Quantum logic and quantum computation

Waegell

et al. 2019

Sci Rep

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Quantum contextuality turns out to be a necessary resource for universal quantum computation and also has applications in quantum communication. Thus it becomes important to generate contextual sets of arbitrary structure and complexity to enable a variety of implementations. In recent years, such generation has been done for contextual sets known as Kochen-Specker sets. Up to now, two approaches have been used for massive generation of non-isomorphic Kochen-Specker sets: exhaustive generation up to a given size and downward generation from master sets and their associated coordinatizations. Master sets were obtained earlier from serendipitous or intuitive connections with polytopes or Pauli operators, and more recently from arbitrary vector components using an algorithm that generates orthogonal vector groupings from them. However, both upward and downward generation face an inherent exponential complexity barrier. In contrast, in this paper we present methods and algorithms that we apply to downward generation that can overcome the exponential barrier in many cases of interest. These involve tailoring and manipulating Kochen-Specker master sets obtained from a small number of simple vector components, filtered by the features of the sets we aim to obtain. Some of the classes of Kochen-Specker sets we generate contain all previously known ones, and others are completely novel. We provide examples of both kinds in 4- and 6-dim Hilbert spaces. We also give a brief introduction for a wider audience and a novice reader.

Section: Methodsmentioning

confidence: 99%

Automated generation of Kochen-Specker sets

Waegell

et al. 2019

Sci Rep

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“…We call a lattice in which all MGEs hold an MGO; i.e., MGO is the largest class of lattices (equational variety) in which all MGEs hold. The simplest known example of an equation implied by an MGE that is independent from all Godowski equations is 19…”

Section: Each Variable Occurs the Same Number Of Times On Each Side Omentioning

confidence: 99%

“…Any Hilbert lattice admits a strong and therefore a full set of states, and the orthoarguesian equations hold in any Hilbert lattice. 17,19 On the other hand, the 16-9 OML in Fig. 9 (b) satisfies orthoarguesian equations and admits a full set of states but does not admit a strong set of states, L42 from Fig.…”

Section: Lattices That Admit Almost No Hilbert Lattice Equationsmentioning

confidence: 99%

“…17 While all OMLs admitting strong sets of states satisfy all Godowski equations, there are examples showing the converse isn't true. 19 (Fig. 10, p. 780) Such examples can be exhaustively generated, but no common structural feature has been recognized so far.…”

Section: Lattices That Admit Almost No Hilbert Lattice Equationsmentioning

confidence: 99%

“…The cycles themselves will allow us to generate new lattice equations following the procedure developed in 19,21,67 , but they do not automatically follow possible geometrical symmetries of the hypergraphs. In the 36-36 case they do, but, e.g., they do not exhibit the left right symmetry of the 35-35 lattice shown in Fig.…”

Section: Properties Of Lattices With Equal Numbers Of Atoms and Bmentioning

confidence: 99%

See 2 more Smart Citations

Graph approach to quantum systems

McKay

Journal of Mathematical Physics

et al. 2010

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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.

Hilbert Lattice Equations

2010

Ann. Henri Poincaré

Self Cite

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.