Minimal complexity of Kochen-Specker sets does not scale with dimension

Abstract

A Kochen-Specker (KS) set is a specific set of projectors and measurement contexts that prove the BellKochen-Specker contextuality theorem. The simplest known KS sets in Hilbert space dimensions d = 3, 4,5,6,8 are reproduced, and several methods by which a new KS set can be constructed using one or more known KS sets in lower dimensions are reviewed and improved. These KS sets and improved methods enable the construction of explicitly critical new KS sets in all dimensions, where critical refers to the irreducibility of the set of contexts.The simplest known critical KS sets are derived in all even dimensions d ≥ 10 with at most 9 contexts and 30 projectors, and in all odd dimensions d ≥ 7 with at most 13 contexts and 39 projectors. These results show that neither the number of contexts nor the number of projectors in a minimal KS set scales with dimension d.

Kochen-Specker sets and Hadamard matrices

“…A Kochen-Specker pair (V, B) in C 6 with |V| = 21 and |B| = 7 was recently discovered [9]. It was noted [2] as the simplest Kochen-Specker pair (KS pair) since it strictly minimizes the cardinality of B among all known KS pairs (V, B), see [11]. This KS pair was originally found by computer search and its internal structure has not been fully studied yet.…”

“…In this paper we focus on proofs of Kochen-Specker theorem that are given by showing that, for n ≥ 3, there does not exist a function f : C n → {0, 1} such that for every orthogonal basis B of C n there exists exactly one vector x ∈ B such that f (x) = 1 (where C n denotes the n-dimensional vector space over the field of complex numbers). This particular approach has been used in many publications, see for example [1,9,10,11] and many references cited therein. The following definition formalizes one common way of constructing such proofs.…”

“…It is quite common in the literature [2,9,11] to refer to a Kochen-Specker pair as Kochen-Specker set, and we will do so sometimes in this paper. Kochen-Specker sets are key tools for proving some fundamental results in quantum theory and they also have various potential applications in quantum information processing [2].…”

Kochen-Specker sets in four-dimensional spaces

“…In this paper we focus on proofs of the KS theorem that are given by showing that, for n ≥ 3, there does not exist a function f : C n → {0, 1} such that for every orthogonal basis B of C n there exists exactly one vector x ∈ B such that f (x) = 1 (where C n denotes the n-dimensional vector space over the field of complex numbers). This particular approach has been used in many publications, see for example [1,5,8] and many references cited therein. Definition 1.1 given below formalizes one common way of constructing such proofs, using a simple parity argument.…”

“…It is quite common in the literature [2,5,8] to refer to a KS pair as a KS set, and we will do so sometimes in this paper.…”