Quantum States Arising from the Pauli Groups, Symmetries and Paradoxes

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 BWn (n = 2m). We focus on the smallest subsets of rays allowing a state proof of the BellKochen-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. 1 Real rays from the multiple qubit Pauli group and BarnesWall lattices An n-dimensional Euclidean lattice L is a discrete additive subgroup of the real vector space R, endowed with the standard Euclidean product, and spanned by a generator matrix M with rows in R. The automorphism group Aut(L) is the set of orthogonal matrices B such that under the conjugation action U = MBM−1 by the generating matrix M , one has (i) det U = ±1 and (ii) U is an integer matrix [1]. One is specifically interested by the family of Barnes-Wall lattices BWn (n = 2 m and m > 1) that generalize the root lattices BW4 ∼= D4 and BW8 ∼= E8 with BW16 (the densest known lattices) and higher [2]. Their automorphismgroup group is the so-called real Clifford group[4, 3]. Let us study the relationship between the Barnes-Wall lattices and the real rays arising from the n-qubit Pauli group Pn [5]. The total number of states/rays appearing as eigenstates shared by the maximal commuting setsmcs of operators in Pm is nL, where n = 2 m and the number of maximal commuting sets is L = ∏m i=1(1 + 2 ), as shown in column 3 of table 1; the corresponding number of real rays is shown in column 4. The rays form maximal orthogonal bases