Equations and Hilbert lattices

Mayet, R.

doi:10.1016/b978-044452870-4/50034-0

Cited by 5 publications

(10 citation statements)

References 21 publications

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“…When we found our nOA, it was unknown whether the same might occur with nOA at the 6OA level. 20 Our result (the aforementioned passing of 3OA through 6OA and failure of 7OA in Peres' lattice) dispels any doubt. It was serendipitous that we obtained this result in this way, because no present-day supercomputer is capable of generating 7OA examples by brute force-at least not with our present algorithms.…”

Section: Discussionmentioning

confidence: 58%

Graph approach to quantum systems

Pavičić

McKay

Megill

et al. 2010

Journal of Mathematical Physics

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Using a graph approach to quantum systems, we show that descriptions of 3-dim Kochen-Specker (KS) setups as well as descriptions of 3-dim spin systems by means of Greechie diagrams (a kind of lattice) that we find in the literature are wrong. Correct lattices generated by McKay-Megill-Pavicic (MMP) hypergraphs and Hilbert subspace equations are given. To enable future exhaustive generation of 3-dim KS setups by means of our recently found stripping technique, bipartite graph generation is used to provide us with lattices with equal numbers of elements and blocks (orthogonal triples of elements)-up to 41 of them. We obtain several new results on such lattices and hypergraphs, in particular on properties such as superposition and orthoraguesian equations.

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Section: Discussionmentioning

confidence: 58%

Graph approach to quantum systems

Pavičić

McKay

Megill

et al. 2010

Journal of Mathematical Physics

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“…Recently, Mayet found a direct proof of this independence that does not need a computer calculation. [8] Nonetheless, the algorithm still provides a useful tool for testing the simultaneous validity of all n-Go equations for individual OMLs that are not of the form required by Mayet's theorem.…”

Section: Resultsmentioning

confidence: 99%

“…We also define a subclass of HL for which Def. 8.1 will later become relevant: 8 Hilbert lattice, QHL, is a Hilbert lattice orthoisomorphic to the set of closed subspaces of the Hilbert space defined over either a real field, or a complex field, or a quaternion skew field.…”

Section: Mayet's E-equations and A Solution To A Related Open Problemmentioning

confidence: 99%

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Hilbert Lattice Equations

Megill

Pavičić

2010

Ann. Henri Poincaré

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Abstract. There are five known classes of lattice equations that hold in every infinite dimensional Hilbert space underlying quantum systems: generalised orthoarguesian, Mayet's EA, Godowski, Mayet-Godowski, and Mayet's E equations. We obtain a result which opens a possibility that the first two classes coincide. We devise new algorithms to generate Mayet-Godowski equations that allow us to prove that the fourth class properly includes the third. An open problem related to the last class is answered. Finally, we show some new results on the Godowski lattices characterising the third class of equations. Mathematics Subject Classification (2000). Primary 46C15; Secondary 06B20.

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“…The first one was algebraic, utilizing Boolean ✻ The application to quantum theory uses the Hilbert spaces defined over real, R, complex, C, or quaternion (quasi), Q, fields. For these fields, in 2006, René Mayet [37] (see also [38]) used a technique similar to the one used for generating MGEs we presented in Sec. 6, to arrive at a new class of E-equations we will present in this section.…”

Section: State Vectors: Mayet's E-equationsmentioning

confidence: 99%