Ranyiliu Chen

Random numbers are used in a wide range of sciences. In many applications, generating unpredictable private random numbers is indispensable. Device-independent quantum random number generation is a framework that makes use of the intrinsic randomness of quantum processes to generate numbers that are fundamentally unpredictable according to our current understanding of physics. While device-independent quantum random number generation is an exceptional theoretical feat, the difficulty of controlling quantum systems makes it challenging to carry out in practice. It is therefore desirable to harness the full power of the quantum degrees of freedom (the dimension) that one can control. It is known that no more than 2log(d) bits of private device-independent randomness can be extracted from a quantum system of local dimension d. In this paper we demonstrate that this bound can be achieved for all dimensions d by providing a family of explicit protocols. In order to obtain our result, we develop new certification techniques that can be of wider interest in device-independent applications for scenarios in which complete certification ('self-testing') is impossible or impractical. With our C*-algebra representation tools, we are able to device-independently certify non-projective measurements for the purpose of randomness generation. Our protocols use a class of measurements we call "balanced informationally complete" (BIC) POVMs, which we anticipate to be useful in scenarios where normally symmetric informationally complete (SIC) POVMs are useful. Moreover, we explicitly construct BIC-POVMs in every dimension, circumventing the problem with SIC-POVMs which are only conjectured to exist in every dimension.

We show that every real-valued projective measurement can be self-tested from correlations. To achieve this, we develop the theory of post-hoc self-testing, which extends existing self-tested strategies to incorporate new measurements. A sufficient and computationally feasible condition for a projective measurement to be post-hoc self-tested by a given strategy is proven. Recent work by Mančinska et al. [arXiv:2103.01729] showed that a strategy containing d+1 two-output projective measurements and the maximally entangled state with the local dimension d is self-tested. Applying the post-hoc self-testing technique to this work results in an extended strategy that can incorporate any real-valued projective measurement. We further study the general theory of iterative post-hoc self-testing whenever the state in the initial strategy is maximally entangled and characterize the iteratively post-hoc self-tested measurements in terms of a Jordan algebra generated by the initial strategy.