The Minimum 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 Bell-Kochen-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 $d\ge 10$ with at most 9 contexts and 30 projectors, and in all odd dimensions $d\ge 7$ 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 $d$.

8 pages, 4 figures, attached supplemental information with 3 additional figures

References in corpus (6)

• Hidden Variables and the Two Theorems of John Bell
• Contextuality supplies the magic for quantum computation
• Negativity and contextuality are equivalent notions of nonclassicality
• Discrete Wigner functions and quantum computational speedup
• Strong Kochen-Specker theorem and incomputability of quantum randomness
• Parity proofs of the Kochen-Specker theorem based on the Lie algebra E8

Cited by in corpus (8)

• Kochen-Specker Contextuality
• Quantum Contextuality
• Kochen-Specker sets and Hadamard matrices
• Automated Generation of Arbitrarily Many Kochen-Specker and Other Contextual Sets in Odd Dimensional Hilbert Spaces
• Non-Kochen-Specker Contextuality
• Vector Generation of Contextual Sets
• Generation of Kochen-Specker contextual sets in higher dimensions by dimensional upscaling whose complexity does not scale with dimension and their applications
• Kochen-Specker sets in four-dimensional spaces