Proofs of the Kochen-Specker theorem based on the N-qubit Pauli group

TL;DR

This paper introduces new, visually intuitive, irreducible proofs of the Kochen-Specker theorem using the N-qubit Pauli group for N >= 4, and provides algorithms to convert these into projectors-based proofs with practical experimental applications.

Contribution

It presents novel observables-based proofs for N-qubit systems, including an infinite family applicable from two qubits upwards, and offers a method to generate multiple projectors-based proofs from them.

Findings

Proofs are presented for N >= 4 qubits, extending previous work for fewer qubits.

A simple algorithm transforms observables-based proofs into numerous projectors-based proofs.

An infinite family of proofs applies to all N ≥ 2 qubits, with minimal experimental contexts.

Abstract

We present a number of observables-based proofs of the Kochen-Specker (KS) theorem based on the N-qubit Pauli group for N >= 4, thus adding to the proofs that have been presented earlier for the two- and three-qubit groups. These proofs have the attractive feature that they can be presented in the form of diagrams from which they are obvious by inspection. They are also irreducible in the sense that they cannot be reduced to smaller proofs by ignoring some subset of qubits and/or observables in them. A simple algorithm is given for transforming any observables-based KS proof into a large number of projectors-based KS proofs; if the observables-based proof has O observables, with each observable occurring in exactly two commuting sets and any two commuting sets having at most one observable in common, the number of associated projectors-based parity proofs is 2^O. We introduce symbols for the observables- and projectors-based KS proofs that capture their important features and also convey a feeling for the enormous variety of both these types of proofs within the N-qubit Pauli group. We discuss an infinite family of observables-based proofs, whose members apply to all numbers of qubits from two up, and show how it can be used to generate projectors-based KS proofs involving only nine bases (or experimental contexts) in any dimension of the form 2^N for N >= 2. Some implications of our results are discussed.