Security of device-independent cryptography

Device-independent (DI) quantum information aims to establish security of a protocol without
making any assumptions on the inner workings of the devices used [1] . Instead, DI protocols
base security directly on the input-output behaviour of the devices by feeding them inputs and
recording 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 key
distribution and the Bell inequality violation certifies that the devices are behaving quantum
mechanically, 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 conditional
min entropy of the raw randomness, or key, conditioned on all side information any
eavesdropper 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 Neumann
entropy of each round. This reduces the theoretical problem to the optimisation of this entropy
over all possible eavesdropping strategies to prove security. This is still a difficult task, since the
von Neumann entropy is non-linear, and hence usual heuristic optimization techniques are not
guaranteed to converge (unlike for instance problems that can be solved with a linear or
semidefinite 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, and
the validity and tightness of such relaxations must be properly characterised. The overarching
goal of this PhD is to develop and evaluate such relaxations, producing bounds that are tight
and computationally cheap to calculate.
The project will consist of theoretical work in deriving and evaluating bounds, and computational
working in solving the resulting optimisation problems. This project utilises several of the skills
developed in project B. The student will conduct this PhD at the University of York.