Quantum entanglement and contextuality with complexifications of $E_8$ root system

TL;DR

The paper develops a 120-state complex quantum-state configuration derived from the $E_8$ root system and connects it to the 40-state Witting model via an 8D real root-space mapping. It introduces the $\tilde{\mathcal{A}_7}$ symmetry group as a 4D unitary representation of the double cover of $A_7$, and constructs 210 orthonormal bases spanning 120 states; contextuality is demonstrated through Kochen–Specker graphs, with a maximal nonorthogonal clique of size 24 and the absence of a 30-state noncontextual partition. The work further studies entanglement between the two configurations, identifying subgroups $\tilde{\mathcal{A}}_5$ and $\tilde{\mathcal{S}}_5$ that preserve entangled states $|\Omega_J\rangle$ and proposing two measurement schemes using $J$-opposite states. Together, these results provide a group-theoretic framework for analyzing nonlocality and contextuality in high-dimensional, $E_8$-root–based quantum-state configurations.

Abstract

The Witting configuration with 40 complex rays was suggested as a possible reformulation of Penrose model with two spin-3/2 systems based on geometry of dodecahedron and used for analysis of nonlocality and contextuality in quantum mechanics. Yet another configuration with 120 quantum states is considered in presented work. Despite of different number of states both configurations can be derived from complexification of 240 minimal vectors of 8D real lattice corresponding to root system of Lie algebra $E_8$. An analysis of properties of suggested configuration of quantum states is provided using many analogies with properties of Witting configuration.