Mermin pentagrams arising from Veldkamp lines for three qubits

TL;DR

This paper explores the geometric structure of Mermin pentagrams in three-qubit systems, revealing their organization into families linked to Veldkamp lines and connecting their geometry to group representations, with implications for quantum contextuality.

Contribution

It introduces a geometric and group-theoretic framework for understanding Mermin pentagrams and their organization, relating them to Veldkamp lines and representation theory, which is a novel approach in quantum contextuality studies.

Findings

12096 possible pentagrams organized into 1008 families

Each family contains a 'double six' configuration

Geometry of configurations linked to $SU(6)$ representation

Abstract

We study the geometry of the space of Mermin pentagrams, objects that are used to rule out the existence of noncontextual hidden variable theories as alternatives to quantum theory. It is shown that this space of 12096 possible pentagrams is organized into 1008 families, with each family containing a "double six" of pentagrams. The 1008 families are connected to a special set of Veldkamp lines in the Veldkamp space of three qubits an object well-known to finite geometers but has only been introduced to physics recently. Due to the transitive action of the symplectic group on this set of Veldkamp lines it is enough to study only one "canonical" double six configuration of pentagrams. We prove that the geometry of this double-six configuration is encapsulated in the weight diagram of the 20 dimensional irreducible representation of the group $SU(6)$. As an interesting by-product of our approach we show that Mermin pentagrams and a class of Mermin squares labelled by three qubit Pauli operators are inherently related. We conjecture that by studying the representation theoretic content of other Veldkamp lines of the Veldkamp space for $N$-qubits makes it possible to find new contextual configurations in a systematic manner.