New Kochen–Specker sets in four dimensions

Pavičić, Mladen; Megill, Norman D.; Merlet, Jean‐Pierre

doi:10.1016/j.physleta.2010.03.019

Cited by 29 publications

(81 citation statements)

References 33 publications

(53 reference statements)

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“…However, we also found KS sets that were not subsets of the 24-24 set (e.g., the 22-11 one, shown in Fig. 3(a) of [31]) and those subsets have the components from a wider set of values (see Table 2 of [31] for the aforementioned 22-11 set). That indicated that there is another class or other classes which contain those sets or both kinds of sets and we started stripping master sets meanwhile discovered [8,37,47,49,50].…”

Section: Tentative 24-24 Class Of 4-dim Ks Sets; Generation Of Kmentioning

confidence: 78%

“…# $ % & ' ( ) * -/ : ; < = > ? @ [ ]ˆ_ ' { | }~ [31]. When all these characters are exhausted we reuse them so as to prefix them by '+', then by '++', and so on.…”

Section: Ks Sets Mmp Hypergraphs Formalism Algorithms and Promentioning

confidence: 99%

“…When dealing with such ASCII line encoding of MMP hypergraphs we call them MMP hypergraphs lines or strings when needed. The order of the strings and characters is irrelevant; gaps in characters are allowed and its number is not limited; tens of thousands of them are not a problem for our programs shortd.c, mmpstrip.c, subgraph.c, vectorfind.c, states01.c and others [9,27,31,[35][36][37].…”

Section: Ks Sets Mmp Hypergraphs Formalism Algorithms and Promentioning

confidence: 99%

“…In this section we shall make use of the results about the 24-24 class of 4-dim KS sets we obtained in [31,36,[39][40][41]] to introduce the main steps and strategy we shall undertake to obtain the results in the subsequent sections.…”

Section: Tentative 24-24 Class Of 4-dim Ks Sets; Generation Of Kmentioning

confidence: 99%

“…1-that they were all subgraphs of the hypergraph we drew for Peres' 24-24 set. Then Pavičić, Megill, and Merlet designed an algorithm for stripping (peeling) edges off the latter hypergraph and obtained 1232 KS subsets [31] (including all 6 criticals from Fig. 1) within less than 2 minutes on a PC.…”

Section: Tentative 24-24 Class Of 4-dim Ks Sets; Generation Of Kmentioning

confidence: 99%

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Arbitrarily exhaustive hypergraph generation of 4-, 6-, 8-, 16-, and 32-dimensional quantum contextual sets

Pavičić¹

2017

Phys. Rev. A

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124

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Quantum contextuality turns out to be a necessary resource for universal quantum computation and important in the field of quantum information processing. It is therefore of interest both for theoretical considerations and for experimental implementation to find new types and instances of contextual sets and develop methods of their optimal generation. We present an arbitrary exhaustive hypergraph-based generation of the most explored contextual sets-Kochen-Specker (KS) ones-in 4, 6, 8, 16, and 32 dimensions. We consider and analyse twelve KS classes and obtain numerous properties of theirs, which we then compare with the results previously obtained in the literature. We generate several thousand times more types and instances of KS sets than previously known. All KS sets in three of the classes and in the upper part of a fourth are novel. We make use of the MMP hypergraph language, algorithms, and programs to generate KS sets strictly following their definition from the Kochen-Specker theorem. This approach proves to be particularly advantageous over the parity-proof-based ones (which prevail in the literature), since it turns out that only a very few KS sets have a parity proof (in six KS classes < 0.01% and in one of them 0%). MMP hypergraph formalism enables a translation of an exponentially complex task of solving systems of nonlinear equations, describing KS vector orthogonalities, into a statistically linearly complex task of evaluating vertex states of hypergraph edges, thus exponentially speeding up the generation of KS sets and enabling us to generate billions of novel instances of them. The MMP hypergraph notation also enables us to graphically represent KS sets and to visually discern their features.

show abstract

Section: Tentative 24-24 Class Of 4-dim Ks Sets; Generation Of Kmentioning

confidence: 78%

“…# $ % & ' ( ) * -/ : ; < = > ? @ [ ]ˆ_ ' { | }~ [31]. When all these characters are exhausted we reuse them so as to prefix them by '+', then by '++', and so on.…”

Section: Ks Sets Mmp Hypergraphs Formalism Algorithms and Promentioning

confidence: 99%

Section: Ks Sets Mmp Hypergraphs Formalism Algorithms and Promentioning

confidence: 99%

Section: Tentative 24-24 Class Of 4-dim Ks Sets; Generation Of Kmentioning

confidence: 99%

Section: Tentative 24-24 Class Of 4-dim Ks Sets; Generation Of Kmentioning

confidence: 99%

See 3 more Smart Citations

Arbitrarily exhaustive hypergraph generation of 4-, 6-, 8-, 16-, and 32-dimensional quantum contextual sets

Pavičić¹

2017

Phys. Rev. A

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124

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GHZ, As Seen Through the Looking Glass

Waegell

Aravind

2021

Fundamental Theories of Physics

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Parity Proofs of the Bell-Kochen-Specker Theorem Based on the 600-cell

et al. 2011

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The set of 60 real rays in four dimensions derived from the vertices of a 600-cell is shown to possess numerous subsets of rays and bases that provide basiscritical parity proofs of the Bell-Kochen-Specker (BKS) theorem (a basis-critical proof is one that fails if even a single basis is deleted from it). The proofs vary considerably in size, with the smallest having 26 rays and 13 bases and the largest 60 rays and 41 bases. There are at least 90 basic types of proofs, with each coming in a number of geometrically distinct varieties. The replicas of all the proofs under the symmetries of the 600-cell yield a total of almost a hundred million parity proofs of the BKS theorem. The proofs are all very transparent and take no more than simple counting to verify. A few of the proofs are exhibited, both in tabular form as well as in the form of MMP hypergraphs that assist in their visualization. A survey of the proofs is given, simple procedures for generating some of them are described and their applications are discussed. It is shown that all four-dimensional parity proofs of the BKS theorem can be turned into experimental disproofs of noncontextuality.

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New Kochen–Specker sets in four dimensions

Cited by 29 publications

References 33 publications

Arbitrarily exhaustive hypergraph generation of 4-, 6-, 8-, 16-, and 32-dimensional quantum contextual sets

Arbitrarily exhaustive hypergraph generation of 4-, 6-, 8-, 16-, and 32-dimensional quantum contextual sets

GHZ, As Seen Through the Looking Glass

Parity Proofs of the Bell-Kochen-Specker Theorem Based on the 600-cell

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