A complete characterization of all-versus-nothing arguments for stabilizer states

Samson Abramsky, Rui Soares Barbosa, Giovanni Carù, Simon Perdrix

Research output: Contribution to journalArticlepeer-review


An important class of contextuality arguments in quantum foundations are the all-versus-nothing (AvN) proofs, generalizing a construction originally due to Mermin. We present a general formulation of AvN arguments and a complete characterization of all such arguments that arise from stabilizer states. We show that every AvN argument for an n-qubit stabilizer state can be reduced to an AvN proof for a three-qubit state that is local Clifford-equivalent to the tripartite Greenberger-Horne-Zeilinger state. This is achieved through a combinatorial characterization 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 Z2 on n-qubit stabilizer states. We also present new insights into the stabilizer formalism and its connections with logic. This article is part of the themed issue ‘Second quantum revolution: foundational questions’.

Original languageEnglish
Article number20160385
Number of pages18
JournalPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2106
Early online date2 Oct 2017
Publication statusPublished - 13 Nov 2017
Externally publishedYes


  • All-versus-nothing arguments
  • Contextuality
  • Graph states
  • Stabilizer states

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