📝 SummarXiv

Abstract

Currently, generalizations of quantum communication protocols from qubits to systems with higher-dimensional state spaces (qudits) typically use mutually unbiased bases (MUB). The construction with maximal number of MUB is known in any dimension equal to a prime power and at least two such bases exist in any dimension. However, in small dimensions, there also exist formally more symmetric systems of states, described by regular complex polytopes, which are a generalization of the idea of Platonic solids to complex spaces. This work considers the application of a model originally proposed by R. Penrose and based on the geometry of dodecahedron and two entangled particles with spin 3/2. In a more general case, two arbitrary quantum systems with four basis states (ququarts) can be used instead. It was later shown that this system with 40 states is equivalent to the Witting configuration and is related to the four-dimensional complex polytope described by Coxeter. Presented paper describes how to use this configuration for a quantum key distribution protocol based on contextuality using some illustrative examples with 40 "quantum cards".