Galois Field Quantum Mechanics

Lewis

et al. 2014

Int. J. Mod. Phys. A

Dedicated to Freeman Dyson on the occasion of his 90th birthdayIn this talk we describe our recent work on discrete quantum theory based on Galois fields. In particular, we discuss how discrete quantum theory sheds new light on the foundations of quantum theory and we review an explicit model of super-quantum correlations we have constructed in this context. We also discuss the larger questions of the origins and foundations of quantum theory, as well as the relevance of super-quantum theory for the quantum theory of gravity.

Section: Projective Linear and Unitary Groupsmentioning

confidence: 83%

Section: Correlations In Classical and Quantum Theories And Beyondmentioning

confidence: 99%

See 1 more Smart Citation

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

Lewis

et al. 2014

Int. J. Mod. Phys. A

“…There are multiple ways to define QM on finite fields as discussed in Refs. [3,4,5,6,7]. The formalism presented here is closest to Refs.…”

Section: Finite Field Quantum Mechanicsmentioning

confidence: 95%

Spekkens’ Toy Model, Finite Field Quantum Mechanics, and the Role of Linearity

Takeuchi

2019

J. Phys.: Conf. Ser.

Self Cite

We map Spekkens' toy model to a quantum mechanics defined over the finite field F5. This allows us to define arbitrary linear combinations of the epistemic states in the model. For Spekkens' elementary system with only 2 2 = 4 ontic states, the mapping is exact and the two models agree completely. However, for a pair of elementary systems there exist interesting differences between the entangled states of the two models.

“…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.…”

Section: Spin Correlationsmentioning

confidence: 99%

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

Lewis

J. Phys. A: Math. Theor.

et al. 2014

Self Cite

We argue that the q = 1 limit of Galois Field Quantum Mechanics, which was constructed on a vector space over the Galois Field Fq = GF (q), corresponds to its 'classical limit,' where superposition of states is disallowed. The limit preserves the projective geometry nature of the state space, and can be understood as being constructed on an appropriately defined analogue of a 'vector' space over the 'field with one element' F 1 .