Diagrammatic security proof for 8-state encoding

Dirac notation is the most common way to describe quantum states and operations on states. It is very convenient and allows for quick visual distinction between vectors, scalars and operators. For quantum processes that involve interactions of multiple systems an even better visualisation has been proposed by Coecke and Kissinger, in the form of a diagrammatic formalism [CK2017]. Their notation expresses formulas in the form of diagrams, somewhat similar to Feynman diagrams, and is more general than the circuit notation for quantum computing. This document consists of two parts. (1) We give a brief summary of the diagrammatic notation of quantum processes, tailored to readers who already know quantum physics and are not interested in general process theory. For this audience our summary is less daunting than the encyclopaedic book by Coecke and Kissinger [CK2017], and on the other hand more accessible than the ultra-compact introduction of [KTW2017]. We deviate a somewhat from [CK2017,KTW2017] in that we do not assume basis states to equal their own complex conjugate; this means that we do not use symmetric notation for basis states, and it leads us to explicitly show arrows on wires where they are usually omitted. (2) We extend the work of Kissinger, Tull and Westerbaan [KTW2017] which gives a diagrammatic security proof for BB84 and 6-state Quantum Key Distribution. Their proof is based on a sequence of diagrammatic manipulations that works when the bases used in the protocol are mutually unbiased. We extend this result to 8-state encoding, which has been proposed as a tool in quantum key recycling protocols [SdV2017,LS2018], and which does not have mutually unbiased bases.