Quantum Communication Scheme Described by a Regular Four-Dimensional Complex Polytope

At present, generalizations of quantum communication protocols from qubits to systems with higher-dimensional state spaces (qudits) commonly employ mutually unbiased bases (MUB), the construction of which is known for any dimension equal to a prime power. However, in low dimensions, there also exist formally more symmetric state systems described by regular complex polytopes, which are a generalization of the concept of regular polytopes to complex spaces. This work considers the application of a model originally proposed by R. Penrose and based on the geometry of the 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 subsequently shown that this system of 40 states is equivalent to the Witting configuration and is related to the four-dimensional complex polytope described by Coxeter. The paper proposes a quantum key distribution protocol based on contextuality, utilizing this configuration.