FOUNDATIONS OF QUANTUM MECHANICS

V. 3. KOCHEN AND SPECKER’S THEOREM 119
Figure V. 3: M.C. Escher, Waterfall. Consider the 3 interpenetrating cubes on the top of the
left pillar. Each cube has 4 lines from the mutual center to its vertices, 6 lines to the centers of
its edges, and 3 lines to the centers of its faces. Three of the lines are shared by all three cubes,
giving 3 · (4 + 6 + 3 ) − 6 = 33 lines. These are Peres’ vectors. (Text Meyer 2003 )
It is interesting to see what the measure (V. 29), according to Von Neumann the probability measure
of quantum mechanics, looks like in this case. For a pure state W = |ψ⟩ ⟨ψ|, with P i = |χ⟩ ⟨χ|
the measure (V. 29) is
µ(P i ) = Tr P i W = ⟨ψ | P i | ψ⟩ = |⟨χ | ψ⟩| 2 (V. 32)
so that in a real space we have
µ(P i ) = |⟨χ | ψ⟩| 2 = cos 2 θ, (V. 33)