Sigurd Anker Laursen Storgaard

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.

In this work, we study a nonlocal game with two questions and three answers per player, which was first considered by Feige in 1991, and show that there is quantum advantage in this game. We prove that the game is a robust self-test for the 3-dimensional maximally entangled state. Furthermore, we show that the game can be seen as the "or" of two games that each do not have quantum advantage. Lastly, we investigate the behavior of the game with respect to parallel repetition in the classical, quantum and non-signalling case and obtain perfect parallel repetition of the non-signalling value if Feige's game is repeated an even amount of times.