansion significantly more practical. It also enables the locality loophole to be closed while expanding randomness in a way that typical spot-checking protocols do not.

Figures (6)

• Figure 1: Graphs of the rates for (a) the one-sided and (b) the two-sided randomness with uniformly chosen inputs. Each of these curves has a non-linear part and the blue curves do not have a linear part.
• Figure 2: Graphs of the conjectured rates and lower bounds for (a) $G_{A|XYE}$ (b) $G_{AB|X=0,Y=0,E}$ with uniformly chosen inputs. For $G_{AB|X=0,Y=0,E}$ we also show a lower bound from Brown et al. BFF2022. We also demonstrate that the lower bound for $G_{AB|X=0,Y=0,E}$ can be tightened by refining the partitioning of the domain for a specific point (due to the increased computation time, we did not do this throughout).
• Figure 3: Graphs of the net rate of certifiable randomness according to the EAT for (a) the spot checking protocol (Protocol \ref{['prot:spotcheck']}), (b) the protocol with recycled input randomness (Protocol \ref{['prot:nonspotcheck']}), and (c) the protocol with biased local random number generators (Protocol \ref{['prot:biased']}), showing the variation with the number of rounds for three different scores, $\omega$. The error parameters used were $\epsilon_S=3.09\times10^{-12}$ and $\epsilon_C=10^{-6}$. For each point on the curve (a) an optimization over $\gamma$ was performed to maximize the randomness; similarly, the values of $\zeta^A=\zeta^B$ were optimized over to generate the curves in (c).
• Figure 4: Graphs of the net rate of certifiable randomness according to the EAT for (a) the spot checking protocol (Protocol \ref{['prot:spotcheck']}), (b) the protocol with recycled input randomness (Protocol \ref{['prot:nonspotcheck']}), and (c) the protocol with biased local random number generators (Protocol \ref{['prot:biased']}), showing the variation with the CHSH score $\omega$. The round numbers, $n$, are indicated in the legend. The error parameters used were $\epsilon_S=3.09\times10^{-12}$ and $\epsilon_C=10^{-6}$. As in Figure \ref{['fig:EAT_n']}, the values of $\gamma$ (for (a)) and $\zeta^A=\zeta^B$ (for (c)) were optimized over for each point.
• Figure 5: (a) Two-sided and (b) one-sided entropy curves conditioned on $X$, $Y$ and $E$ with uniform input distribution.

Theorems & Definitions (100)

• Lemma 1
• proof
• Lemma 2
• proof
• Conjecture 1
• Definition 1
• Definition 2
• Definition 3
• Lemma 3
• Theorem 1: Naimark's theorem