Mermin pentagrams arising from Veldkamp lines for three qubits

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.

arXiv comment28 pages, 10 figures. Will be published in Journal of Physics A: Mathematical and Theoretical, Volume 50, Number 9

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Cited by in corpus (11)

• Magic informationally complete POVMs with permutations
• The magic three-qubit Veldkamp line: A finite geometric underpinning for form theories of gravity and black hole entropy
• Contextuality degree of quadrics in multi-qubit symplectic polar spaces
• Taxonomy of Polar Subspaces of Multi-Qubit Symplectic Polar Spaces of Small Rank
• New and improved bounds on the contextuality degree of multi-qubit configurations
• A Finite-Geometric Classification of Three-Qubit Mermin Pentagrams
• Magic Three-Qubit Veldkamp Line and Veldkamp Space of the Doily
• A Combinatorial Grassmannian Representation of the Magic Three-Qubit Veldkamp Line
• A new heuristic approach for contextuality degree estimates and its four- to six-qubit portrayals
• Hexagons govern three-qubit contextuality
• Veldkamp Spaces: From (Dynkin) Diagrams to (Pauli) Groups