## About

80

Publications

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Introduction

I work on the mathematical foundations of physics, especially quantum mechanics, and its logical aspects. My weapons of choice are category theory, functional analysis, and order theory; specifically, monoidal categories, operator algebras, and orthomodular lattices. I suppose you could say that my ultimate goal is to really understand the category of Hilbert spaces, in particular categorical aspects of a choice of basis.

Additional affiliations

October 2015 - present

November 2009 - November 2012

August 2009 - September 2015

Education

August 2005 - August 2009

August 2000 - August 2005

August 1999 - August 2005

## Publications

Publications (80)

We introduce a construction that turns a category of pure state spaces and
operators into a category of observable algebras and superoperators. For
example, it turns the category of finite-dimensional Hilbert spaces into the
category of finite-dimensional C*-algebras and completely positive maps. In
particular, the new category contains both quantu...

New scientific paradigms typically consist of an expansion of the conceptual language with which we describe the world. Over the past decade, theoretical physics and quantum information theory have turned to category theory to model and reason about quantum protocols. This new use of categorical and algebraic tools allows a more conceptual and insi...

The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*-algebra of observables A induces a topos \({\mat...

We provide a universal construction of the category of finite-dimensional C*-algebras and completely positive trace-nonincreasing maps from the rig category of finite-dimensional Hilbert spaces and unitaries. This construction, which can be applied to any dagger rig category, is described in three steps, each associated with their own universal pro...

We study the two dual quantum information effects to manipulate the amount of information in quantum computation: hiding and allocation. The resulting type-and-effect system is fully expressive for irreversible quantum computing, including measurement. We provide universal categorical constructions that semantically interpret this arrow metalanguag...

We study the two dual quantum information effects to manipulate the amount of information in quantum computation: hiding and allocation. The resulting type-and-effect system is fully expressive for irreversible quantum computing, including measurement. We provide universal categorical constructions that semantically interpret this arrow metalanguag...

Every small monoidal category with universal (finite) joins of central idempotents is monoidally equivalent to the category of global sections of a sheaf of (sub)local monoidal categories on a topological space. Every small stiff monoidal category monoidally embeds into such a category of global sections. These representation results are functorial...

We present a universal construction that relates reversible dynamics on open systems to arbitrary dynamics on closed systems: the well-pointed restriction affine completion of a monoidal restriction category. This categorical completion encompasses both quantum channels, via Stinespring dilation, and classical computing, via Bennett's method. Moreo...

A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We show that under mild conditions subunits endow any monoidal category with a kind of topological intuition: there...

We study internal structures in regular categories using monoidal methods. Groupoids in a regular Goursat category can equivalently be described as special dagger Frobenius monoids in its monoidal category of relations. Similarly, connectors can equivalently be described as Frobenius structures with a ternary multiplication. We study such ternary F...

We develop a direct method to recover an orthoalgebra from its poset of Boolean subalgebras. For this a new notion of direction is introduced. Directions are also used to characterize in purely order-theoretic terms those posets that are isomorphic to the poset of Boolean subalgebras of an orthoalgebra. These posets are characterized by simple cond...

The Ehresmann-Schein-Nambooripad theorem gives a structure theorem for inverse monoids: they are inductive groupoids. A particularly nice case due to Jarek is that commutative inverse monoids become semilattices of abelian groups. It has also been categorified by DeWolf-Pronk to a structure theorem for inverse categories as locally complete inducti...

We study internal structures in regular categories using monoidal methods. Groupoids in a regular Goursat category can equivalently be described as special dagger Frobenius monoids in its monoidal category of relations. Similarly, connectors can equivalently be described as Frobenius structures with a ternary multiplication. We study such ternary F...

A C*-algebra is determined to a great extent by the partial order of its commutative C*-subalgebras. We study order-theoretic properties of this directed-complete partially ordered (dcpo). Many properties coincide: the dcpo is, equivalently, algebraic, continuous, meet-continuous, atomistic, quasi-algebraic or quasi-continuous, if and only if the C...

Topos quantum mechanics, developed by Isham et. al., creates a topos of presheaves over the poset V(N) of abelian von Neumann subalgebras of the von Neumann algebra N of bounded operators associated to a physical system, and established several results, including: (a) a connection between the Kochen-Specker theorem and the non-existence of a global...

Reversible computing models settings in which all processes can be reversed. Applications include low-power computing, quantum computing, and robotics. It is unclear how to represent side-effects in this setting, because conventional methods need not respect reversibility. We model reversible effects by adapting Hughes’ arrows to dagger arrows and...

A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We show that under mild conditions subunits endow any monoidal category with topological intuition: there are well-b...

We study the monoidal dagger category of Hilbert C*-modules over a commutative C*-algebra from the perspective of categorical quantum mechanics. The dual objects are the finitely presented projective Hilbert C*-modules. Special dagger Frobenius structures correspond to bundles of uniformly finite-dimensional C*-algebras. A monoid is dagger Frobeniu...

Reversible computing models settings in which all processes can be reversed. Applications include low-power computing, quantum computing, and robotics. It is unclear how to represent side-effects in this setting, because conventional methods need not respect reversibility. We model reversible effects by adapting Hughes' arrows to dagger arrows and...

The structure of the category of matroids and strong maps is investigated: it
has coproducts and equalizers, but not products or coequalizers; there are
functors from the categories of graphs and vector spaces, the latter being
faithful; there is a functor to the
category of geometric lattices, that is nearly full; there are various
adjunctions and...

We develop a notion of limit for dagger categories, that we show is suitable in the following ways: it subsumes special cases known from the literature; dagger limits are unique up to unitary isomorphism; a wide class of dagger limits can be built from a small selection of them; dagger limits of a fixed shape can be phrased as dagger adjoints to a...

We present a modular semantic account of Bayesian inference algorithms for probabilistic programming lan- guages, as used in data science and machine learning. Sophisticated inference algorithms are often explained in terms of composition of smaller parts. However, neither their theoretical justification nor their implemen- tation reflects this mod...

We present a modular semantic account of Bayesian inference algorithms for probabilistic programming languages, as used in data science and machine learning. Sophisticated inference algorithms are often explained in terms of composition of smaller parts. However, neither their theoretical justification nor their implementation reflects this modular...

We give a construction that identifies the collection of pure processes (i.e. those which are deterministic, or without randomness) within a theory containing both pure and mixed processes. Working in the framework of symmetric monoidal categories, we define a pure subcategory. This definition arises elegantly from the categorical notion of a weak...

Higher-order probabilistic programming languages allow programmers to write sophisticated models in machine learning and statistics in a succinct and structured way, but step outside the standard measure-theoretic formalization of probability theory. Programs may use both higher-order functions and continuous distributions, or even define a probabi...

We investigate how a C*-algebra could consist of functions on a noncommutative set: a discretization of a C*-algebra $A$ is a $*$-homomorphism $A \to M$ that factors through the canonical inclusion $C(X) \subseteq \ell^\infty(X)$ when restricted to a commutative C*-subalgebra. Any C*-algebra admits an injective but nonfunctorial discretization, as...

We study the semantic foundation of expressive probabilistic programming languages, that support higher-order functions, continuous distributions, and soft constraints (such as Anglican, Church, and Venture). We define a metalanguage (an idealised version of Anglican) for probabilistic computation with the above features, develop both operational a...

Does information play a significant role in the foundations of physics? We investigate whether information-theoretic constraints characterize quantum theory. In a C*-algebraic framework, this is known to hold via three equivalences: no broadcasting and noncommutativity; no bit commitment and nonlocality; no signalling and kinematic independence. Bu...

The theory of monads on categories equipped with a dagger (a contravariant identity-on-objects involutive endofunctor) works best when everything respects the dagger: the monad and adjunctions should preserve the dagger, and the monad and its algebras should satisfy the so-called Frobenius law. Then any monad resolves as an adjunction, with extrema...

We study the semantic foundation of expressive probabilistic programming
languages, that support higher-order functions, continuous distributions, and
soft constraints (such as Anglican, Church, and Venture). We define a
metalanguage (an idealised version of Anglican) for probabilistic computation
with the above features, develop both operational a...

This volume contains the proceedings of the 12th International Workshop on
Quantum Physics and Logic (QPL 2015), which was held July 15-17, 2015 at Oxford
University. The goal of this workshop series is to bring together researchers
working on mathematical foundations of quantum physics, quantum computing,
spatio-temporal causal structures, and rel...

Categories of relations over a regular category form a family of models of
quantum theory. Using regular logic, many properties of relations over sets
lift to these models, including the correspondence between Frobenius structures
and internal groupoids. Over compact Hausdorff spaces, this lifting gives
continuous symmetric encryption. Over a regul...

There are two ways to turn a categorical model for pure quantum theory into
one for mixed quantum theory, both resulting in a category of completely
positive maps. One has quantum systems as objects, whereas the other also
allows classical systems on an equal footing. The former has been axiomatized
using environment structures. We extend this axio...

We extend categorical semantics of monadic programming to reversible
computing, by considering monoidal closed dagger categories: the dagger gives
reversibility, whereas closure gives higher-order expressivity. We demonstrate
that Frobenius monads model the appropriate notion of coherence between the
dagger and closure by reinforcing Cayley's theor...

Operator algebras provide uniform semantics for deterministic, reversible,
probabilistic, and quantum computing, where intermediate results of partial
computations are given by commutative subalgebras. We study this setting using
domain theory, and show that a given operator algebra is scattered if and only
if its associated partial order is, equiv...

Interpretational problems with quantum mechanics can be phrased precisely by
only talking about empirically accessible information. This prompts a
mathematical reformulation of quantum mechanics in terms of classical
mechanics. We survey this programme in terms of algebraic quantum theory.

The C*-algebra of bounded operators on the separable Hilbert space cannot be
mapped to a W*-algebra in such a way that each unital commutative C*-subalgebra
C(X) factors normally through $\ell^\infty(X)$. Consequently, there is no
faithful functor discretizing C*-algebras to W*-algebras this way.

We introduce the CP*–construction on a dagger compact closed category as a generalisation of Selinger's CPM-construction. While the latter takes a dagger compact closed category and forms its category of "abstract matrix algebras" and completely positive maps, the CP*-construction forms its category of "abstract C*-algebras" and completely positive...

Any functor from the category of C*-algebras to the category of locales that
assigns to each commutative C*-algebra its Gelfand spectrum must be trivial on
algebras of nxn-matrices for n at least 3. This obstruction also applies to
other spectra such as those named after Zariski, Stone, and Pierce. We extend
these no-go results to functors with val...

In many a traditional physics textbook, a quantum measurement is defined as a projective measurement represented by a Hermitian operator. In quantum information theory, however, the concept of a measurement is dealt with in complete generality and we are therefore forced to confront the more general notion of positive-operator valued measures (POVM...

There are two ways to describe the interaction between classical and quantum
information categorically: one based on completely positive maps between
Frobenius algebras, the other using symmetric monoidal 2-categories. This paper
makes a first step towards combining the two. The integrated approach allows a
unified description of quantum teleportat...

We characterise piecewise Boolean domains, that is, those domains that arise
as Boolean subalgebras of a piecewise Boolean algebra. This leads to equivalent
descriptions of the category of piecewise Boolean algebras: either as piecewise
Boolean domains equipped with an orientation, or as full structure sheaves on
piecewise Boolean domains.

We prove that AW*-algebras are determined by their projections, their
symmetries, and the action of the latter on the former. We introduce active
lattices, which are formed from these three ingredients. More generally, we
prove that the category of AW*-algebras is equivalent to a full subcategory of
active lattices. Crucial ingredients are an equiv...

We aim to characterize the category of injective *-homomorphisms between commutative C * -subalgebras of a given C * -algebra A. We reduce this problem to finding a weakly terminal commutative subalgebra of A, and solve the latter for various C * -algebras, including all commutative ones and all type I von Neumann algebras. This addresses a natural...

Joint measurability of sharp quantum observables is determined pairwise, and
so can be captured in a graph. We prove the converse: any graph, whose vertices
represent sharp observables, and whose edges represent joint measurability, is
realised by quantum theory. This leads us to show that joint measurability is
not stable under Neumark dilation, i...

Any associative bilinear multiplication on the set of n-by-n matrices over
some field of characteristic not two, that makes the same vectors orthogonal
and has the same trace as ordinary matrix multiplication, must be ordinary
matrix multiplication or its opposite.

The recently introduced CP*-construction unites quantum channels and
classical systems, subsuming the earlier CPM-construction in categorical
quantum mechanics. We compare this construction to two earlier attempts at
solving this problem: freely adding biproducts to CPM, and freely splitting
idempotents in CPM. The CP*-construction embeds the forme...

Quantum logic aims to capture essential quantum mechanical structure in order-theoretic terms. The Achilles' heel of quantum logic is the absence of a canonical description of composite systems, given descriptions of their components. We introduce a framework in which order-theoretic structure comes with a primitive composition operation. The order...

We study the functor l^2 from the category of partial injections to the category of Hilbert spaces. The former category is finitely accessible, and both categories are enriched over algebraic domains. The functor preserves daggers, monoidal structures, enrichment, and various (co)limits, but has no adjoints. Up to unitaries, its direct image consis...

The main purpose of this erratum is to correct a claim made in “On the functor ℓ2” (Computation, Logic, Games, and Quantum Foundations, Lecture Notes in Computer Science Volume 7860, 2013, pp 107–121) in Lemma 5.9. Namely, positive operators on Hilbert space are not necessarily isomorphisms, but merely bimorphisms, i.e. both monic and epic; this is...

We study classical structures in various categories of completely positive
morphisms: on sets and relations, on cobordisms, on a free dagger compact
category, and on Hilbert spaces. As an application, we prove that quantum maps
with commuting Kraus operators can be sequentialized. Hence such protocols are
precisely as robust under general dephasing...

Every commuting set of normal matrices with entries in an AW*-algebra can be
simultaneously diagonalized. To establish this, a dimension theory for properly
infinite projections in AW*-algebras is developed. As a consequence, passing to
matrix rings is a functor on the category of AW*-algebras.

Every partial algebra is the colimit of its total subalgebras. We prove this
result for partial Boolean algebras (including orthomodular lattices) and the
new notion of partial C*-algebras (including noncommutative C*-algebras), and
variations such as partial complete Boolean algebras and partial AW*-algebras.
The first two results are related by t...

We relate notions of complementarity in three layers of quantum mechanics:
(i) von Neumann algebras, (ii) Hilbert spaces, and (iii) orthomodular lattices.
Taking a more general categorical perspective of which the above are instances,
we consider dagger monoidal kernel categories for (ii), so that (i) become
(sub)endohomsets and (iii) become subobj...

A central theme in current work in quantum information and quantum
foundations is to see quantum mechanics as occupying one point in a space of
possible theories, and to use this perspective to understand the special
features and properties which single it out, and the possibilities for
alternative theories. Two formalisms which have been used in t...

Following Birkhoff and von Neumann, quantum logic has traditionally been
based on the lattice of closed linear subspaces of some Hilbert space, or, more
generally, on the lattice of projections in a von Neumann algebra A.
Unfortunately, the logical interpretation of these lattices is impaired by
their nondistributivity and by various other problems...

Sequences of commuting quantum operators can be parallelized using entanglement. This transformation is behind some optimal quantum metrology protocols and recent results on quantum circuit complexity. We show that dephasing quantum maps in arbitrary dimension can also be parallelized. This implies that for general dephasing noise the protocol with...

This dissertation studies the logic behind quantum physics, using category theory as the principal tool and conceptual guide. To do so, principles of quantum mechanics are modeled categorically. These categorical quantum models are justified by an embedding into the category of Hilbert spaces, the traditional formalism of quantum physics. In partic...

We functorially characterize groupoids as special dagger Frobenius algebras
in the category of sets and relations. This is then generalized to a non-unital
setting, by establishing an adjunction between H*-algebras in the category of
sets and relations, and locally cancellative regular semigroupoids. Finally, we
study a universal passage from the f...

Two fundamental contributions to categorical quantum mechanics are presented.
First, we generalize the CP-construction, that turns any dagger compact
category into one with completely positive maps, to arbitrary dimension.
Second, we axiomatize when a given category is the result of this construction.

We tentatively propose two guiding principles for the construction of theories of physics, which should be satisfied by a possible future theory of quantum gravity. These principles are inspired by those that led Einstein to his theory of general relativity, viz. his principle of general covariance and his equivalence principle, as well as by the t...

We compare two influential ways of defining a generalized notion of space. The first, inspired by Gelfand duality, states that the category of 'noncommutative spaces' is the opposite of the category of C*-algebras. The second, loosely generalizing Stone duality, maintains that the category of 'pointfree spaces' is the opposite of the category of fr...

Any functor from the category of C*-algebras to the category of locales which assigns to each commutative C*-algebra its Gelfand spectrum must be trivial on algebras of n-by-n matrices for n>=3. The same obstruction
applies to the Zariski, Stone, and Pierce spectra. The possibility of spectra in categories other than that of locales is briefly disc...

A certain class of Frobenius algebras has been used to characterize orthonormal bases and observables on finite-dimensional Hilbert spaces. The presence of units in these algebras means that they can only be realized finite-dimensionally. We seek a suitable generalization, which will allow arbitrary bases and observables to be described within cate...

This paper investigates quantum logic from the perspective of cate-
gorical logic, and starts from minimal assumptions, namely the existence
of involutions/daggers and kernels. The resulting structures turn out to
(1) encompass many examples of interest, such as categories of relations,
partial injections, Hilbert spaces (also modulo phase), and Bo...

The aim of this chapter is to construct new foundations for quantum logic and
quantum spaces. This is accomplished by merging algebraic quantum theory and
topos theory (encompassing the theory of locales or frames, of which toposes in a
sense form the ultimate generalization). In a nutshell, the relation between these
fields is as follows.
First, o...

The microcosm principle, advocated by Baez and Dolan and formalized for Lawvere theories lately by three of the authors, has been applied to coalgebras in order to describe compositional behavior sys- tematically. Here we further illustrate the usefulness of the approach by extending it to a many-sorted setting. Then we can show that the coalge- br...

Arrows are an extension of the well-established notion of a monad in functional programming languages. This article presents several examples and constructions, and develops
denotational semantics of arrows as monoids in categories of bifunctors C^op x C -> C. Observing similarities to monads -- which are monoids in categories of endofunctors C ->...

A decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Doering and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*-algebraic approach to quantum theory w...

We axiomatically define (pre-)Hilbert categories. The axioms resemble those for monoidal Abelian categories with the addition of an involutive functor. We then prove embedding theorems: any locally small pre-Hilbert category whose monoidal unit is a simple generator embeds (weakly) monoidally into the category of pre-Hilbert spaces and adjointable...

Compact categories have lately seen renewed interest via applications to
quantum physics. Being essentially finite-dimensional, they cannot accomodate
(co)limit-based constructions. For example, they cannot capture protocols such
as quantum key distribution, that rely on the law of large numbers. To overcome
this limitation, we introduce the notion...

A category with biproducts is enriched over (commutative) additive monoids. A category with tensor
products is enriched over scalar multiplication actions. A symmetric monoidal category with biproducts
is enriched over semimodules. We show that these extensions of enrichment (e.g. from hom-sets to homsemimodules) are functorial, and use them to mak...

Monads are by now well-established as programming construct in functional languages. Recently, the notion of “Arrow” was introduced by Hughes as an extension, not with one, but with two type parameters. At ﬁrst, these Arrows may look somewhat arbitrary. Here we show that they are categorically fairly civilised, by showing
that they correspond to mo...

We investigate whether polyhedral models suffice for accurate silhouette computation. Therefore, we set up a theoretical framework, a mathematical foundation, to compare various algorithms computing silhouettes on spatial and topological accuracy. Within this carefully constructed environment, we can argue on a formal basis why some silhouette-comp...

A central theme in current work in quantum information and quantum foundations is to see quantum mechanics as occupying one point in a space of possible theories, and to use this perspective to understand the special features and properties which single it out, and the possibilities for alternative theories. Two formalisms which have been used in t...