A Categorical Model for Classical and Quantum Block Designs

Classical block designs are important combinatorial structures with a wide range of applications in Computer Science and Statistics. Here we give a new abstract description of block designs based on the arrow category construction. We show that models of this structure in the category of matrices and natural numbers recover the traditional classical combinatorial objects, while models in the category of completely positive maps yield a new definition of quantum designs. We show that this generalizes both a previous notion of quantum designs given by Zauner and the traditional description of block designs. Furthermore, we demonstrate that there exists a functor which relates every categorical block design to a quantum one.Comment: In Proceedings ACT 2023, arXiv:2312.08138. 19 page

Globular: an online proof assistant for higher-dimensional rewriting

This article introduces Globular, an online proof assistant for the formalization and veri cation of proofs in higher-dimensional category theory. The tool produces graphical visualizations of higher-dimensional proofs, assists in their construction with a point-and- click interface, and performs type checking to prevent incorrect rewrites. Hosted on the web, it has a low barrier to use, and allows hyperlinking of formalized proofs directly from research papers. It allows the formalization of proofs from logic, topology and algebra which are not formalizable by other methods, and we give several examples

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

Completeness of dagger-categories and the complex numbers

The complex numbers are an important part of quantum theory, but are difficult to motivate from a theoretical perspective. We describe a simple formal framework for theories of physics, and show that if a theory of physics presented in this manner satisfies certain completeness properties, then it necessarily includes the complex numbers as a mathematical ingredient. Central to our approach are the techniques of category theory, and we introduce a new category-theoretical tool, called the dagger-limit, which governs the way in which systems can be combined to form larger systems. These dagger-limits can be used to characterize the dagger-functor on the category of finite-dimensional Hilbert spaces, and so can be used as an equivalent definition of the inner product. One of our main results is that in a nontrivial monoidal dagger-category with all finite dagger-limits and a simple tensor unit, the semiring of scalars embeds into an involutive field of characteristic 0 and orderable fixed field.Comment: 39 pages. Accepted for publication in the Journal of Mathematical Physic

Feasibility and Acceptability of Mindfulness-Based Cognitive Therapy Compared with Mindfulness-Based Stress Reduction and Treatment as Usual in People with Depression and Cardiovascular Disorders: A three-arm Randomised Controlled Trial

This is the author accepted manuscript. The final version is available from Springer via the DOI in this record.Depression co-occurs in 20 % of people with cardiovascular disorders, can persist for years, and predicts worse physical health outcomes. While psychosocial treatments have been shown to treat acute depression effectively in those with comorbid cardiovascular disorders, to date, there has been no evaluation of approaches aiming to prevent relapse and treat residual depression symptoms in this group. Consequently, the current study aimed to examine the feasibility and acceptability of a randomised controlled trial design evaluating an adapted version of mindfulness-based cognitive therapy (MBCT) designed specifically for people with co-morbid depression and cardiovascular disorders. A 3-arm feasibility randomised controlled trial was conducted, comparing MBCT adapted for people with cardiovascular disorders plus treatment as usual (TAU), mindfulness-based stress reduction (MBSR) plus TAU, and TAU alone. Participants completed a set of selfreport measures of depression severity, anxiety, quality of life, illness perceptions, mindfulness, self-compassion and affect and had their blood pressure taken immediately before, after, and three months following the intervention. Those in the adapted-MBCT arm additionally underwent a qualitative interview to gather their views about the adapted intervention. 3,400 potentially eligible participants were approached when attending an outpatient appointment at a cardiology clinic or via a GP letter following a case note search. 242 (7.1 %) were interested in taking part, 59 (1.7 %) were screened as being suitable, and 33 (<1 %) were eventually randomised to the three groups. Of 11 participants randomised to adapted MBCT, seven completed the full course, levels of home mindfulness practice were high, and positive qualitative feedback about the intervention was given. Twenty-nine out of 33 randomised participants completed all the assessment measures at all three-time points. The means PHQ-9 scores for the MBCT-HeLM group were lower at post-intervention and at the three-month follow-up compared to the MBSR and TAU groups. The sample was heterogeneous in terms of whether they reported current depression or had a history of depression and the time since the onset of cardiovascular disorders (one to 25 years). The adapted MBCT intervention was feasible and acceptable to participants, however, certain aspects of the trial design were not. In particular, low recruitment rates were achieved and there was a high withdrawal rate between screening and randomisation. Moreover, the heterogeneity in the sample was high, meaning the adapted intervention was unlikely to be well tailored to all the participants needs. This suggests that if the decision is made to move to a definitive trial, study recruitment procedures will need to be revised to recruit a target sample that optimally matches the adapted intervention.This research was, in part, supported by the National Institute for Health Research (NIHR) Collaboration for Leadership in Applied Health Research and Care South West Peninsula. The views expressed are those of the authors and not necessarily those of the NHS, the NIHR or the Department of Health. Also, the research was carried out as part of a PhD funded by King Saud University, Saudi Arabian Ministry of Higher Education

Multidimensional collaboration; reflections on action research in a clinical context

This paper reflects on the challenges and benefits of multidimensional collaboration in an action research study to evaluate and improve preoperative education for patients awaiting colorectal surgery. Three cycles of planning, acting,observing and reflecting were designed to evaluate practice and implement change in this interactive setting, calling for specific and distinct collaborations. Data collection includes: observing educational interactions; administering patient evaluation questionnaires; interviewing healthcare staff, patients and carers; patient and carer focus groups; and examining written and audiovisual educational materials. The study revolves around and depends on multi-dimensional collaborations. Reflecting on these collaborations highlights the diversity of perspectives held by all those engaged in the study and enhances the action research lessons. Successfully maintaining the collaborations recognises the need for negotiation, inclusivity, comprehension, brokerage,and problem-solving. Managing the potential tensions is crucial to the successful implementation of changes introduced to practice and thus has important implications for patients’ well-being. This paper describes the experiences from an action research project involving new and specific collaborations, focusing on a particular healthcare setting. It exemplifies the challenges of the collaborative action research process and examines how both researchers and practitioners might reflect on the translation of theory into educational practices within a hospital colorectal department. Despite its context-specific features, the reflections on the types of challenges faced and lessons learned provide implications for action researchers in diverse healthcare settings across the world

Categorical formulation of quantum algebras

We describe how dagger-Frobenius monoids give the correct categorical description of certain kinds of finite-dimensional 'quantum algebras'. We develop the concept of an involution monoid, and use it to construct a correspondence between finite-dimensional C*-algebras and certain types of dagger-Frobenius monoids in the category of Hilbert spaces. Using this technology, we recast the spectral theorems for commutative C*-algebras and for normal operators into an explicitly categorical language, and we examine the case that the results of measurements do not form finite sets, but rather objects in a finite Boolean topos. We describe the relevance of these results for topological quantum field theory.Comment: 34 pages, to appear in Communications in Mathematical Physic

Picturing classical and quantum Bayesian inference

We introduce a graphical framework for Bayesian inference that is sufficiently general to accommodate not just the standard case but also recent proposals for a theory of quantum Bayesian inference wherein one considers density operators rather than probability distributions as representative of degrees of belief. The diagrammatic framework is stated in the graphical language of symmetric monoidal categories and of compact structures and Frobenius structures therein, in which Bayesian inversion boils down to transposition with respect to an appropriate compact structure. We characterize classical Bayesian inference in terms of a graphical property and demonstrate that our approach eliminates some purely conventional elements that appear in common representations thereof, such as whether degrees of belief are represented by probabilities or entropic quantities. We also introduce a quantum-like calculus wherein the Frobenius structure is noncommutative and show that it can accommodate Leifer's calculus of `conditional density operators'. The notion of conditional independence is also generalized to our graphical setting and we make some preliminary connections to the theory of Bayesian networks. Finally, we demonstrate how to construct a graphical Bayesian calculus within any dagger compact category.Comment: 38 pages, lots of picture

Use of mixed methods designs in substance research: a methodological necessity in Nigeria

The utility of mixed methods (qualitative and quantitative) is becoming increasingly accepted in health sciences, but substance studies are yet to substantially benefit from such utilities. While there is a growing number of mixed methods alcohol articles concerning developed countries, developing nations are yet to embrace this method. In the Nigerian context, the importance of mixed methods research is yet to be acknowledged. This article therefore, draws on alcohol studies to argue that mixed methods designs will better equip scholars to understand, explore, describe and explain why alcohol consumption and its related problems are increasing in Nigeria. It argues that as motives for consuming alcohol in contemporary Nigeria are multiple, complex and evolving, mixed method approaches that provide multiple pathways for proffering solutions to problems should be embraced