Tight Analytic Bound on the Trade-Off between Device-Independent Randomness and Nonlocality

Two parties sharing entangled quantum systems can generate correlations that cannot be produced using only shared classical resources. These nonlocal correlations are a fundamental feature of quantum theory but also have practical applications. For instance, they can be used for device-independent random number generation, whose security is certified independently of the operations performed inside the devices. The amount of certifiable randomness that can be generated from some given nonlocal correlations is a key quantity of interest. Here, we derive tight analytic bounds on the maximum certifiable randomness as a function of the nonlocality as expressed using the Clauser-Horne-Shimony-Holt (CHSH) value. We show that for every CHSH value greater than the local value (2) and up to $3\sqrt{3}/2\approx 2.598$ there exist quantum correlations with that CHSH value that certify a maximal two bits of global randomness. Beyond this CHSH value the maximum certifiable randomness drops. We give a second family of Bell inequalities for CHSH values above $3\sqrt{3}/2$, and show that they certify the maximum possible randomness for the given CHSH value. Our work hence provides an achievable upper bound on the amount of randomness that can be certified for any CHSH value. We illustrate the robustness of our results, and how they could be used to improve randomness generation rates in practice, using a Werner state noise model.

Figure 1

The relationship between maximum randomness and CHSH value in the noiseless bipartite two-input two-output scenario. Plotted is the maximum achievable randomness for all quantum strategies that achieve a particular CHSH value (blue), and a reliable lower bound certified by the same CHSH value (orange) using analysis from [30]. For the interval of values $(2,3\sqrt{3}/2]$ two bits of randomness are certified by the family of strategies in Eq. (3), and for the region $[3\sqrt{3}/2,2\sqrt{2}]$ the maximum is certified by the family of strategies in Eq. (5). Note that it is not the case that a CHSH value guarantees rates given by the blue curve; the blue curve gives a tight upper bound on achievable rates in a noiseless scenario.

Figure 2

Noise comparison for Bell expressions that certify maximum randomness in the bipartite two-input two-output scenario. Our constructions (blue) are compared to the tilted Bell inequalities from [22], using both the numerical technique from [30] (orange), and the min-entropy [22] (green). All these curves have been generated by optimizing the Bell expression (within the relevant families) at each value of the noise, and the new analysis shows improved rates of randomness generation over the CHSH statistic (red).