Czech Technical University in Virtual, Czech Republic, voracva1@fel.cvut.cz
(joint work with Mirko Navara)
The Bell–Kochen–Specker theorem is an important no-go theorem in quantum mechanics, which proves the incompatibility of quantum physics with local hidden-variable theories. The Kochen–Specker proof is based on a construction of a set of 117 three-dimensional vectors admitting no {0,1}-coloring.
There are numerous generalizations of the Kochen–Specker theorem, e.g., using rational vectors, or replacing a {0,1}-coloring with a ℤ2-coloring. It was shown in [1], that there is no ℤ2-coloring of vectors in ℝ4 and consequently no ℤ2-coloring in dimensions greater than or equal to 5, see [2]. We show that there is no ℤ2-coloring even for ℝ3, which was an open question, formulated, e.g., in [3,4]. The existence of a ℤ2-coloring in ℝ2 is trivial, hence we answer the only remaining case.
[1] peres, a. Generalized Kochen-Specker theorem. Foundations of Physics 26, 6 (1996), 807–812.
[2] navara, m., and pták, p. For n ≥ 5 there is no nontrivial Z2-measure on L(Rn). International Journal of Theoretical Physics 43, 7 (2004), 1595–1598.
[3] navara, m. Mathematical questions related to non-existence of hidden variables. American Institute of Physics Conference Proceedings 1101 (2009), 119–126.
[4] harding, j., jager, e., and smith, d. Group-valued measures on the lattice of closed subspaces of a Hilbert space. International Journal of Theoretical Physics 44 (2005), 539–548.