Entanglement as a semantic resource

The characteristic holistic features of the quantum theoretic formalism and the intriguing notion of entanglement can be applied to a field that is far from microphysics: logical semantics. Quantum computational logics are new forms of quantum logic that have been suggested by the theory of quantum logical gates in quantum computation. In the standard semantics of these logics, sentences denote quantum information quantities: systems of qubits (quregisters) or, more generally, mixtures of quregisters (qumixes), while logical connectives are interpreted as special quantum logical gates (which have a characteristic reversible and dynamic behavior). In this framework, states of knowledge may be entangled, in such a way that our information about the whole determines our information about the parts; and the procedure cannot be, generally, inverted. In spite of its appealing properties, the standard version of the quantum computational semantics is strongly "Hilbert-space dependent". This certainly represents a shortcoming for all applications, where real and complex numbers do not generally play any significant role (as happens, for instance, in the case of natural and of artistic languages). We propose an abstract version of quantum computational semantics, where abstract qumixes, quregisters and registers are identified with some special objects (not necessarily living in a Hilbert space), while gates are reversible functions that transform qumixes into qumixes. In this framework, one can give an abstract definition of the notions of superposition and of entangled pieces of information, quite independently of any numerical values. We investigate three different forms of abstract holistic quantum computational logic

A Study of Entanglement in a Categorical Framework of Natural Language

In both quantum mechanics and corpus linguistics based on vector spaces, the notion of entanglement provides a means for the various subsystems to communicate with each other. In this paper we examine a number of implementations of the categorical framework of Coecke, Sadrzadeh and Clark (2010) for natural language, from an entanglement perspective. Specifically, our goal is to better understand in what way the level of entanglement of the relational tensors (or the lack of it) affects the compositional structures in practical situations. Our findings reveal that a number of proposals for verb construction lead to almost separable tensors, a fact that considerably simplifies the interactions between the words. We examine the ramifications of this fact, and we show that the use of Frobenius algebras mitigates the potential problems to a great extent. Finally, we briefly examine a machine learning method that creates verb tensors exhibiting a sufficient level of entanglement.Comment: In Proceedings QPL 2014, arXiv:1412.810

Compositional Distributional Semantics with Compact Closed Categories and Frobenius Algebras

This thesis contributes to ongoing research related to the categorical compositional model for natural language of Coecke, Sadrzadeh and Clark in three ways: Firstly, I propose a concrete instantiation of the abstract framework based on Frobenius algebras (joint work with Sadrzadeh). The theory improves shortcomings of previous proposals, extends the coverage of the language, and is supported by experimental work that improves existing results. The proposed framework describes a new class of compositional models that find intuitive interpretations for a number of linguistic phenomena. Secondly, I propose and evaluate in practice a new compositional methodology which explicitly deals with the different levels of lexical ambiguity (joint work with Pulman). A concrete algorithm is presented, based on the separation of vector disambiguation from composition in an explicit prior step. Extensive experimental work shows that the proposed methodology indeed results in more accurate composite representations for the framework of Coecke et al. in particular and every other class of compositional models in general. As a last contribution, I formalize the explicit treatment of lexical ambiguity in the context of the categorical framework by resorting to categorical quantum mechanics (joint work with Coecke). In the proposed extension, the concept of a distributional vector is replaced with that of a density matrix, which compactly represents a probability distribution over the potential different meanings of the specific word. Composition takes the form of quantum measurements, leading to interesting analogies between quantum physics and linguistics.Comment: Ph.D. Dissertation, University of Oxfor

Evaluating Composition Models for Verb Phrase Elliptical Sentence Embeddings

Ellipsis is a natural language phenomenon where part of a sentence is missing and its information must be recovered from its surrounding context, as in “Cats chase dogs and so do foxes.”. Formal semantics has different methods for resolving ellipsis and recovering the missing information, but the problem has not been considered for distributional semantics, where words have vector embeddings and combinations thereof provide embeddings for sentences. In elliptical sentences these combinations go beyond linear as copying of elided information is necessary. In this paper, we develop different models for embedding VP-elliptical sentences. We extend existing verb disambiguation and sentence similarity datasets to ones containing elliptical phrases and evaluate our models on these datasets for a variety of non-linear combinations and their linear counterparts. We compare results of these compositional models to state of the art holistic sentence encoders. Our results show that non-linear addition and a non-linear tensor-based composition outperform the naive non-compositional baselines and the linear models, and that sentence encoders perform well on sentence similarity, but not on verb disambiguation

A first-order epistemic quantum computational semantics with relativistic-like epistemic effects

Quantum computation has suggested new forms of quantum logic, called quantum computational logics. In these logics well-formed formulas are supposed to denote pieces of quantum information: possible pure states of quantum systems that can store the information in question. At the same time, the logical connectives are interpreted as quantum logical gates: unitary operators that process quantum information in a reversible way, giving rise to quantum circuits. Quantum computational logics have been mainly studied as sentential logics (whose alphabet consists of atomic sentences and of logical connectives). In this article we propose a semantic characterization for a first-order epistemic quantum computational logic, whose language can express sentences like "Alice knows that everybody knows that she is pretty". One can prove that (unlike the case of logical connectives) both quantifiers and epistemic operators cannot be generally represented as (reversible) quantum logical gates. The "act of knowing" and the use of universal (or existential) assertions seem to involve some irreversible "theoretic jumps", which are similar to quantum measurements. Since all epistemic agents are characterized by specific epistemic domains (which contain all pieces of information accessible to them), the unrealistic phenomenon of logical omniscience is here avoided: knowing a given sentence does not imply knowing all its logical consequences

Semantic distillation: a method for clustering objects by their contextual specificity

Techniques for data-mining, latent semantic analysis, contextual search of databases, etc. have long ago been developed by computer scientists working on information retrieval (IR). Experimental scientists, from all disciplines, having to analyse large collections of raw experimental data (astronomical, physical, biological, etc.) have developed powerful methods for their statistical analysis and for clustering, categorising, and classifying objects. Finally, physicists have developed a theory of quantum measurement, unifying the logical, algebraic, and probabilistic aspects of queries into a single formalism. The purpose of this paper is twofold: first to show that when formulated at an abstract level, problems from IR, from statistical data analysis, and from physical measurement theories are very similar and hence can profitably be cross-fertilised, and, secondly, to propose a novel method of fuzzy hierarchical clustering, termed \textit{semantic distillation} -- strongly inspired from the theory of quantum measurement --, we developed to analyse raw data coming from various types of experiments on DNA arrays. We illustrate the method by analysing DNA arrays experiments and clustering the genes of the array according to their specificity.Comment: Accepted for publication in Studies in Computational Intelligence, Springer-Verla

An Holistic Extension for Classical Logic via Quantum Fredkin Gate

An holistic extension for classical propositional logic is introduced in the framework of quantum computation with mixed states. The mentioned extension is obtained by applying the quantum Fredkin gate to non-factorizable bipartite states. In particular, an extended notion of classical contradiction is studied in this holistic framework