Two fundamental solutions to the rigid Kochen-Specker set problem and the solution to the minimal Kochen-Specker set problem under one assumption

TL;DR

This paper identifies two fundamental rigid Kochen-Specker sets in quantum theory, providing new insights into their structure and proposing a conjecture that 31 elements form the minimal KS set in three-dimensional quantum systems.

Contribution

The paper introduces two fundamental rigid KS sets derived from core quantum structures and offers a conjecture on the minimal KS set size in bc, advancing understanding of quantum contextuality.

Findings

Two fundamental rigid KS sets are defined from core quantum structures.

No KS set with 30 elements can be formed from the minimal contextuality set by certain operations.

A conjecture that the minimal KS set in bc has 31 elements.

Abstract

Recent results show that Kochen-Specker (KS) sets of observables are fundamental to quantum information, computation, and foundations beyond previous expectations. Among KS sets, those that are unique up to unitary transformations (i.e., "rigid") are especially important. The problem is that we do not know any rigid KS set in $\mathbb{C}^3$, the smallest quantum system that allows for KS sets. Moreover, none of the existing methods for constructing KS sets leads to rigid KS sets in $\mathbb{C}^3$. Here, we show that two fundamental structures of quantum theory define two rigid KS sets. One of these structures is the super-symmetric informationally complete positive-operator-valued measure. The other is the minimal state-independent contextuality set. The second construction provides a clue to solve the minimal KS problem, the most important open problem in this field. We prove that there is no KS set of 30 elements that can be obtained from the minimal state-independent contextuality set by completing bases and adding elements that are orthogonal to two previous elements. We conjecture that 31 is the solution to the minimal KS set problem.