Maximal device-independent randomness in every dimension

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 $2 \log(d)$ 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.