Certifying optimal device-independent quantum randomness in quantum networks

TL;DR

A family of multipartite Bell inequalities that allows to certify optimal quantum randomness and self-test GHZ (Greenberger-Horne-Zeilinger) states, which are inspired from the stabilizer group of the GHZ state are presented.

Abstract

Bell nonlocality provides a device-independent (DI) way to certify quantum randomness, based on which true random numbers can be extracted from the observed correlations without detail characterizations on devices for quantum state preparation and measurement. However, the efficiency of current strategies for DI randomness certification is still heavily constrained when it comes to non-maximal Bell values, especially for multiple parties. Here, we present a family of multipartite Bell inequalities that allows to certify optimal quantum randomness and self-test GHZ (Greenberger-Horne-Zeilinger) states, which are inspired from the stabilizer group of the GHZ state. Due to the simple representation of stabilizer group for GHZ states, this family of Bell inequalities is of simple structure and can be easily expanded to more parties. Compared with the Mermin-type inequalities, this family of Bell inequality is more efficient in certifying quantum randomness when non-maximal Bell values achieved. Meanwhile, the general analytical upper bound for the Holevo quantity is presented, and achieves better performance compared with the MABK (Mermin-Ardehali-Belinskii-Klyshko) inequality, Parity-CHSH (Clauser-Horne-Shimony-Holt) inequality and Holz inequality at $N=3$, which is of particular interests for experimental researches on DI quantum cryptography in quantum networks.

Figures (4)

• Figure 1: Diagram of a typical device-independent quantum random number generation protocol. The entropy bound is derived by the DI quantum randomness certification protocol (perhaps with the help of technics for multiple rounds experiments arnon2018practicalzhang2018certifyingwang2025one). Then, the true random bit string is extracted through the extractor $\text{Ext}(O,Z)$ with raw bit string and entropy bound as inputs $O$ by consuming a short random seed $Z$. $x_i$ and $a_i$ are input and output for the lab of party $P_i$. $\{p(a_1a_2\cdots a_N|x_1x_2\cdots x_N)\}$ is the probability distribution for the measurement results among parties $P_1$, $P_2$, $\cdots$, $P_N$. $H_{\infty}^{global}(x_1x_2\cdots x_N|E)$ denotes min-entropy for given measurement inputs $x_1, x_2,\cdots,x_N$ under quantum eavesdropping operation $E$. Black dots within $P_i$ and Eve are untrusted quantum systems. Blue arrows are directions for data flows indicating data inputs and outputs for the Extractor.
• Figure 2: Guessing probability versus $\alpha$ at $N=6$ when maximal Bell values achieved. By adjusting $\alpha$, the guessing probability approaches the optimal values which are indicated by dashed lines in each subfigure.
• Figure 3: Global randomness versus Bell values for $N=3, 4$ at $\alpha=10$ and $\alpha=30$ under quantum attacks, respectively. The orange dashed dot line, orange real line and orange dashed line in Fig. \ref{['compare']} represent min-entropy, guessing probability and optimal guessing probability of $N=3$. The gray dashed x line and gray real x line in Fig. \ref{['compare']} represent min-entropy and guessing probability in Ref. woodhead2018randomness, respectively. The blue dashed dot line, real line and dashed line in Fig. \ref{['performance_N=4']} represent min-entropy, guessing probability and optimal guessing probability of $N=4$, respectively. The results are calculated with the NPA hierarchy method at level 3.
• Figure 4: Comparison on the performance of the one-outcome conditional von Neumann entropy bounds versus Bell violation for $N=3$ parties. The orange solid line is the entropy bound by this work at $\alpha=1$. The blue dashed line is the entropy bound based on the MABK inequality grasselli2021entropy with $H(X|E)\geq 1-h\left(\frac{1}{2}+\frac{1}{2}\sqrt{\frac{m^2}{8}-1}\right)$. The pink dashed dotted line is the entropy bound based on the Parity-CHSH inequality with $H(A_0|E)\geq 1-h\left(\frac{1}{2}+\frac{1}{2}\sqrt{m^2-1}\right)$. The green circle solid line is the entropy bound based on the Holz inequality with $H(A_0|E)\geq 1-h\left(\frac{1}{4}(m+1+\sqrt{m^2+2m-3}\right)$. Here, $m$ is the Bell value for Bell inequalities correspondingly grasselli2023boosting.

Theorems & Definitions (4)

• Theorem 1
• Corollary 1
• Theorem 2
• proof