Expanding bipartite Bell inequalities for maximum multi-partite randomness

TL;DR

This work advances device-independent randomness certification in multipartite quantum systems by introducing expanded Bell inequalities seeded with bipartite self-tests. The authors prove a decoupling lemma showing Eve becomes uncorrelated at maximal quantum violation, enabling direct entropy-based randomness rates. They show that for odd numbers of parties, maximum MABK violation yields N bits of randomness, while for even numbers there is a threshold $m_N^*$ beyond which randomness decreases with violation; they construct new expanded Bell expressions that certify exactly $N$ bits of randomness for all even $N$ up to the maximum MABK value, and analyze the asymptotic behavior as $N$ grows. They also develop a two-parameter seed approach to map the randomness-versus-violation trade-off more generally, supported by analytical results and numerical evidence, with implications for robust multipartite DIRE protocols and multi-party cryptography.

Abstract

Nonlocal tests on multi-partite quantum correlations form the basis of protocols that certify randomness in a device-independent (DI) way. Such correlations admit a rich structure, making the task of choosing an appropriate test difficult. For example, extremal Bell inequalities are tight witnesses of nonlocality, but achieving their maximum violation places constraints on the underlying quantum system, which can reduce the rate of randomness generation. As a result there is often a trade-off between maximum randomness and the amount of violation of a given Bell inequality. Here, we explore this trade-off for more than two parties. More precisely, we study the maximum amount of randomness that can be certified by correlations with a particular violation of the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality. For any even number of parties, we find that maximum randomness cannot occur beyond a threshold quantum violation, which increases with the number of parties, and we give a conjectured form of the maximum randomness in terms of the MABK value. We also show that maximum randomness can be obtained for any MABK violation for odd numbers of parties. To obtain our results, we derive new families of Bell inequalities certifying maximum randomness from a technique for randomness certification, which we call "expanding Bell inequalities". Our technique allows a bipartite Bell expression to be used as a seed, and transformed into a multi-partite Bell inequality tailored for randomness certification, showing how intuition learned in the bipartite case can find use in more complex scenarios.

Figures (7)

• Figure 1: Using the family of strategies in \ref{['eq:evenStrat']} for even $N$ we plot the MABK value renormalized by the maximum quantum value over all strategies (i.e., $2^{(N-1)/2}$) in terms of $\theta \in [\pi/4,3\pi/4]$, and $\theta \in [5\pi/4,7\pi/4]$ respectively. The dashed lines indicate where the strategy becomes local. All points in this interval, excluding the boundaries and center point (since $\pi/2$ and $3\pi/2$ are not in $\mathcal{G}$), correspond to strategies that certify $N$ bits of randomness device-independently using our expanded Bell expressions.
• Figure 2: A similar plot to that of Fig. \ref{['fig:merm1']} for odd $N$ and second measurement angle of all parties $\theta \in [\pi/4,3\pi/4]$, and $\theta \in [5\pi/4,7\pi/4]$ respectively. All points in this interval, excluding the boundaries, correspond to strategies that can certify $N$ bits of randomness device-independently, using our expanded Bell expressions, or using the MABK inequality for the center points. For $N = 3,7,11,15$, we use the MABK expression given by \ref{['eq:MABK']}, and for $N = 5,9,13,17$ we use the same expression after relabelling every parties inputs followed by their first measurement's output.
• Figure 3: Nonlocality, as measured using local dilution, of the strategies in \ref{['eq:evenStrat']}, for second measurement angle of all parties $\theta \in [0,\pi]$. All values of $\theta \in (\pi/4,\pi/2) \cup (\pi/2,3\pi/4)$ correspond to strategies which can certify $N$ bits of maximum randomness using our technique for expanding Bell expressions. $\theta = \pi/2$ can also correspond to maximum randomness when $N$ is odd by testing the MABK inequality.
• Figure 4: The conjectured curves of maximum device-independent randomness versus MABK value, where the MABK value is normalized by its maximum quantum value $2^{(N-1)/2}$. When $N$ is odd, maximum global randomness can be achieved for any MABK value between the local and quantum bound, indicated by solid lines. The blue crosses indicate the conjectured maximum MABK value for which maximum randomness can be achieved when $N$ is even, $m_{N}^{*}$, which tends to the maximum quantum value as $N \rightarrow \infty$. To the left of the blue crosses, $N$ bits of randomness can be achieved for MABK values between the local bound (not shown on this plot) and $m_{N}^{*}$; since this is the global maximum, it is optimal, indicated by solid lines. To the right of the blue crosses, dashed lines indicate lower bounds on the trade-off between maximum randomness and MABK value from $m_{N}^{*}$ to the maximum quantum value, which are conjectured to be tight. The case of $N=2$ was proven in Ref WBC, which we reproduce with the results of the present paper.
• Figure 5: Upper bounds on the trade-off between asymptotic global DI randomness and violation of non-trivial extremal Bell inequalities, for the tripartite scenario with two binary measurements per party. The violation has been normalized by the maximum quantum value. $S_{1}$ and $S_{2}$ are given by \ref{['eq:S1']} and \ref{['eq:S2']} respectively, and $M_{3}$ is the MABK expression.

Theorems & Definitions (45)

• Definition 1: Bipartite self-test
• Definition 2: Expanded Bell expressions
• Lemma 1
• Lemma 2: Decoupling lemma
• Corollary 1
• Proposition 1: Maximum randomness for $N$ odd DharaMaxRand
• Lemma 3: $I_{\theta}$ family of self-tests
• Lemma 4: Bell inequality for maximum randomness
• Proposition 2: Maximum randomness certification
• Proposition 3