ARNAULT François

We describe a protocol for generating random numbers based on the existence of quantum violations of a free version of Clauser-Horne-Shimony-Holt inequality for qutrit, namely CHSH-3. Our method uses semidefinite programming relaxations to compute such violations. In a standard setting the CHSH-3 inequality involves two separated qutrits and compatible measurement, that is, commuting with each other, yielding the known quantum bound of $1+\sqrt{11/3} \approx 2.9149$ . In our framework, $d$-dimensional quantum systems (qudits) where $d$ is not fixed {\it a priori}, and measurement operators possibly not compatible, are allowed. This loss of constraints yields a higher value of $4$ for the maximum expectation of CHSH-3, attained for $d=3 $ and non-commutative measurements for one party. Based on such upper bound on the violation of free CHSH-3, which appear to be a non-algebraic bound, we develop a random number generator with only one party. Our protocol generates a maximal entropy and its security is based, through self testing arguments, on the attainability of the maximal violation of the free CHSH-3 for quantum systems.