Vector Generation of Quantum Contextual Sets in Even Dimensional Hilbert Spaces

Abstract

Recently, quantum contextuality has been proved to be the source of quantum computation's power. That, together with multiple recent contextual experiments, prompts improving the methods of generation of contextual sets and finding their features. The most elaborated contextual sets, which offer blueprints for contextual experiments and computational gates, are the Kochen-Specker (KS) sets. In this paper, we show a method of vector generation that supersedes previous methods. It is implemented by means of algorithms and programs that generate hypergraphs embodying the Kochen-Specker property and that are designed to be carried out on supercomputers. We show that vector component generation of KS hypergraphs exhausts all possible vectors that can be constructed from chosen vector components, in contrast to previous studies that used incomplete lists of vectors and therefore missed a majority of hypergraphs. Consequently, this unified method is far more efficient for generations of KS sets and their implementation in quantum computation and quantum communication. Several new KS classes and their features have been found and are elaborated on in the paper. Greechie diagrams are discussed. A detailed and complete blueprint of a particular 21-11 KS set with a complex coordinatization is presented in Appendix A, in contrast to the one from the published version of this paper where only a few of its states were given.

Hypergraph Contextuality

“…A class of state-independent contextual (SIC) [ 33 ] sets that have been elaborated on the most in the literature are the Kochen-Specker (KS) sets [ 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 ]. They boil down to a list of n -dim vectors and their n -tuples of orthogonalities, such that one cannot assign definite binary values to them.…”

Automated generation of Kochen-Specker sets

“…For instance, in Ref. [43] (Table 2) we were not able to obtain smaller KS sets from the 4D 2316-3052, or 6D 11808-314446, or 8D 3280-1361376, or 16D, or 32D master sets directly. The main reason is that we cannot generate smaller sets from a master set by a brute force-the programs are too slow for that.…”

“…Taken together, the main aim of Ref. [43] was to classify KS sets in even dimensions via vector-component generated master sets that would contain all known KS sets, while the main aim of the present paper is to provide a functional and feasible method of generating KS sets of chosen sizes and properties from such master sets.…”

“…In the present paper, we present an optimized method of generating KS sets from tweaked KS master sets engineered via the method outlined in Ref. [43]. In the latter method, we start with random vector components and build n-tuples of mutually orthogonal vectors which results a in KS masters with a specific coordinatization.…”

Quantum Probability and Randomness

“…The article by Pavicic and Megill [ 23 ], Vector Generation of Quantum Contextual Sets in Even Dimensional Hilbert Spaces , is a novel contribution to quantum contextuality theory. As is well known, the most elaborated contextual sets, which offer blueprints for contextual experiments and computational gates, are the Kochen–Specker sets.…”