Spin and rotations in Galois field quantum mechanics

Abstract

Abstract. We discuss the properties of Galois Field Quantum Mechanics constructed on a vector space over the finite Galois field GF (q). In particular, we look at 2-level systems analogous to spin, and discuss how SO(3) rotations could be embodied in such a system. We also consider two-particle 'spin' correlations and show that the Clauser-Horne-Shimony-Holt (CHSH) inequality is nonetheless not violated in this model.

Quantum ${{\mathbb{F}}_{{\rm un}}}$: theq= 1 limit of Galois field quantum mechanics, projective geometry and the field with one element

“…It is straightforward to show that these probabilities cannot be reproduced by any classical hidden variable theory. 40,41 Thus, GQM is 'quantum' in this sense. It should be noted, though, that GQM also has a common feature with CM when we look at the Clauser-Horne-Shimony-Holt (CHSH) version of Bell's inequality.…”

Quantum Systems Based Upon Galois Fields — From Sub-Quantum to Super-Quantum Correlations

“…This was the prime motivation for our search for a simple model with super-quantum correlations in the context of discrete quantum theories over Galois fields. [2][3][4][5] Note that the CHSH inequality relies on the knowledge of expectation values and not probabilities. Though predicting probabilities and predicting expectation values may seem like the same thing, it turns out they are not necessarily when "mutations" are introduced.…”

“…Mappings of these 'spin' directions to actual directions in 3D space are discussed in Ref. 3. Applying Eq.…”

“…Note that in this model, the expectation values are predicted but the probabilities are not. For instance, from ϕ ( U |σ 3 ⊗σ 3 |U ) = −1 we can conclude that the probabilities that the measurement ofσ 3…”

“…In the following, we review our recent work on "mutant" quantum theories constructed on discrete and finite vector spaces over Galois fields. [2][3][4][5] (See also Refs. 6-9.)…”