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Bob Coecke

University of Oxford, Department of Computer Science, Emeritus

Perimeter Institute for Theoretical Physics, Quantum Foundations, Distinguished Visiting Research Chair

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I am Chief Scientist at Quantinuum, head Quantinuum's Oxford-based Quantum NLP & Compositional Intelligence sub-team, Distinguished Visiting Research Chair at the Perimeter Institute for Theoretical Physics, Emeritus Fellow at Wolfson College Oxford, and Visiting Fellow at the Computer Science Department and the Mathematical Institute of Oxford University. Previously I was Professor of Quantum Foundations, Logics and Structures at the Department of Computer Science at Oxford University, where I was 20 years, and co-founded and led a multi-disciplinary Quantum Group that grew to 50 members and I supervised close to 70 PhD students. I am still supervising, at Oxford and elsewhere, and also still teach at Oxford's Mathematical Institute. I pioneered Categorical Quantum Mechanics (now in AMS's MSC2020 classification), ZX-calculus, DisCoCat natural language meaning, mathematical foundations for resource theories, Quantum Natural Language Processing, and DisCoCirc natural language meaning. I co-authored Picturing Quantum Processes, with Aleks Kissinger, a book providing a fully diagrammatic treatment of quantum theory and its applications. I co-authored Quantum in Pictures, with Stefano Gogioso, which does the same, but now accessible to people with no maths background. I co-authored some 200 research papers. I obtained approx. 35 grants, including from NFWO, EPSRC, Leverhulme, EU, ONR, AFOSR, FQXi, JTF. I still hold grants with the latter two. I am a founding father of the QPL (Quantum Physics and Logic) and ACT (Applied Category Theory) communities, of the diamond-open-access journal Compositionality, and Cambridge University Press' Applied Category Theory book series. I was the first person to have Quantum Foundations as part of his academic title. My work has been headlined by various media outlets, including Forbes, New Scientist, PhysicsWorld, ComputerWeekly.
Phone: +44 7881 333990
Address: Quantinuum (Oxford Campus), 17 Beaumont Street, OX1 2NA Oxford, UK.

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Papers by Bob Coecke

Mathematical Foundations for Distributed Compositional Model of Meaning. Lambek Festschrift

Tutorial: Graphical Calculus for Quantum Circuits

Reversible Computation, 2013

We explain the graphical zx-calculus for reasoning about qubits without any reference to the unde... more

Partiality in physics

Computing Research Repository - CORR, 2003

We revisit the standard axioms of domain theory with emphasis on their relation to the concept of... more We revisit the standard axioms of domain theory with emphasis on their relation to the concept of partiality, explain how this idea arises naturally in probability theory and quantum mechanics, and then search for a mathematical setting capable of providing a satisfactory unification of the two. finite string is partial, the infinite strings are total. In the domain (IR, ⊑), the collection of compact intervals (a, b) of the real line ordered by reverse inclusion, an interval like (p, q) with p < q rational is partial, while a one point interval (x) representing a real number is total. In the domain (n, ⊑), the n dimensional mixed states in the spectral order (to be defined later), a pure state is total, while mixed states which are not pure are partial. In all the cases above, total elements coincide with elements which are maximal in the given order. As we can see, the partiality idea arises naturally in both computer science and physics. The idea is important in computer scienc...

Domain theory and quantum mechanics

We introduce a partial order on classical and quantum states which reveals that these sets are ac... more We introduce a partial order on classical and quantum states which reveals that these sets are actually domains: Directed complete partially ordered sets with an intrinsic notion of approximation. The operational significance of the orders involved conclusively establishes that physical information has a natural domain theoretic structure. In the same way that the order on a domain provides a rigorous qualitative definition of information, a special type of mapping on a domain called a measurement provides a formal account of ...

Categorical quantum mechanics meets the Pavia principles: towards a representation theorem for CQM constructions (position paper)

QPL 2011, Oct 27, 2011

This position paper serves two purposes:(1) In the light of recent reconstructions of quantum the... more This position paper serves two purposes:(1) In the light of recent reconstructions of quantum theory, this paper shows the need for a representation theorem for categorical quantum mechanics (CQM), which establishes how conceptually-meaningful constructions on processes and their compositions lead to Hilbert space structure, without any reference of instrumentalist concepts such as measurement.(2) As a first step towards this goal we show how several of the instrumentalist principles underpinning Pavia's reconstruction of finitary ...

Toy quantum categories

Electronic notes in theoretical computer science, Feb 10, 2011

We show that Rob Spekken's toy quantum theory arises as an instance of our categorical appro... more We show that Rob Spekken's toy quantum theory arises as an instance of our categorical approach to quantum axiomatics, as a (proper) subcategory of the dagger compact category FRel of finite sets and relations with the cartesian product as tensor, where observables correspond to dagger Frobenius algebras. This in particular implies that the quantum-like properties of the toy model are in fact very general categorytheoretic properties. We also show the remarkable fact that we can already interpret complementary quantum ...

The Compositional Structure of Multipartite Quantum Entanglement

Automata, Languages and Programming, 2010

While multipartite quantum states constitute a (if not the) key resource for quantum computations... more While multipartite quantum states constitute a (if not the) key resource for quantum computations and protocols, obtaining a high-level, structural understanding of entanglement involving arbitrarily many qubits is a longstanding open problem in quantum computer science. In this paper we expose the algebraic and graphical structure of the GHZ-state and the W-state, as well as a purely graphical distinction that characterises the behaviours of these states. In turn, this structure yields a compositional graphical model for expressing general multipartite states. We identify those states, named Frobenius states, which canonically induce an algebraic structure, namely the structure of a commutative Frobenius algebra (CFA). We show that all SLOCC-maximal tripartite qubit states are locally equivalent to Frobenius states. Those that are SLOCC-equivalent to the GHZ-state induce special commutative Frobenius algebras, while those that are SLOCC-equivalent to the W-state induce what we call anti-special commutative Frobenius algebras. From the SLOCC-classification of tripartite qubit states follows a representation theorem for two dimensional CFAs. Together, a GHZ and a W Frobenius state form the primitives of a graphical calculus. This calculus is expressive enough to generate and reason about arbitrary multipartite states, which are obtained by "composing" the GHZ-and W-states, giving rise to a rich graphical paradigm for general multipartite entanglement.

Early Greek Thought and Perspectives for the Interpretation of Quantum Mechanics: Preliminaries to an Ontological Approach

Metadebates on Science, 1999

It will be shown in this article that an ontological approach for some problems related to the in... more It will be shown in this article that an ontological approach for some problems related to the interpretation of Quantum Mechanics (QM) could emerge from a re-evaluation of the main paradox of early Greek thought: the paradox of Being and non-Being, and the solutions presented to it by Plato and Aristotle. More well known are the derivative paradoxes of Zeno: the paradox of motion and the paradox of the One and the Many. They stem from what was perceived by classical philosophy to be the fundamental enigma for thinking about the world: the seemingly contradictory results that followed from the coincidence of being and non-being in the world of change and motion as we experience it, and the experience of absolute existence here and now. The most clear expression of both stances can be found, again following classical thought, in the thinking of Heraclitus of Ephesus and Parmenides of Elea. The problem put forward by these paradoxes reduces for both Plato and Aristotle to the possibility of the existence of stable objects as a necessary condition for knowledge. Hence the primarily ontological nature of the solutions they proposed: Plato's Theory of Forms and Aristotle's metaphysics and logic. Plato's and Aristotle's systems are argued here to do on the ontological level essentially the same: to introduce stability in the world by introducing the notion of a separable, stable object, for which a 'principle of contradiction' is valid: an object cannot be and not-be at the same place at the same time. So it becomes possible to forbid contradiction on an epistemological level, and thus to guarantee the certainty of knowledge that seemed to be threatened before. After leaving Aristotelian metaphysics, early modern science had to cope with these problems: it did so by introducing "space" as the seat of stability, and "time" as the theater of motion. But the ontological structure present in this solution remained the same. Therefore the fundamental notion 'separable system', related to the notions observation and measurement, themselves related to the modern concepts of space and time, appears to be intrinsically problematic, because it is inextricably connected to classical logic on the ontological level. We see therefore the problems dealt with by quantum logic not as merely formal, and the problem of 'non-locality' as related to it, indicating the need to rethink the notions system, entity, as well as the implications of the operation 'measurement', which is seen here as an application of classical logic (including its ontological consequences) on the material world. 54 Here, 'induction' does not refer to any of its usual philosophical significances, but rather refers to physical theories like electro-magnetism.

Spekkens’s toy theory as a category of processes

Proceedings of Symposia in Applied Mathematics, 2012

We provide two mathematical descriptions of Spekkens's toy qubit theory, an inductively one in te... more We provide two mathematical descriptions of Spekkens's toy qubit theory, an inductively one in terms of a small set of generators, as well as an explicit closed form description. It is a subcategory MSpek of the category of finite sets, relations and the cartesian product. States of maximal knowledge form a subcategory Spek. This establishes the consistency of the toy theory, which has previously only been constructed for at most four systems. Our model also shows that the theory is closed under both parallel and sequential composition of operations (= symmetric monoidal structure), that it obeys map-state duality (= compact closure), and that states and effects are in bijective correspondence (= dagger structure). From the perspective of categorical quantum mechanics, this provides an interesting alternative model which enables us to describe many quantum phenomena in a discrete manner, and to which mathematical concepts such as basis structures, and complementarity thereof, still apply. Hence, the framework of categorical quantum mechanics has delivered on its promise to encompass theories other than quantum theory.

Three qubit entanglement within graphical Z/X-calculus

Electronic Proceedings in Theoretical Computer Science, 2011

The compositional techniques of categorical quantum mechanics are applied to analyse 3-qubit quan... more The compositional techniques of categorical quantum mechanics are applied to analyse 3-qubit quantum entanglement. In particular the graphical calculus of complementary observables and corresponding phases due to Duncan and one of the authors is used to construct representative members of the two genuinely tripartite SLOCC classes of 3-qubit entangled states, GHZ and W. This nicely illustrates the respectively pairwise and global tripartite entanglement found in the Wand GHZ-class states. A new concept of supplementarity allows us to characterise inhabitants of the W class within the abstract diagrammatic calculus; these method extends to more general multipartite qubit states.

Time Asymmetry of Probabilities Versus Relativistic Causal Structure: An Arrow of Time

Physical Review Letters, 2012

There is an incompatibility between the symmetries of causal structure in relativity theory and t... more There is an incompatibility between the symmetries of causal structure in relativity theory and the signaling abilities of probabilistic devices with inputs and outputs: while time-reversal in relativity will not introduce the ability to signal between spacelike separated regions, this is not the case for probabilistic devices with space-like separated input-output pairs. We explicitly describe a nonsignaling device which becomes a perfect signaling device under time-reversal, where time-reversal can be conceptualized as playing backwards a videotape of an agent manipulating the device. This leads to an arrow of time that is identifiable when studying the correlations of events for spacelike separated regions. Somewhat surprisingly, although time-reversal of Popuscu-Rörlich boxes also allows agents to signal, it does not yield a perfect signaling device. Finally, we realize time-reversal using post-selection, which could lead experimental implementation.

Interacting quantum observables: categorical algebra and diagrammatics

New Journal of Physics, 2011

This paper has two tightly intertwined aims: (i) To introduce an intuitive and universal graphica... more This paper has two tightly intertwined aims: (i) To introduce an intuitive and universal graphical calculus for multi-qubit systems, the zx-calculus, which greatly simplifies derivations in the area of quantum computation and information. (ii) To axiomatise complementarity of quantum observables within a general framework for physical theories in terms of dagger symmetric monoidal categories. We also axiomatize phase shifts within this framework. Using the well-studied canonical correspondence between graphical calculi and dagger symmetric monoidal categories, our results provide a purely graphical formalisation of complementarity for quantum observables. Each individual observable, represented by a commutative special dagger Frobenius algebra, gives rise to an abelian group of phase shifts, which we call the phase group. We also identify a strong form of complementarity, satisfied by the Z and X spin observables, which yields a scaled variant of a bialgebra.

Causal Categories: Relativistically Interacting Processes

Foundations of Physics, 2012

A symmetric monoidal category naturally arises as the mathematical structure that organizes physi... more A symmetric monoidal category naturally arises as the mathematical structure that organizes physical systems, processes, and composition thereof, both sequentially and in parallel. This structure admits a purely graphical calculus. This paper is concerned with the encoding of a fixed causal structure within a symmetric monoidal category: causal dependencies will correspond to topological connectedness in the graphical language. We show that correlations, either classical or quantum, force terminality of the tensor unit. We also show that well-definedness of the concept of a global state forces the monoidal product to be only partially defined, which in turn results in a relativistic covariance theorem. Except for these assumptions, at no stage do we assume anything more than purely compositional symmetric-monoidal categorical structure. We cast these two structural results in terms of a mathematical entity, which we call a causal category. We provide methods of constructing causal categories, and we study the consequences of these methods for the general framework of categorical quantum mechanics.

Graphical Calculus for Quantum Key Distribution (Extended Abstract)

Electronic Notes in Theoretical Computer Science, 2011

Controlled complementary measurements are key to quantum key distribution protocols, among many o... more Controlled complementary measurements are key to quantum key distribution protocols, among many other things. We axiomatize controlled complementary measurements within symmetric monoidal categories, which provides them with a corresponding graphical calculus. We study the BB84 and Ekert91 protocols within this calculus, including the case where there is an intercept-resend attack.

Towards a Compositional Distributional Model of Meaning

comlab.ox.ac.uk

We propose a mathematical framework for a unifica-tion of the distributional theory of meaning in... more

Algebraic and Topological Methods in Non-Classical Logics III (TANCL'07)

atlas-conferences.com, 2007

Atlas home || Conferences | Abstracts | about Atlas ALGEBRAIC AND TOPOLOGICAL METHODS IN NON-CLAS... more Atlas home || Conferences | Abstracts | about Atlas ALGEBRAIC AND TOPOLOGICAL METHODS IN NON-CLASSICAL LOGICS III (TANCL'07) August 5-9, 2007 St Anne's College, University of Oxford Oxford, England. Organizers Mai Gehrke and Hilary Priestley. Conference Homepage. Abstracts. SAMSON ABRAMSKY Domain Theory in Logical Form Revisited: a 20-year Retrospective Stefano Aguzzoli Goedel algebras free over finite distributive ...

Diagrammatic Reasoning about Meaning of Sentences

The symbolic [5] and distributional [8] theories of meaning are somewhat orthogonal with competin... more The symbolic [5] and distributional [8] theories of meaning are somewhat orthogonal with competing pros and cons: the former is compositional but only qualitative, the latter is non-compositional but quantitative. Following [9] in the context of Cognitive Science, where a similar problem exists between the connectionist and symbolic models of mind,[3] argued for the use of the tensor product of vector spaces and pairing the vectors of meaning with their syntactic roles. This abstract summarizes the framework developed in [2], which builds on ...

Partiality in physics

arXiv preprint quant-ph/0312044, Dec 4, 2003

Abstract: We revisit the standard axioms of domain theory with emphasis on their relation to the ... more Abstract: We revisit the standard axioms of domain theory with emphasis on their relation to the concept of partiality, explain how this idea arises naturally in probability theory and quantum mechanics, and then search for a mathematical setting capable of providing a satisfactory unification of the two.

Reasoning about meaning in natural language with compact closed categories and Frobenius algebras

Logic and Algebraic Structures in Quantum Computing

Compact closed categories have found applications in modeling quantum information protocols by Ab... more Compact closed categories have found applications in modeling quantum information protocols by Abramsky-Coecke. They also provide semantics for Lambek's pregroup algebras, applied to formalizing the grammatical structure of natural language, and are implicit in a distributional model of word meaning based on vector spaces. Specifically, in previous work Coecke-Clark-Sadrzadeh used the product category of pregroups with vector spaces and provided a distributional model of meaning for sentences. We recast this theory in terms of strongly monoidal functors and advance it via Frobenius algebras over vector spaces. The former are used to formalize topological quantum field theories by Atiyah and Baez-Dolan, and the latter are used to model classical data in quantum protocols by Coecke-Pavlovic-Vicary. The Frobenius algebras enable us to work in a single space in which meanings of words, phrases, and sentences of any structure live. Hence we can compare meanings of different language constructs and enhance the applicability of the theory. We report on experimental results on a number of language tasks and verify the theoretical predictions.

Toy Quantum Categories (Extended Abstract)

Electronic Notes in Theoretical Computer Science, 2011

We show that Rob Spekken's toy quantum theory arises as an instance of our categorical approach t... more We show that Rob Spekken's toy quantum theory arises as an instance of our categorical approach to quantum axiomatics, as a (proper) subcategory of the dagger compact category FRel of finite sets and relations with the cartesian product as tensor, where observables correspond to dagger Frobenius algebras. This in particular implies that the quantum-like properties of the toy model are in fact very general categorytheoretic properties. We also show the remarkable fact that we can already interpret complementary quantum observables on the two-element set in FRel.

Mathematical Foundations for Distributed Compositional Model of Meaning. Lambek Festschrift

Tutorial: Graphical Calculus for Quantum Circuits

Reversible Computation, 2013

We explain the graphical zx-calculus for reasoning about qubits without any reference to the unde... more

Partiality in physics

Computing Research Repository - CORR, 2003

We revisit the standard axioms of domain theory with emphasis on their relation to the concept of... more We revisit the standard axioms of domain theory with emphasis on their relation to the concept of partiality, explain how this idea arises naturally in probability theory and quantum mechanics, and then search for a mathematical setting capable of providing a satisfactory unification of the two. finite string is partial, the infinite strings are total. In the domain (IR, ⊑), the collection of compact intervals (a, b) of the real line ordered by reverse inclusion, an interval like (p, q) with p < q rational is partial, while a one point interval (x) representing a real number is total. In the domain (n, ⊑), the n dimensional mixed states in the spectral order (to be defined later), a pure state is total, while mixed states which are not pure are partial. In all the cases above, total elements coincide with elements which are maximal in the given order. As we can see, the partiality idea arises naturally in both computer science and physics. The idea is important in computer scienc...

Domain theory and quantum mechanics

We introduce a partial order on classical and quantum states which reveals that these sets are ac... more We introduce a partial order on classical and quantum states which reveals that these sets are actually domains: Directed complete partially ordered sets with an intrinsic notion of approximation. The operational significance of the orders involved conclusively establishes that physical information has a natural domain theoretic structure. In the same way that the order on a domain provides a rigorous qualitative definition of information, a special type of mapping on a domain called a measurement provides a formal account of ...

Categorical quantum mechanics meets the Pavia principles: towards a representation theorem for CQM constructions (position paper)

QPL 2011, Oct 27, 2011

This position paper serves two purposes:(1) In the light of recent reconstructions of quantum the... more This position paper serves two purposes:(1) In the light of recent reconstructions of quantum theory, this paper shows the need for a representation theorem for categorical quantum mechanics (CQM), which establishes how conceptually-meaningful constructions on processes and their compositions lead to Hilbert space structure, without any reference of instrumentalist concepts such as measurement.(2) As a first step towards this goal we show how several of the instrumentalist principles underpinning Pavia's reconstruction of finitary ...

Toy quantum categories

Electronic notes in theoretical computer science, Feb 10, 2011

We show that Rob Spekken's toy quantum theory arises as an instance of our categorical appro... more We show that Rob Spekken's toy quantum theory arises as an instance of our categorical approach to quantum axiomatics, as a (proper) subcategory of the dagger compact category FRel of finite sets and relations with the cartesian product as tensor, where observables correspond to dagger Frobenius algebras. This in particular implies that the quantum-like properties of the toy model are in fact very general categorytheoretic properties. We also show the remarkable fact that we can already interpret complementary quantum ...

The Compositional Structure of Multipartite Quantum Entanglement

Automata, Languages and Programming, 2010

While multipartite quantum states constitute a (if not the) key resource for quantum computations... more While multipartite quantum states constitute a (if not the) key resource for quantum computations and protocols, obtaining a high-level, structural understanding of entanglement involving arbitrarily many qubits is a longstanding open problem in quantum computer science. In this paper we expose the algebraic and graphical structure of the GHZ-state and the W-state, as well as a purely graphical distinction that characterises the behaviours of these states. In turn, this structure yields a compositional graphical model for expressing general multipartite states. We identify those states, named Frobenius states, which canonically induce an algebraic structure, namely the structure of a commutative Frobenius algebra (CFA). We show that all SLOCC-maximal tripartite qubit states are locally equivalent to Frobenius states. Those that are SLOCC-equivalent to the GHZ-state induce special commutative Frobenius algebras, while those that are SLOCC-equivalent to the W-state induce what we call anti-special commutative Frobenius algebras. From the SLOCC-classification of tripartite qubit states follows a representation theorem for two dimensional CFAs. Together, a GHZ and a W Frobenius state form the primitives of a graphical calculus. This calculus is expressive enough to generate and reason about arbitrary multipartite states, which are obtained by "composing" the GHZ-and W-states, giving rise to a rich graphical paradigm for general multipartite entanglement.

Early Greek Thought and Perspectives for the Interpretation of Quantum Mechanics: Preliminaries to an Ontological Approach

Metadebates on Science, 1999

It will be shown in this article that an ontological approach for some problems related to the in... more It will be shown in this article that an ontological approach for some problems related to the interpretation of Quantum Mechanics (QM) could emerge from a re-evaluation of the main paradox of early Greek thought: the paradox of Being and non-Being, and the solutions presented to it by Plato and Aristotle. More well known are the derivative paradoxes of Zeno: the paradox of motion and the paradox of the One and the Many. They stem from what was perceived by classical philosophy to be the fundamental enigma for thinking about the world: the seemingly contradictory results that followed from the coincidence of being and non-being in the world of change and motion as we experience it, and the experience of absolute existence here and now. The most clear expression of both stances can be found, again following classical thought, in the thinking of Heraclitus of Ephesus and Parmenides of Elea. The problem put forward by these paradoxes reduces for both Plato and Aristotle to the possibility of the existence of stable objects as a necessary condition for knowledge. Hence the primarily ontological nature of the solutions they proposed: Plato's Theory of Forms and Aristotle's metaphysics and logic. Plato's and Aristotle's systems are argued here to do on the ontological level essentially the same: to introduce stability in the world by introducing the notion of a separable, stable object, for which a 'principle of contradiction' is valid: an object cannot be and not-be at the same place at the same time. So it becomes possible to forbid contradiction on an epistemological level, and thus to guarantee the certainty of knowledge that seemed to be threatened before. After leaving Aristotelian metaphysics, early modern science had to cope with these problems: it did so by introducing "space" as the seat of stability, and "time" as the theater of motion. But the ontological structure present in this solution remained the same. Therefore the fundamental notion 'separable system', related to the notions observation and measurement, themselves related to the modern concepts of space and time, appears to be intrinsically problematic, because it is inextricably connected to classical logic on the ontological level. We see therefore the problems dealt with by quantum logic not as merely formal, and the problem of 'non-locality' as related to it, indicating the need to rethink the notions system, entity, as well as the implications of the operation 'measurement', which is seen here as an application of classical logic (including its ontological consequences) on the material world. 54 Here, 'induction' does not refer to any of its usual philosophical significances, but rather refers to physical theories like electro-magnetism.

Spekkens’s toy theory as a category of processes

Proceedings of Symposia in Applied Mathematics, 2012

We provide two mathematical descriptions of Spekkens's toy qubit theory, an inductively one in te... more We provide two mathematical descriptions of Spekkens's toy qubit theory, an inductively one in terms of a small set of generators, as well as an explicit closed form description. It is a subcategory MSpek of the category of finite sets, relations and the cartesian product. States of maximal knowledge form a subcategory Spek. This establishes the consistency of the toy theory, which has previously only been constructed for at most four systems. Our model also shows that the theory is closed under both parallel and sequential composition of operations (= symmetric monoidal structure), that it obeys map-state duality (= compact closure), and that states and effects are in bijective correspondence (= dagger structure). From the perspective of categorical quantum mechanics, this provides an interesting alternative model which enables us to describe many quantum phenomena in a discrete manner, and to which mathematical concepts such as basis structures, and complementarity thereof, still apply. Hence, the framework of categorical quantum mechanics has delivered on its promise to encompass theories other than quantum theory.

Three qubit entanglement within graphical Z/X-calculus

Electronic Proceedings in Theoretical Computer Science, 2011

The compositional techniques of categorical quantum mechanics are applied to analyse 3-qubit quan... more The compositional techniques of categorical quantum mechanics are applied to analyse 3-qubit quantum entanglement. In particular the graphical calculus of complementary observables and corresponding phases due to Duncan and one of the authors is used to construct representative members of the two genuinely tripartite SLOCC classes of 3-qubit entangled states, GHZ and W. This nicely illustrates the respectively pairwise and global tripartite entanglement found in the Wand GHZ-class states. A new concept of supplementarity allows us to characterise inhabitants of the W class within the abstract diagrammatic calculus; these method extends to more general multipartite qubit states.

Time Asymmetry of Probabilities Versus Relativistic Causal Structure: An Arrow of Time

Physical Review Letters, 2012

There is an incompatibility between the symmetries of causal structure in relativity theory and t... more There is an incompatibility between the symmetries of causal structure in relativity theory and the signaling abilities of probabilistic devices with inputs and outputs: while time-reversal in relativity will not introduce the ability to signal between spacelike separated regions, this is not the case for probabilistic devices with space-like separated input-output pairs. We explicitly describe a nonsignaling device which becomes a perfect signaling device under time-reversal, where time-reversal can be conceptualized as playing backwards a videotape of an agent manipulating the device. This leads to an arrow of time that is identifiable when studying the correlations of events for spacelike separated regions. Somewhat surprisingly, although time-reversal of Popuscu-Rörlich boxes also allows agents to signal, it does not yield a perfect signaling device. Finally, we realize time-reversal using post-selection, which could lead experimental implementation.

Interacting quantum observables: categorical algebra and diagrammatics

New Journal of Physics, 2011

This paper has two tightly intertwined aims: (i) To introduce an intuitive and universal graphica... more This paper has two tightly intertwined aims: (i) To introduce an intuitive and universal graphical calculus for multi-qubit systems, the zx-calculus, which greatly simplifies derivations in the area of quantum computation and information. (ii) To axiomatise complementarity of quantum observables within a general framework for physical theories in terms of dagger symmetric monoidal categories. We also axiomatize phase shifts within this framework. Using the well-studied canonical correspondence between graphical calculi and dagger symmetric monoidal categories, our results provide a purely graphical formalisation of complementarity for quantum observables. Each individual observable, represented by a commutative special dagger Frobenius algebra, gives rise to an abelian group of phase shifts, which we call the phase group. We also identify a strong form of complementarity, satisfied by the Z and X spin observables, which yields a scaled variant of a bialgebra.

Causal Categories: Relativistically Interacting Processes

Foundations of Physics, 2012

A symmetric monoidal category naturally arises as the mathematical structure that organizes physi... more A symmetric monoidal category naturally arises as the mathematical structure that organizes physical systems, processes, and composition thereof, both sequentially and in parallel. This structure admits a purely graphical calculus. This paper is concerned with the encoding of a fixed causal structure within a symmetric monoidal category: causal dependencies will correspond to topological connectedness in the graphical language. We show that correlations, either classical or quantum, force terminality of the tensor unit. We also show that well-definedness of the concept of a global state forces the monoidal product to be only partially defined, which in turn results in a relativistic covariance theorem. Except for these assumptions, at no stage do we assume anything more than purely compositional symmetric-monoidal categorical structure. We cast these two structural results in terms of a mathematical entity, which we call a causal category. We provide methods of constructing causal categories, and we study the consequences of these methods for the general framework of categorical quantum mechanics.

Graphical Calculus for Quantum Key Distribution (Extended Abstract)

Electronic Notes in Theoretical Computer Science, 2011

Controlled complementary measurements are key to quantum key distribution protocols, among many o... more Controlled complementary measurements are key to quantum key distribution protocols, among many other things. We axiomatize controlled complementary measurements within symmetric monoidal categories, which provides them with a corresponding graphical calculus. We study the BB84 and Ekert91 protocols within this calculus, including the case where there is an intercept-resend attack.

Towards a Compositional Distributional Model of Meaning

comlab.ox.ac.uk

We propose a mathematical framework for a unifica-tion of the distributional theory of meaning in... more

Algebraic and Topological Methods in Non-Classical Logics III (TANCL'07)

atlas-conferences.com, 2007

Atlas home || Conferences | Abstracts | about Atlas ALGEBRAIC AND TOPOLOGICAL METHODS IN NON-CLAS... more Atlas home || Conferences | Abstracts | about Atlas ALGEBRAIC AND TOPOLOGICAL METHODS IN NON-CLASSICAL LOGICS III (TANCL'07) August 5-9, 2007 St Anne's College, University of Oxford Oxford, England. Organizers Mai Gehrke and Hilary Priestley. Conference Homepage. Abstracts. SAMSON ABRAMSKY Domain Theory in Logical Form Revisited: a 20-year Retrospective Stefano Aguzzoli Goedel algebras free over finite distributive ...

Diagrammatic Reasoning about Meaning of Sentences

The symbolic [5] and distributional [8] theories of meaning are somewhat orthogonal with competin... more The symbolic [5] and distributional [8] theories of meaning are somewhat orthogonal with competing pros and cons: the former is compositional but only qualitative, the latter is non-compositional but quantitative. Following [9] in the context of Cognitive Science, where a similar problem exists between the connectionist and symbolic models of mind,[3] argued for the use of the tensor product of vector spaces and pairing the vectors of meaning with their syntactic roles. This abstract summarizes the framework developed in [2], which builds on ...

Partiality in physics

arXiv preprint quant-ph/0312044, Dec 4, 2003

Abstract: We revisit the standard axioms of domain theory with emphasis on their relation to the ... more Abstract: We revisit the standard axioms of domain theory with emphasis on their relation to the concept of partiality, explain how this idea arises naturally in probability theory and quantum mechanics, and then search for a mathematical setting capable of providing a satisfactory unification of the two.

Reasoning about meaning in natural language with compact closed categories and Frobenius algebras

Logic and Algebraic Structures in Quantum Computing

Compact closed categories have found applications in modeling quantum information protocols by Ab... more Compact closed categories have found applications in modeling quantum information protocols by Abramsky-Coecke. They also provide semantics for Lambek's pregroup algebras, applied to formalizing the grammatical structure of natural language, and are implicit in a distributional model of word meaning based on vector spaces. Specifically, in previous work Coecke-Clark-Sadrzadeh used the product category of pregroups with vector spaces and provided a distributional model of meaning for sentences. We recast this theory in terms of strongly monoidal functors and advance it via Frobenius algebras over vector spaces. The former are used to formalize topological quantum field theories by Atiyah and Baez-Dolan, and the latter are used to model classical data in quantum protocols by Coecke-Pavlovic-Vicary. The Frobenius algebras enable us to work in a single space in which meanings of words, phrases, and sentences of any structure live. Hence we can compare meanings of different language constructs and enhance the applicability of the theory. We report on experimental results on a number of language tasks and verify the theoretical predictions.

Toy Quantum Categories (Extended Abstract)

Electronic Notes in Theoretical Computer Science, 2011

We show that Rob Spekken's toy quantum theory arises as an instance of our categorical approach t... more We show that Rob Spekken's toy quantum theory arises as an instance of our categorical approach to quantum axiomatics, as a (proper) subcategory of the dagger compact category FRel of finite sets and relations with the cartesian product as tensor, where observables correspond to dagger Frobenius algebras. This in particular implies that the quantum-like properties of the toy model are in fact very general categorytheoretic properties. We also show the remarkable fact that we can already interpret complementary quantum observables on the two-element set in FRel.