Security of device-independent cryptography

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Device-independent (DI) quantum information aims to establish security of a protocol withoutmaking any assumptions on the inner workings of the devices used [1] . Instead, DI protocolsbase security directly on the input-output behaviour of the devices by feeding them inputs andrecording the outputs, building up a conditional distribution which should violate a Bell inequality.Two important tasks that can be made device-independent are randomness expansion and keydistribution and the Bell inequality violation certifies that the devices are behaving quantummechanically, and can be used to quantify the amount of randomness or key.Proving the security of such protocols boils down to finding tight lower bounds on the conditionalmin entropy of the raw randomness, or key, conditioned on all side information anyeavesdropper may have. Recently, a novel technique called the entropy accumulation theorem(EAT) [2] has been developed, which reduces the problem to an estimation of the von Neumannentropy of each round. This reduces the theoretical problem to the optimisation of this entropyover all possible eavesdropping strategies to prove security. This is still a difficult task, since thevon Neumann entropy is non-linear, and hence usual heuristic optimization techniques are notguaranteed to converge (unlike for instance problems that can be solved with a linear orsemidefinite program). Consequently, relaxed versions of these problems are typically studied[3,4] which have better computational properties at the possible expense of a looser bound, andthe validity and tightness of such relaxations must be properly characterised. The overarchinggoal of this PhD is to develop and evaluate such relaxations, producing bounds that are tightand computationally cheap to calculate.The project will consist of theoretical work in deriving and evaluating bounds, and computationalworking in solving the resulting optimisation problems. This project utilises several of the skillsdeveloped in project B. The student will conduct this PhD at the University of York.