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One of the distinguishing features of quantum theory is that its measurementoutcomes are usually unpredictable or, equivalently, random. Moreover, thisrandomness is certifiable with minimal assumptions in the so-calleddevice-independent (DI) paradigm, where a device's behavior does not need to bepresupposed but can be verified through the statistics it produces. In thiswork, we explore various forms of randomness that are certifiable in thissetting, where two users can perform two binary-outcome measurements on theirshared entangled state. In this case, even though theClauser-Horne-Shimony-Holt (CHSH) Bell-inequality violation is a pre-requisitefor the generation of DI certifiable randomness, the CHSH value alone does notgenerally give a tight bound on the certifiable randomness. Here, we determinethe certifiable randomness when zero-probability constraints are incorporatedinto the task of DI randomness expansion for the standard local and globalrandomness and the so-called "blind" randomness. Asymptotically, we observeconsistent improvements in the amount of DI certifiable randomness (of allkinds) as we increase the number zero constraints for a wide range of givenCHSH Bell violations. However, if we further optimize over the allowed CHSHvalues, then benefits of these additional constraints over the standardCHSH-based protocol are only found in the case of global and blind randomness.In contrast, in the regimes of finite data, these zero constraints only give aslight improvement in the local randomness rate when compared with all existingprotocols.

Incorporating Zero-Probability Constraints to Device-Independent Randomness Expansion