Automated generation of Kochen-Specker sets

Abstract

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.

Hypergraph Contextuality

“…There is a one-to-one correspondence between KS n -tuples of vectors and MMPH edges when they are all of their maximal size, as established in Reference [ 35 , 46 , 47 , 48 ] and between KS vectors and MMPH vertices with coordinatization within an MMPH with maximal edges.…”

“…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.…”

What Is So Special about Quantum Clicks?

“…The early 1990s saw an ongoing flurry of papers recasting the Kochen-Specker proof with ever smaller numbers of, or more symmetric, configurations of observables (see Refs. [ 35 , 44 , 143 , 162 , 167 , 168 , 169 , 170 , 171 , 172 , 173 , 174 , 175 , 176 , 177 , 178 , 179 , 180 , 181 , 182 , 183 , 184 , 185 , 186 ] for an incomplete list). The “most compact” proofs (in terms of the number of vectors and their associated observables) should contain no less than 22 vectors in three-dimensional space [ 187 ], and no less than 18 vectors for dimension four [ 179 ] and higher [ 180 , 188 ].…”

“…It consists of 9 contexts, with each of the 18 atoms tightly intertwined in two contexts. The challenge in such (mostly automated) computations is twofold: to generate (and exclude a sufficient number) of “candidate graphs”; and subsequently to find a faithful orthogonal representation [ 124 , 125 , 126 , 185 , 186 ].…”

Vector generation of contextual sets

“…Our program VECFIND provides us with master sets generated by vector components. As shown in [51] {-1,0,1} components give us a master set with 40 vertices and 32 edges, denoted as 40-32. It consists of two disconnected subsets: a 24-24 KS set and a 16-8 non-KS set, as shown in Fig.…”