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arXiv · 1705.08459

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

Samson Abramsky·Rui Soares Barbosa·Giovanni Carù·Simon Perdrix
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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.

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Samson Abramsky, Rui Soares Barbosa, Giovanni Carù, Simon Perdrix. 2017-05-23. A complete characterisation of All-versus-Nothing arguments for stabiliser states. https://doi.org/10.1098/rsta.2016.0385

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