@MISC{A95theoremfor, author = {Michael Kernaghan A and Asher Peres B}, title = {theorem for eight-dimensional space}, year = {1995} }

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Abstract

A Kochen-Specker contradiction is produced with 36 vectors in a real eight-dimensional Hilbert space. These vectors can be combined into 30 distinct projection operators ( 14 of rank 2, and 16 of rank 1). A state-specific variant of this contradiction requires only 13 vectors, a remarkably low number for eight dimensions. The Kochen-Specker theorem [ l] asserts that, in a Hilbert space with a finite number of dimensions, d 2 3, it is possible to produce a set of n projection operators, representing yes-no questions about a quantum system, such that none of the 2 ” possible answers is compatible with the sum rules of quantum mechanics. Namely, if a subset of mutually orthogonal projection operators sums up to the unit matrix, one and only one of the answers is yes. The physical meaning of this theorem is that there is no way of introducing noncontextual “hidden” variables [ 21 which would ascribe definite outcomes to these n yes-no tests. This conclusion holds irrespective of the quantum state of the system being tested. It is also possible to formulate a “state-specific ” version of this theorem, valid for systems which have been prepared in a known pure state. In that case, the projection operators are chosen in a way adapted to the known state. A smaller number of questions is then sufficient to obtain incompatibility with the quantum mechanical sum rules. An even smaller number is needed if strict sum rules are replaced by weaker probabilistic arguments