 
Summary: Found Phys (2010) 40: 10961101
DOI 10.1007/s1070101094342
Isomorphism between the Peres and Penrose Proofs
of the BKS Theorem in Three Dimensions
Elizabeth Gould · P.K. Aravind
Received: 24 September 2009 / Accepted: 15 February 2010 / Published online: 2 March 2010
© Springer Science+Business Media, LLC 2010
Abstract It is shown that the 33 complex rays in three dimensions used by Penrose
to prove the BellKochenSpecker theorem have the same orthogonality relations as
the 33 real rays of Peres, and therefore provide an isomorphic proof of the theorem. It
is further shown that the Peres and Penrose rays are just two members of a continuous
threeparameter family of unitarily inequivalent rays that prove the theorem.
Keywords KochenSpecker theorem · Bell's theorem · Foundations of quantum
mechanics
Some time back Peres [1] gave a proof of the BellKochenSpecker (BKS) theorem
[2, 3] using 33 real rays (or directions) in three dimensions. Penrose [4] later gave a
different proof of the theorem using 33 complex rays in three dimensions. Penrose
pointed out that his set of rays is essentially complex (i.e., there is no basis in which
the components of all the rays can be made real) and that there is no Hilbert space
rotation that will take his rays into those of Peres. It might therefore be thought that
