Quantum States Arising from the Pauli Groups, Symmetries and Paradoxes

TL;DR

This paper explores the geometric and algebraic structures of quantum states derived from Pauli groups, focusing on Bell-Kochen-Specker theorem proofs and their symmetries in multi-qubit systems.

Contribution

It identifies the structure of minimal BKS proofs using rays from Pauli groups and reveals symmetry signatures in the bases' geometric arrangements.

Findings

Real rays form Barnes-Wall lattices

Characteristic signatures in basis distances

Analysis of small BKS-proof classes

Abstract

We investigate multiple qubit Pauli groups and the quantum states/rays arising from their maximal bases. Remarkably, the real rays are carried by a Barnes-Wall lattice $BW_n$ ($n=2^m$). We focus on the smallest subsets of rays allowing a state proof of the Bell-Kochen-Specker theorem (BKS). BKS theorem rules out realistic non-contextual theories by resorting to impossible assignments of rays among a selected set of maximal orthogonal bases. We investigate the geometrical structure of small BKS-proofs $v-l$ involving $v$ rays and $l$ $2n$-dimensional bases of $n$-qubits. Specifically, we look at the classes of parity proofs 18-9 with two qubits (A. Cabello, 1996), 36-11 with three qubits (M. Kernaghan & A. Peres, 1995) and related classes. One finds characteristic signatures of the distances among the bases, that carry various symmetries in their graphs.