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

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.Comment: 18 pages, 6 figure

On the Cohomology of Contextuality

Recent work by Abramsky and Brandenburger used sheaf theory to give a mathematical formulation of non-locality and contextuality. By adopting this viewpoint, it has been possible to define cohomological obstructions to the existence of global sections. In the present work, we illustrate new insights into different aspects of this theory. We shed light on the power of detection of the cohomological obstruction by showing that it is not a complete invariant for strong contextuality even under symmetry and connectedness restrictions on the measurement cover, disproving a previous conjecture. We generalise obstructions to higher cohomology groups and show that they give rise to a refinement of the notion of cohomological contextuality: different "levels" of contextuality are organised in a hierarchy of logical implications. Finally, we present an alternative description of the first cohomology group in terms of torsors, resulting in a new interpretation of the cohomological obstructions.Comment: In Proceedings QPL 2016, arXiv:1701.0024

Hardy is (almost) everywhere: nonlocality without inequalities for almost all entangled multipartite states

We show that all $n$-qubit entangled states, with the exception of tensor products of single-qubit and bipartite maximally-entangled states, admit Hardy-type proofs of non-locality without inequalities or probabilities. More precisely, we show that for all such states, there are local, one-qubit observables such that the resulting probability tables are logically contextual in the sense of Abramsky and Brandenburger, this being the general form of the Hardy-type property. Moreover, our proof is constructive: given a state, we show how to produce the witnessing local observables. In fact, we give an algorithm to do this. Although the algorithm is reasonably straightforward, its proof of correctness is non-trivial. A further striking feature is that we show that $n+2$ local observables suffice to witness the logical contextuality of any $n$-qubit state: two each for two for the parties, and one each for the remaining $n-2$ parties.Comment: 23 pages. Submitted for publicatio

A Deep Incremental Boltzmann Machine for Modeling Context in Robots

Context is an essential capability for robots that are to be as adaptive as possible in challenging environments. Although there are many context modeling efforts, they assume a fixed structure and number of contexts. In this paper, we propose an incremental deep model that extends Restricted Boltzmann Machines. Our model gets one scene at a time, and gradually extends the contextual model when necessary, either by adding a new context or a new context layer to form a hierarchy. We show on a scene classification benchmark that our method converges to a good estimate of the contexts of the scenes, and performs better or on-par on several tasks compared to other incremental models or non-incremental models.Comment: 6 pages, 5 figures, International Conference on Robotics and Automation (ICRA 2018

Topos Quantum Logic and Mixed States

The topos approach to the formulation of physical theories includes a new form of quantum logic. We present this topos quantum logic, including some new results, and compare it to standard quantum logic, all with an eye to conceptual issues. In particular, we show that topos quantum logic is distributive, multi-valued, contextual and intuitionistic. It incorporates superposition without being based on linear structures, has a built-in form of coarse-graining which automatically avoids interpretational problems usually associated with the conjunction of propositions about incompatible physical quantities, and provides a material implication that is lacking from standard quantum logic. Importantly, topos quantum logic comes with a clear geometrical underpinning. The representation of pure states and truth-value assignments are discussed. It is briefly shown how mixed states fit into this approach.Comment: 25 pages; to appear in Electronic Notes in Theoretical Computer Science (6th Workshop on Quantum Physics and Logic, QPL VI, Oxford, 8.--9. April 2009), eds. B. Coecke, P. Panangaden, P. Selinger (2010