# A complete characterisation of All-versus-Nothing arguments for stabiliser states

## TL;DR

This paper provides a complete characterization of All-versus-Nothing (AvN) contextuality proofs for stabiliser states, showing they can be reduced to GHZ states and developing a computational method for generating all such proofs.

## Contribution

It introduces a combinatorial AvN triple theorem, enabling reduction of AvN proofs to GHZ states and facilitating systematic generation of all AvN arguments for stabiliser states.

## Key findings

- All AvN arguments for n-qubit stabiliser states reduce to GHZ state proofs.
- Developed a computational method to generate all AvN arguments in $\
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The AvN triple theorem links graph states with AvN proofs.

## Abstract

An important class of contextuality arguments in quantum foundations are the All-versus-Nothing (AvN) proofs, generalising a construction originally due to Mermin. We present a general formulation of All-versus-Nothing arguments, and a complete characterisation of all such arguments which arise from stabiliser states. We show that every AvN argument for an n-qubit stabiliser state can be reduced to an AvN proof for a three-qubit state which is local Clifford-equivalent to the tripartite GHZ state. This is achieved through a combinatorial characterisation of AvN arguments, the AvN triple Theorem, whose proof makes use of the theory of graph states. This result enables the development of a computational method to generate all the AvN arguments in $\mathbb{Z}_2$ on n-qubit stabiliser states. We also present new insights into the stabiliser formalism and its connections with logic.