Mate Farkas

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.

The degree of experimentally attainable nonlocality, as gauged by the loophole-free or effective violation of Bell inequalities, remains severely limited due to inefficient detectors. We address an experimentally motivated question: Which quantum strategies attain the maximal loophole-free nonlocality in the presence of inefficient detectors? For any Bell inequality and any specification of detection efficiencies, the optimal strategies are those that maximally violate a tilted version of the Bell inequality in ideal conditions. In the simplest scenario, we demonstrate that the quantum strategies that maximally violate the doubly-tilted versions of Clauser-Horne-Shimony-Holt inequality are unique up to local isometries. We utilize Jordan's lemma and Grobner basis-based proof technique to analytically derive self-testing statements for the entire family of doubly-tilted CHSH inequalities and numerically demonstrate their robustness. These results enable us to reveal the insufficiency of even high levels of the Navascues-Pironio-Acin hierarchy to saturate the maximum quantum violation of these inequalities.

Performing a measurement on one half of an entangled pair of systems remotely prepares the other half in an ensemble of quantum states. Now consider any set of ensembles that mix to the same density operator. The Schrödinger--HJW theorem states that one can remotely prepare any chosen ensemble from this set by a choice of measurement performed on one half of a fixed entangled state. We generalise this result to show that any set of ensembles can be remotely prepared if one is allowed to post-select on the outcome of the preparing measurement. The probability that the remote preparation is successful is then lower bounded in terms of the distance between the ensembles. In a prepare-and-measure quantum key distribution protocol the distance between the ensembles represents the amount of information that is leaked about the choice of ensemble, e.g. the choice of basis in the BB84 protocol. Using our generalised result, we can prove the security of such protocols from only the fundamental assumption of how much information is leaked about the choice of ensemble/basis, i.e. from bounded basis-dependency.