A categorical framework for the quantum harmonic oscillator

This paper describes how the structure of the state space of the quantum harmonic oscillator can be described by an adjunction of categories, that encodes the raising and lowering operators into a commutative comonoid. The formulation is an entirely general one in which Hilbert spaces play no special role. Generalised coherent states arise through the hom-set isomorphisms defining the adjunction, and we prove that they are eigenstates of the lowering operators. Surprisingly, generalised exponentials also emerge naturally in this setting, and we demonstrate that coherent states are produced by the exponential of a raising morphism acting on the zero-particle state. Finally, we examine all of these constructions in a suitable category of Hilbert spaces, and find that they reproduce the conventional mathematical structures.Comment: 44 pages, many figure

Predicate Transformers and Linear Logic, yet another denotational model

International audienceIn the refinement calculus, monotonic predicate transformers are used to model specifications for (imperative) programs. Together with a natural notion of simulation, they form a category enjoying many algebraic properties. We build on this structure to make predicate transformers into a de notational model of full linear logic: all the logical constructions have a natural interpretation in terms of predicate transformers (i.e. in terms of specifications). We then interpret proofs of a formula by a safety property for the corresponding specification

Generating reversible circuits from higher-order functional programs

Boolean reversible circuits are boolean circuits made of reversible elementary gates. Despite their constrained form, they can simulate any boolean function. The synthesis and validation of a reversible circuit simulating a given function is a difficult problem. In 1973, Bennett proposed to generate reversible circuits from traces of execution of Turing machines. In this paper, we propose a novel presentation of this approach, adapted to higher-order programs. Starting with a PCF-like language, we use a monadic representation of the trace of execution to turn a regular boolean program into a circuit-generating code. We show that a circuit traced out of a program computes the same boolean function as the original program. This technique has been successfully applied to generate large oracles with the quantum programming language Quipper.Comment: 21 pages. A shorter preprint has been accepted for publication in the Proceedings of Reversible Computation 2016. The final publication is available at http://link.springer.co

Subsystems of a finite quantum system and Bell-like inequalities

YesThe set of subsystems Sigma(m) of a finite quantum system Sigma(n) with variables in Z(n) together with logical connectives, is a Heyting algebra. The probabilities tau(m vertical bar rho(n)) Tr vertical bar B(m)rho(n)] (where B(m) is the projector to Sigma(m)) are compatible with associativity of the join in the Heyting algebra, only if the variables belong to the same chain. Consequently, contextuality in the present formalism, has the chains as contexts. Various Bell-like inequalities are discussed. They are violated, and this proves that quantum mechanics is a contextual theory

On Linear Information Systems

Scott's information systems provide a categorically equivalent, intensional description of Scott domains and continuous functions. Following a well established pattern in denotational semantics, we define a linear version of information systems, providing a model of intuitionistic linear logic (a new-Seely category), with a "set-theoretic" interpretation of exponentials that recovers Scott continuous functions via the co-Kleisli construction. From a domain theoretic point of view, linear information systems are equivalent to prime algebraic Scott domains, which in turn generalize prime algebraic lattices, already known to provide a model of classical linear logic

On coalgebras with internal moves

In the first part of the paper we recall the coalgebraic approach to handling the so-called invisible transitions that appear in different state-based systems semantics. We claim that these transitions are always part of the unit of a certain monad. Hence, coalgebras with internal moves are exactly coalgebras over a monadic type. The rest of the paper is devoted to supporting our claim by studying two important behavioural equivalences for state-based systems with internal moves, namely: weak bisimulation and trace semantics. We continue our research on weak bisimulations for coalgebras over order enriched monads. The key notions used in this paper and proposed by us in our previous work are the notions of an order saturation monad and a saturator. A saturator operator can be intuitively understood as a reflexive, transitive closure operator. There are two approaches towards defining saturators for coalgebras with internal moves. Here, we give necessary conditions for them to yield the same notion of weak bisimulation. Finally, we propose a definition of trace semantics for coalgebras with silent moves via a uniform fixed point operator. We compare strong and weak bisimilation together with trace semantics for coalgebras with internal steps.Comment: Article: 23 pages, Appendix: 3 page

A Metalanguage for Guarded Iteration

Notions of guardedness serve to delineate admissible recursive definitions in various settings in a compositional manner. In recent work, we have introduced an axiomatic notion of guardedness in symmetric monoidal categories, which serves as a unifying framework for various examples from program semantics, process algebra, and beyond. In the present paper, we propose a generic metalanguage for guarded iteration based on combining this notion with the fine-grain call-by-value paradigm, which we intend as a unifying programming language for guarded and unguarded iteration in the presence of computational effects. We give a generic (categorical) semantics of this language over a suitable class of strong monads supporting guarded iteration, and show it to be in touch with the standard operational behaviour of iteration by giving a concrete big-step operational semantics for a certain specific instance of the metalanguage and establishing soundness and (computational) adequacy for this case.Comment: extended version for the special issu

Findings from the SASA! Study: a cluster randomized controlled trial to assess the impact of a community mobilization intervention to prevent violence against women and reduce HIV risk in Kampala, Uganda

Background Intimate partner violence (IPV) and HIV are important and interconnected public health concerns. While it is recognized that they share common social drivers, there is limited evidence surrounding the potential of community interventions to reduce violence and HIV risk at the community level. The SASA! study assessed the community-level impact of SASA!, a community mobilization intervention to prevent violence and reduce HIV-risk behaviors. Methods From 2007 to 2012 a pair-matched cluster randomized controlled trial (CRT) was conducted in eight communities (four intervention and four control) in Kampala, Uganda. Cross-sectional surveys of a random sample of community members, 18- to 49-years old, were undertaken at baseline (n = 1,583) and four years post intervention implementation (n = 2,532). Six violence and HIV-related primary outcomes were defined a priori. An adjusted cluster-level intention-to-treat analysis compared outcomes in intervention and control communities at follow-up. Results The intervention was associated with significantly lower social acceptance of IPV among women (adjusted risk ratio 0.54, 95% confidence interval (CI) 0.38 to 0.79) and lower acceptance among men (0.13, 95% CI 0.01 to 1.15); significantly greater acceptance that a woman can refuse sex among women (1.28, 95% CI 1.07 to 1.52) and men (1.31, 95% CI 1.00 to 1.70); 52% lower past year experience of physical IPV among women (0.48, 95% CI 0.16 to 1.39); and lower levels of past year experience of sexual IPV (0.76, 95% CI 0.33 to 1.72). Women experiencing violence in intervention communities were more likely to receive supportive community responses. Reported past year sexual concurrency by men was significantly lower in intervention compared to control communities (0.57, 95% CI 0.36 to 0.91). Conclusions This is the first CRT in sub-Saharan Africa to assess the community impact of a mobilization program on the social acceptability of IPV, the past year prevalence of IPV and levels of sexual concurrency. SASA! achieved important community impacts, and is now being delivered in control communities and replicated in 15 countries

Observed communication semantics for classical processes

Classical Linear Logic (CLL) has long inspired readings of its proofs as communicating processes. Wadler's CP calculus is one of these readings. Wadler gave CP an operational semantics by selecting a subset of the cut-elimination rules of CLL to use as reduction rules. This semantics has an appealing close connection to the logic, but does not resolve the status of the other cut-elimination rules, and does not admit an obvious notion of observational equivalence. We propose a new operational semantics for CP based on the idea of observing communication, and use this semantics to define an intuitively reasonable notion of observational equivalence. To reason about observational equivalence, we use the standard relational denotational semantics of CLL. We show that this denotational semantics is adequate for our operational semantics. This allows us to deduce that, for instance, all the cut-elimination rules of CLL are observational equivalences

Topological fault-tolerance in cluster state quantum computation

We describe a fault-tolerant version of the one-way quantum computer using a cluster state in three spatial dimensions. Topologically protected quantum gates are realized by choosing appropriate boundary conditions on the cluster. We provide equivalence transformations for these boundary conditions that can be used to simplify fault-tolerant circuits and to derive circuit identities in a topological manner. The spatial dimensionality of the scheme can be reduced to two by converting one spatial axis of the cluster into time. The error threshold is 0.75% for each source in an error model with preparation, gate, storage and measurement errors. The operational overhead is poly-logarithmic in the circuit size.Comment: 20 pages, 12 figure