A comonadic view of simulation and quantum resources

We study simulation and quantum resources in the setting of the sheaf-theoretic approach to contextuality and non-locality. Resources are viewed behaviourally, as empirical models. In earlier work, a notion of morphism for these empirical models was proposed and studied. We generalize and simplify the earlier approach, by starting with a very simple notion of morphism, and then extending it to a more useful one by passing to a co-Kleisli category with respect to a comonad of measurement protocols. We show that these morphisms capture notions of simulation between empirical models obtained via `free' operations in a resource theory of contextuality, including the type of classical control used in measurement-based quantum computation schemes.Comment: To appear in Proceedings of LiCS 201

The Quantum Monadology

The modern theory of functional programming languages uses monads for encoding computational side-effects and side-contexts, beyond bare-bone program logic. Even though quantum computing is intrinsically side-effectful (as in quantum measurement) and context-dependent (as on mixed ancillary states), little of this monadic paradigm has previously been brought to bear on quantum programming languages. Here we systematically analyze the (co)monads on categories of parameterized module spectra which are induced by Grothendieck's "motivic yoga of operations" -- for the present purpose specialized to HC-modules and further to set-indexed complex vector spaces. Interpreting an indexed vector space as a collection of alternative possible quantum state spaces parameterized by quantum measurement results, as familiar from Proto-Quipper-semantics, we find that these (co)monads provide a comprehensive natural language for functional quantum programming with classical control and with "dynamic lifting" of quantum measurement results back into classical contexts. We close by indicating a domain-specific quantum programming language (QS) expressing these monadic quantum effects in transparent do-notation, embeddable into the recently constructed Linear Homotopy Type Theory (LHoTT) which interprets into parameterized module spectra. Once embedded into LHoTT, this should make for formally verifiable universal quantum programming with linear quantum types, classical control, dynamic lifting, and notably also with topological effects.Comment: 120 pages, various figure

Game Comonads & Generalised Quantifiers

Game comonads, introduced by Abramsky, Dawar and Wang and developed by Abramsky and Shah, give an interesting categorical semantics to some Spoiler-Duplicator games that are common in finite model theory. In particular they expose connections between one-sided and two-sided games, and parameters such as treewidth and treedepth and corresponding notions of decomposition. In the present paper, we expand the realm of game comonads to logics with generalised quantifiers. In particular, we introduce a comonad graded by two parameter $n \leq k$ such that isomorphisms in the resulting Kleisli category are exactly Duplicator winning strategies in Hella's $n$-bijection game with $k$ pebbles. We define a one-sided version of this game which allows us to provide a categorical semantics for a number of logics with generalised quantifiers. We also give a novel notion of tree decomposition that emerges from the construction

Proceedings of JAC 2010. Journées Automates Cellulaires

The second Symposium on Cellular Automata “Journ´ees Automates Cellulaires” (JAC 2010) took place in Turku, Finland, on December 15-17, 2010. The first two conference days were held in the Educarium building of the University of Turku, while the talks of the third day were given onboard passenger ferry boats in the beautiful Turku archipelago, along the route Turku–Mariehamn–Turku. The conference was organized by FUNDIM, the Fundamentals of Computing and Discrete Mathematics research center at the mathematics department of the University of Turku. The program of the conference included 17 submitted papers that were selected by the international program committee, based on three peer reviews of each paper. These papers form the core of these proceedings. I want to thank the members of the program committee and the external referees for the excellent work that have done in choosing the papers to be presented in the conference. In addition to the submitted papers, the program of JAC 2010 included four distinguished invited speakers: Michel Coornaert (Universit´e de Strasbourg, France), Bruno Durand (Universit´e de Provence, Marseille, France), Dora Giammarresi (Universit` a di Roma Tor Vergata, Italy) and Martin Kutrib (Universit¨at Gie_en, Germany). I sincerely thank the invited speakers for accepting our invitation to come and give a plenary talk in the conference. The invited talk by Bruno Durand was eventually given by his co-author Alexander Shen, and I thank him for accepting to make the presentation with a short notice. Abstracts or extended abstracts of the invited presentations appear in the first part of this volume. The program also included several informal presentations describing very recent developments and ongoing research projects. I wish to thank all the speakers for their contribution to the success of the symposium. I also would like to thank the sponsors and our collaborators: the Finnish Academy of Science and Letters, the French National Research Agency project EMC (ANR-09-BLAN-0164), Turku Centre for Computer Science, the University of Turku, and Centro Hotel. Finally, I sincerely thank the members of the local organizing committee for making the conference possible. These proceedings are published both in an electronic format and in print. The electronic proceedings are available on the electronic repository HAL, managed by several French research agencies. The printed version is published in the general publications series of TUCS, Turku Centre for Computer Science. We thank both HAL and TUCS for accepting to publish the proceedings.Siirretty Doriast

Closing Bell: Boxing black box simulations in the resource theory of contextuality

This chapter contains an exposition of the sheaf-theoretic framework for contextuality emphasising resource-theoretic aspects, as well as some original results on this topic. In particular, we consider functions that transform empirical models on a scenario S to empirical models on another scenario T, and characterise those that are induced by classical procedures between S and T corresponding to 'free' operations in the (non-adaptive) resource theory of contextuality. We construct a new 'hom' scenario built from S and T, whose empirical models induce such functions. Our characterisation then boils down to being induced by a non-contextual model. We also show that this construction on scenarios provides a closed structure on the category of measurement scenarios.Comment: Corrected a mistake in Theorem 44 and other fixes stemming from it. This supersedes the published version and should be considered the version of referenc

Continuous-variable nonlocality and contextuality

Contextuality is a non-classical behaviour that can be exhibited by quantum systems. It is increasingly studied for its relationship to quantum-over-classical advantages in informatic tasks. To date, it has largely been studied in discrete variable scenarios, where observables take values in discrete and usually finite sets. Practically, on the other hand, continuous-variable scenarios offer some of the most promising candidates for implementing quantum computations and informatic protocols. Here we set out a framework for treating contextuality in continuous-variable scenarios. It is shown that the Fine--Abramsky--Brandenburger theorem extends to this setting, an important consequence of which is that nonlocality can be viewed as a special case of contextuality, as in the discrete case. The contextual fraction, a quantifiable measure of contextuality that bears a precise relationship to Bell inequality violations and quantum advantages, can also be defined in this setting. It is shown to be a non-increasing monotone with respect to classical operations that include binning to discretise data. Finally, we consider how the contextual fraction can be formulated as an infinite linear program, and calculated with increasing accuracy using semi-definite programming approximations.Comment: 27 pages including 6 pages supplemental material, 2 figure

Corrected Bell and Noncontextuality Inequalities for Realistic Experiments

Contextuality is a feature of quantum correlations. It is crucial from a foundational perspective as a nonclassical phenomenon, and from an applied perspective as a resource for quantum advantage. It is commonly defined in terms of hidden variables, for which it forces a contradiction with the assumptions of parameter-independence and determinism. The former can be justified by the empirical property of non-signalling or non-disturbance, and the latter by the empirical property of measurement sharpness. However, in realistic experiments neither empirical property holds exactly, which leads to possible objections to contextuality as a form of nonclassicality, and potential vulnerabilities for supposed quantum advantages. We introduce measures to quantify both properties, and introduce quantified relaxations of the corresponding assumptions. We prove the continuity of a known measure of contextuality, the contextual fraction, which ensures its robustness to noise. We then bound the extent to which these relaxations can account for contextuality, via corrections terms to the contextual fraction (or to any noncontextuality inequality), culminating in a notion of genuine contextuality, which is robust to experimental imperfections. We then show that our result is general enough to apply or relate to a variety of established results and experimental setups.Comment: 20 pages + 14 pages of appendices, 3 figure

A bundle perspective on contextuality: Empirical models and simplicial distributions on bundle scenarios

This paper provides a bundle perspective to contextuality by introducing new categories of contextuality scenarios based on bundles of simplicial complexes and simplicial sets. The former approach generalizes earlier work on the sheaf-theoretic perspective on contextuality, and the latter extends simplicial distributions, a more recent approach to contextuality formulated in the language of simplicial sets. After constructing our bundle categories, we also construct functors that relate them and natural isomorphisms that allow us to compare the notions of contextuality formulated in two languages. We are motivated by applications to the resource theory of contextuality, captured by the morphisms in these categories. In this paper, we develop the main formalism and leave applications to future work.Comment: 50 pages, 2 figure