The algebra of entanglement and the geometry of composition Amar Hadzihasanovic Wolfson College University of Oxford arXiv:1709.08086v2 [math.CT] 27 Sep 2017 A thesis submitted for the degree of Doctor of Philosophy Trinity 2017 Abstract String diagrams turn algebraic equations into topological moves. These moves have often-recurring shapes, involving the sliding of one diagram past another. In the past, this fact has mostly been considered in terms of its computational convenience; this thesis investigates its deeper reasons. In the first part of the thesis, we individuate, at its root, the dual nature of polygraphs — freely generated higher categories — as presentations of higher algeb- raic theories, and as combinatorial descriptions of “directed spaces”: CW complexes whose cells have their boundary subdivided into an input and an output region. Op- erations of polygraphs modelled on operations of topological spaces, including an asymmetric tensor product refining the topological product of spaces, can then be used as the foundation of a compositional universal algebra, where sliding moves of string diagrams for an algebraic theory arise from tensor products of sub-theories. Such compositions come automatically with higher-dimensional coherence cells. We provide several examples of compositional reconstructions of higher algebraic theor- ies, including theories of braids, homomorphisms of algebras, and distributive laws of monads, from operations on polygraphs. In this regard, the standard formalism of polygraphs, based on strict ω-categories, suffers from some technical problems and inadequacies, including the difficulty of computing tensor products. As a solution, we propose a notion of regular polygraph, barring cell boundaries that are degenerate, that is, not homeomorphic to a disk of the appropriate dimension. We develop the theory of globular posets, based on ideas of poset topology, in order to specify a category of shapes for cells of regular poly- graphs. We prove that these shapes satisfy our non-degeneracy requirement, and show how to calculate their tensor products. Then, we introduce a notion of weak unit for regular polygraphs, allowing us to recover weakly degenerate boundaries in low dimensions. We prove that the existence of weak units is equivalent to the existence of cells satisfying certain divisibility properties — an elementary notion of equivalence cell — which prompts new questions on the relation between units and equivalences in higher categories. In the second part of the thesis, we turn to specific applications of diagrammatic algebra to quantum theory. First, we re-evaluate certain aspects of quantum theory from the point of view of categorical universal algebra, which leads us to define a 2-dimensional refinement of the category of Hilbert spaces. Then, we focus on the problem of axiomatising fragments of quantum theory, and present the ZW calculus, the first complete diagrammatic axiomatisation of the theory of qubits. The ZW calculus has specific advantages over its predecessors, the ZX calculi, including the existence of a computationally meaningful normal form, and of a frag- ment whose diagrams can be interpreted physically as setups of fermionic oscillat- ors. Moreover, the choice of its generators reflects an operational classification of entangled quantum states, which is not well-understood for states of more than 3 qubits: our result opens the way for a compositional understanding of entanglement for generic multipartite states. We conclude with preliminary results on generalisa- tions of the ZW calculus to quantum systems of arbitrary finite dimension, including the definition of a universal set of generators in each dimension. Preface This thesis is a report of some results, and the state of my thoughts on a variety of subjects, after four years spent as a doctoral student in the Oxford Quantum Group. The overarching theme is the relationship between algebra, or computation, and geometry, or topology; another recurring theme is compositionality. That algebra is useful for geometry is a notion by now ingrained in mathematical thinking at any level. The converse is somewhat less mainstream, driven recently to the spotlight by the contributions of higher/homotopical algebra and higher category theory to a number of fields, including physics via topological quantum field theory. Even so, in many cases geometry inspires at most a change of environment for algebraic structures, where the algebraic concepts are interpreted in a less rigid context — “weaker”, in higher-categorical jargon — yet by themselves remain atomic and unvaried. That is, we may have “homomorphisms up to homotopy”, but the concept of homomorphism is still fundamentally algebraic. Is it? Being already interested, as a student, in category theory and mathematical physics, at some point in 2012 I saw a variant of = (1) in a paper of Bob Coecke’s on categorical quantum mechanics, and I was introduced to string diagrams. The fact that adjunctions, the omnipresent concept of categorical algebra, is encapsulated in graphical equations such as (1) struck me immediately; the following year I applied to become Bob’s student, largely because of the appeal that the diagrammatic approach, central to categorical quantum mechanics, had on me. Later on, thanks to Dan Marsden, I was exposed to = (2) and related equations, capturing the notion of algebra homomorphism in great gener- ality; similar pictures describe modules and their intertwiners. They endow the basic i ingredients of algebra with specific geometries, comprising planar isotopy (1) and “slid- ing rules” such as (2), and hinting at a connection at the most fundamental level. Sliding rules are particularly common in diagrammatic algebra, and after a while I was com- pelled to ask: why these specific pictures? Are there general combinatorial-geometric processes by which equations of a certain shape arise? With time, this question, and others spawned by its partial answers, have become the main focus of my research; they are the subject of the first three chapters, to which the geometry of composition in the title refers. The key to understanding came from an entirely separate line of applications of topology to algebra, namely, homotopical methods in rewriting theory, and in particular their recent reworking into the theory of polygraphs, due to Albert Burroni and others. A polygraph is a presentation of a higher category or algebraic theory, but it is also a combinatorial model of a CW complex, made up of abstract “directed cells”: it is my claim, in this thesis, that the specific efficacy of string diagrams and their higher-dimensional generalisations rests on this analogy. On the other hand, the algebra of entanglement refers to certain algebraic theories, relevant to quantum information theory, on which I worked before and besides focussing on the questions above. Entanglement is a property of quantum systems that trans- lates, diagrammatically, into connectedness, hence a topological property; yet Bob and Aleks Kissinger linked it to the properties of certain algebras in the category of Hilbert spaces (Section 4.3). It seemed an ideal testing ground for the dual algebraic-topological nature of string diagrams, and it led me to develop the ZW calculus, the first complete diagrammatic axiomatisation of the theory of qubits, which is the main subject of the last two chapters. In fact, the bulk of my work on the ZW calculus came before and was a catalyst for the broader scope of the first part. Even from this short summary, then, it should be clear that the structure of the thesis does not reflect chronology in any meaningful way. The path to the current shape of this work has been (unsurprisingly) tortuous, full of wrong intuitions and all-out changes of mind; more will come in the future, no doubt. So, before leaving you to the post-processed flow of arguments, I will attempt a quick recollection of my actual thought process through its main turns, and thus provide an alternative summary of the thesis. I hope that the knowledge of some abandoned ideas may complement the surviving ones. Early in 2014, I followed Bob’s suggestion to look into the “GHZ/W calculus”, as an early sketch of the ZW calculus was called. I had recently learned about its precursor, the ZX calculus (Section 4.2), whose completeness for the stabiliser fragment of quantum mechanics was proved by Miriam Backens around that time, and what I especially liked about it was the undirectedness: the generators were symmetrical with respect to the swapping and transposition of wires, so the string diagrams could effectively be manipu- lated as labelled graphs. The GHZ/W calculus did not initially share this property, and I decided that I had to make it undirected as well. My intuition, at the time, was that these graph-like string diagrams reveal a deeper, ii undirected geometry of processes, which gets lost in the “bureaucracy” of categorical algebra, with its need for direction and composition functors and structural morphisms; in this I was influenced by Bob’s comments in this direction, and those of Jean-Yves Girard, and others, on proof nets being a kind of de-bureaucratised proof theory. In other words, the geometry is fundamental, while the categorical algebra is imposed on it to express compositions combinatorially. In the early summer of 2014 I discovered a normal form for ZW diagrams, and in August 2014 I had a complete axiomatisation, together with a proof of completeness for what is called the vanilla ZW calculus in Section 5.1. The proof for the general R-labelled case, Theorem 5.27, came only a couple of years later, not because it is substantially different or more complicated, but because I had not tried. The case R = C of Theorem 5.27 is completeness for the theory of qubits, what had escaped the ZX calculi for nearly a decade: it means that any calculation on pure quantum states of a number of qubits that can be performed with vector calculus, or matrix calculus, can also be performed diagrammatically, using the axioms of the ZW calculus.