arXiv subjects

Bob Coecke

Publications and source records attributed to Bob Coecke.

At least 73 records · Page 4Linked to original sources

arXiv2018

Double Dilation $\neq$ Double Mixing (extended abstract)

Density operators are one of the key ingredients of quantum theory. They can be constructed in two ways: via a convex sum of 'doubled kets' (i.e. mixing), and by tracing out part of a 'doubled' two-system ket (i.e. dilation). Both constructions can be iterated, yielding new mathematical species that have already found applications outside physics. However, as we show in this paper, the iterated constructions no longer yield the same mathematical species. Hence, the constructions 'mixing' and 'dilation' themselves are by no means equivalent. Concretely, when applying the Choi-Jamiolkowski isomorphism to the second iteration, dilation produces arbitrary symmetric bipartite states, while mixing only yields the disentangled ones. All results are proven using diagrams, and hence they hold not only for quantum theory, but also for a much more general class of process theories.

quant-ph↗

arXiv2018

Proceedings 14th International Conference on Quantum Physics and Logic

This volume contains the proceedings of the 14th International Conference on Quantum Physics and Logic (QPL 2017), which was held July 3-7, 2017 at the LUX Cinema Nijmegen, the Netherlands, and was hosted by Radboud University. QPL is a conference that brings together researchers working on mathematical foundations of quantum physics, quantum computing, and related areas, with a focus on structural perspectives and the use of logical tools, ordered algebraic and category-theoretic structures, formal languages, semantical methods, and other computer science techniques applied to the study of physical behaviour in general. This conference also welcomes work that applies structures and methods inspired by quantum theory to other fields (including computer science).

cs.LO↗

arXiv2017

The time-reverse of any causal theory is eternal noise

We consider a very general class of theories, process theories, which capture the underlying structure common to most theories of physics as we understand them today (be they established, toy or speculative theories). Amongst these theories, we will be focusing on those which are `causal', in the sense that they are intrinsically compatible with the causal structure of space-time -- as required by relativity. We demonstrate that there is a sharp contrast between these theories and the corresponding time-reversed theories, where time is taken to flow backwards from the future to the past. While the former typically feature a rich gamut of allowed states, the latter only allow for a single state: eternal noise. We illustrate this result by considering of the time-reverse of quantum theory. We also derive a strengthening of the result in PRL 108, 200403 on signalling in time-reversed theories.

quant-ph↗

arXiv2017

Interacting Conceptual Spaces I : Grammatical Composition of Concepts

The categorical compositional approach to meaning has been successfully applied in natural language processing, outperforming other models in mainstream empirical language processing tasks. We show how this approach can be generalized to conceptual space models of cognition. In order to do this, first we introduce the category of convex relations as a new setting for categorical compositional semantics, emphasizing the convex structure important to conceptual space applications. We then show how to construct conceptual spaces for various types such as nouns, adjectives and verbs. Finally we show by means of examples how concepts can be systematically combined to establish the meanings of composite phrases from the meanings of their constituent parts. This provides the mathematical underpinnings of a new compositional approach to cognition.

cs.LO↗

arXiv2017

Equivalence of relativistic causal structure and process terminality

In general relativity, `causal structure' refers to the partial order on space-time points (or regions) that encodes time-like relationships. Recently, quantum information and quantum foundations saw the emergence of a `causality principle'. In the form used in this paper, which we call `process terminality', it states that when the output of a process is discarded, then the process itself may as well be discarded. While causal structure and process terminality at first seem to be entirely different notions, they become equivalent when making explicit what the partial order actually encodes, that is, that an event in the past can influence one in its future, but not vice-versa. The framework in which we establish this equivalence is that of process theories. We show how several previous results are instances of this result and comment on how this framework could provide an ideal, minimal canvas for crafting theories of quantum gravity.

quant-ph↗

arXiv2017

Leaks: quantum, classical, intermediate, and more

We introduce the notion of a leak for general process theories, and identify quantum theory as a theory with minimal leakage, while classical theory has maximal leakage. We provide a construction that adjoins leaks to theories, an instance of which describes the emergence of classical theory by adjoining decoherence-leaks to quantum theory. Finally, we show that defining a notion of purity for processes in general process theories has to make reference to the leaks of that theory ---a feature missing in standard definitions--- hence, we propose a refined definition and study the resulting notion of purity for quantum, classical and intermediate theories.

quant-ph↗

arXiv2017

Double Dilation $\neq$ Double Mixing

Density operators are one of the key ingredients of quantum theory. They can be constructed in two ways: via a convex sum of `doubled kets' (i.e. mixing), and by tracing out part of a `doubled' two-system ket (i.e. dilation). Both constructions can be iterated, yielding new mathematical species that have already found applications outside physics. However, as we show in this paper, the iterated constructions no longer yield the same mathematical species. Hence, the constructions `mixing' and `dilation' themselves are by no means equivalent. Concretely, when applying the Choi-Jamiolkowski isomorphism to the second iteration, dilation produces arbitrary symmetric bipartite states, while mixing only yields the disentangled ones. All results are proven using diagrams, and hence they hold not only for quantum theory, but also for a much more general class of process theories.

quant-ph↗

arXiv2016

Compositional Distributional Cognition

We accommodate the Integrated Connectionist/Symbolic Architecture (ICS) of [32] within the categorical compositional semantics (CatCo) of [13], forming a model of categorical compositional cognition (CatCog). This resolves intrinsic problems with ICS such as the fact that representations inhabit an unbounded space and that sentences with differing tree structures cannot be directly compared. We do so in a way that makes the most of the grammatical structure available, in contrast to strategies like circular convolution. Using the CatCo model also allows us to make use of tools developed for CatCo such as the representation of ambiguity and logical reasoning via density matrices, structural meanings for words such as relative pronouns, and addressing over- and under-extension, all of which are present in cognitive processes. Moreover the CatCog framework is sufficiently flexible to allow for entirely different representations of meaning, such as conceptual spaces. Interestingly, since the CatCo model was largely inspired by categorical quantum mechanics, so is CatCog.

cs.AI↗

arXiv2016

Dual Density Operators and Natural Language Meaning

Density operators allow for representing ambiguity about a vector representation, both in quantum theory and in distributional natural language meaning. Formally equivalently, they allow for discarding part of the description of a composite system, where we consider the discarded part to be the context. We introduce dual density operators, which allow for two independent notions of context. We demonstrate the use of dual density operators within a grammatical-compositional distributional framework for natural language meaning. We show that dual density operators can be used to simultaneously represent: (i) ambiguity about word meanings (e.g. queen as a person vs. queen as a band), and (ii) lexical entailment (e.g. tiger -> mammal). We provide a proof-of-concept example.

cs.CL↗

arXiv2016

Interacting Conceptual Spaces

We propose applying the categorical compositional scheme of [6] to conceptual space models of cognition. In order to do this we introduce the category of convex relations as a new setting for categorical compositional semantics, emphasizing the convex structure important to conceptual space applications. We show how conceptual spaces for composite types such as adjectives and verbs can be constructed. We illustrate this new model on detailed examples.

cs.AI↗

arXiv2016

Quantum Algorithms for Compositional Natural Language Processing

We propose a new application of quantum computing to the field of natural language processing. Ongoing work in this field attempts to incorporate grammatical structure into algorithms that compute meaning. In (Coecke, Sadrzadeh and Clark, 2010), the authors introduce such a model (the CSC model) based on tensor product composition. While this algorithm has many advantages, its implementation is hampered by the large classical computational resources that it requires. In this work we show how computational shortcomings of the CSC approach could be resolved using quantum computation (possibly in addition to existing techniques for dimension reduction). We address the value of quantum RAM (Giovannetti,2008) for this model and extend an algorithm from Wiebe, Braun and Lloyd (2012) into a quantum algorithm to categorize sentences in CSC. Our new algorithm demonstrates a quadratic speedup over classical methods under certain conditions.

cs.CL↗

arXiv2016

Process-theoretic characterisation of the Hermitian adjoint

We show that the physical principle "the adjoint associates to each state a `test' for that state" fully characterises the Hermitian adjoint for pure quantum theory, therefore providing the adjoint with operational meaning beyond its standard mathematical definition. Also, we show that for general process theories, which all admit a diagrammatic representation, this physical principle induces a reflection operation.

quant-ph↗

arXiv2016

Categorical Quantum Mechanics I: Causal Quantum Processes

We derive the category-theoretic backbone of quantum theory from a process ontology. More specifically, we treat quantum theory as a theory of systems, processes and their interactions. In this first part of a three-part overview, we first present a general theory of diagrams, and in particular, of string diagrams. We discuss why diagrams are a very natural starting point for developing scientific theories. Then we define process theories, and define a very general notion of quantum type. We show how our process ontology enables us to assert causality, that is, compatibility of quantum theory and relativity theory, and we prove the no-signalling theorem. Other notable contributions include new, elegant derivations of the no-broadcasting theorem, unitarity of evolution, and Stinespring dilation, all for any 'quantum' type in a general class of process theories.

quant-ph↗

arXiv2016

Categorical Quantum Mechanics II: Classical-Quantum Interaction

This is the second part of a three-part overview, in which we derive the category-theoretic backbone of quantum theory from a process ontology, treating quantum theory as a theory of systems, processes and their interactions. In this part we focus on classical-quantum interaction. Classical and quantum systems are treated as distinct types, of which the respective behavioural properties are specified in terms of processes and their compositions. In particular, classicality is witnessed by 'spiders' which fuse together whenever they connect. We define mixedness and show that pure processes are extremal in the space of all processes, and we define entanglement and show that quantum theory indeed exhibits entanglement. We discuss the classification of tripartite qubit entanglement and show that both the GHZ-state and the W-state come from spider-like families of processes, which differ only in how they behave when they are connected by two or more wires. We define measurements and provide fully-comprehensive descriptions of several quantum protocols involving classical data flow. Finally, we give a notion of 'genuine quantumness', from which special processes called 'phase spiders' arise, and get a first glimpse of quantum non-locality.

quant-ph↗

arXiv2016

From quantum foundations via natural language meaning to a theory of everything

In this paper we argue for a paradigmatic shift from `reductionism' to `togetherness'. In particular, we show how interaction between systems in quantum theory naturally carries over to modelling how word meanings interact in natural language. Since meaning in natural language, depending on the subject domain, encompasses discussions within any scientific discipline, we obtain a template for theories such as social interaction, animal behaviour, and many others.

cs.CL↗

arXiv2016

Harmonic Grammar in a DisCo Model of Meaning

The model of cognition developed in (Smolensky and Legendre, 2006) seeks to unify two levels of description of the cognitive process: the connectionist and the symbolic. The theory developed brings together these two levels into the Integrated Connectionist/Symbolic Cognitive architecture (ICS). Clark and Pulman (2007) draw a parallel with semantics where meaning may be modelled on both distributional and symbolic levels, developed by Coecke et al, 2010 into the Distributional Compositional (DisCo) model of meaning. In the current work, we revisit Smolensky and Legendre (S&L)'s model. We describe the DisCo framework, summarise the key ideas in S&L's architecture, and describe how their description of harmony as a graded measure of grammaticality may be applied in the DisCo model.

cs.AI↗

arXiv2016

Graded Entailment for Compositional Distributional Semantics

The categorical compositional distributional model of natural language provides a conceptually motivated procedure to compute the meaning of sentences, given grammatical structure and the meanings of its words. This approach has outperformed other models in mainstream empirical language processing tasks. However, until recently it has lacked the crucial feature of lexical entailment -- as do other distributional models of meaning. In this paper we solve the problem of entailment for categorical compositional distributional semantics. Taking advantage of the abstract categorical framework allows us to vary our choice of model. This enables the introduction of a notion of entailment, exploiting ideas from the categorical semantics of partial knowledge in quantum computation. The new model of language uses density matrices, on which we introduce a novel robust graded order capturing the entailment strength between concepts. This graded measure emerges from a general framework for approximate entailment, induced by any commutative monoid. Quantum logic embeds in our graded order. Our main theorem shows that entailment strength lifts compositionally to the sentence level, giving a lower bound on sentence entailment. We describe the essential properties of graded entailment such as continuity, and provide a procedure for calculating entailment strength.

cs.CL↗

arXiv2015

Distributional Sentence Entailment Using Density Matrices

Categorical compositional distributional model of Coecke et al. (2010) suggests a way to combine grammatical composition of the formal, type logical models with the corpus based, empirical word representations of distributional semantics. This paper contributes to the project by expanding the model to also capture entailment relations. This is achieved by extending the representations of words from points in meaning space to density operators, which are probability distributions on the subspaces of the space. A symmetric measure of similarity and an asymmetric measure of entailment is defined, where lexical entailment is measured using von Neumann entropy, the quantum variant of Kullback-Leibler divergence. Lexical entailment, combined with the composition map on word representations, provides a method to obtain entailment relations on the level of sentences. Truth theoretic and corpus-based examples are provided.

cs.CL↗