Nonlocal Games and Quantum Permutation Groups

Lupini, Martino; Mančinska, Laura; Roberson, David E.

doi:10.48550/arxiv.1712.01820

Cited by 15 publications

(54 citation statements)

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“…G aut (Γ) = {e}) can have quantum symmetries. There is some "asymptotic" evidence that no asymmetric graph has quantum symmetries, see [LMR17] and our results below support this hypothesis for small graphs.…”

Section: Introductionsupporting

confidence: 66%

“…(i) The ratio between graphs having quantum symmetries and those having no quantum symmetries is about 50 : 50. As it is known that almost all graphs have no symmetries and no quantum symmetries [LMR17], this effect is just a distortion for small n. However, it could be interesting to observe until which n this phenomenon occurs.…”

Section: Application Of the Algorithm To Graphs On Up To Seven Verticesmentioning

confidence: 98%

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Existence of quantum symmetries for graphs on up to seven vertices: a computer based approach

Eder,

Levandovskyy,

Schanz

et al. 2019

Preprint

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The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica capturing the quantum symmetries of the graph. In general, there are more quantum symmetries than symmetries and it is a non-trivial task to determine when this is the case for a given graph: The question is whether or not the algebra associated to the quantum automorphism group is commutative. We use Gröbner base computations in order to tackle this problem; the implementation uses Gap and the Singular package Letterplace. We determine the existence of quantum symmetries for all connected, undirected graphs without multiple edges and without self-edges, for up to seven vertices. As an outcome, we infer within our regime that a classical automorphism group of order one or two is an obstruction for the existence of quantum symmetries.

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Section: Introductionsupporting

confidence: 66%

Section: Application Of the Algorithm To Graphs On Up To Seven Verticesmentioning

confidence: 98%

Existence of quantum symmetries for graphs on up to seven vertices: a computer based approach

Eder,

Levandovskyy,

Schanz

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…Example 6.12. There exist 15 graphs of type 6 for p = 31 which are all of the form C 31 (5, 6, U ), where U is any union of (2, 10, 12), (4,7,11), (3,13,15) and (8, 9, 14) (except union of the fourth). In all the cases, E = {±1, ±5, ±6} so we just need to study C 31 (5,6).…”

Section: Applications To Circulant P-graphsmentioning

confidence: 99%

Study of quantum symmetries for vertex-transitive graphs using intertwiner spaces

Chassaniol

2019

Preprint

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“…This has been studied in [5] for some graphs on p vertices, p prime, and more recently the author showed in [16] that the Petersen graph does not have quantum symmetry. Also Lupini, Mančinska and Roberson proved that almost all graphs have trivial quantum automorphim group in [12], which implies that almost all graphs do not have quantum symmetry.…”

Section: Introductionmentioning

confidence: 99%