# The Minimum Complexity of Kochen-Specker Sets Does Not Scale with Dimension

## TL;DR

This paper demonstrates that the minimal size of Kochen-Specker sets, which are used to prove quantum contextuality, does not increase with the dimension of the Hilbert space, challenging previous assumptions.

## Contribution

It introduces new methods for constructing critical Kochen-Specker sets in all dimensions and shows that their minimal size does not scale with dimension.

## Key findings

- Minimal KS sets in all dimensions have bounded size
- New construction methods for KS sets are developed
- Minimal KS set size does not grow 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 $d\geq10$ with at most 9 contexts and 30 projectors, and in all odd dimensions $d\geq 7$ 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 $d$.