Graphical Classification of Entangled Qutrits

Honda, Kentaro

doi:10.4204/eptcs.95.11

Cited by 5 publications

(2 citation statements)

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“…In [CK10], Coecke and Kissinger observed that there exists, in fact, a commutative Frobenius algebra (c , d , m , u ) on the qubit, whose spider with three outputs and no inputs is proportional to the | W state. This is defined by This was partially extended to the case of qutrits, that is, 3-dimensional Hilbert spaces, where an intermediate class between special and anti-special appears, in [Hon12].…”

Section: The Entanglement Classification Problemmentioning

confidence: 99%

The algebra of entanglement and the geometry of composition

Hadzihasanovic

2017

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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 algebraic theories, and as combinatorial descriptions of "directed spaces": CW complexes whose cells have their boundary subdivided into an input and an output region. Operations 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 theories, 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 polygraphs. 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 fragment whose diagrams can be interpreted physically as setups of fermionic oscillators. Moreover, the choice of its generators reflects an operational clas...

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Section: The Entanglement Classification Problemmentioning

confidence: 99%

The algebra of entanglement and the geometry of composition

Hadzihasanovic

2017

Preprint

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“…For dimensions 3 and above, only some of these algebras extend to Frobenius algebras, and it becomes a straightforward task to enumerate those that are. The classification of Frobenius states for dimension 3 was completed this year by Honda [32]. As in the case for two dimensions, a…”

Section: Classifying Frobenius Statesmentioning

confidence: 99%

Pictures of Processes: Automated Graph Rewriting for Monoidal Categories and Applications to Quantum Computing

Kissinger

2012

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This work is about diagrammatic languages, how they can be represented, and what they in turn can be used to represent. More specifically, it focuses on representations and applications of string diagrams. String diagrams are used to represent a collection of processes, depicted as "boxes" with multiple (typed) inputs and outputs, depicted as "wires". If we allow plugging input and output wires together, we can intuitively represent complex compositions of processes, formalised as morphisms in a monoidal category.While string diagrams are very intuitive, existing methods for defining them rigorously rely on topological notions that do not extend naturally to automated computation. The first major contribution of this dissertation is the introduction of a discretised version of a string diagram called a string graph. String graphs form a partial adhesive category, so they can be manipulated using double-pushout graph rewriting. Furthermore, we show how string graphs modulo a rewrite system can be used to construct free symmetric traced and compact closed categories on a monoidal signature.The second contribution is in the application of graphical languages to quantum information theory. We use a mixture of diagrammatic and algebraic techniques to prove a new classification result for strongly complementary observables. Namely, maximal sets of strongly complementary observables of dimension D must be of size no larger than 2, and are in 1-to-1 correspondence with the Abelian groups of order D. We also introduce a graphical language for multipartite entanglement and illustrate a simple graphical axiom that distinguishes the two maximally-entangled tripartite qubit states: GHZ and W. Notably, we illustrate how the algebraic structures induced by these operations correspond to the (partial) arithmetic operations of addition and multiplication on the complex projective line.The third contribution is a description of two software tools developed in part by the author to implement much of the theoretical content described here. The first tool is Quantomatic, a desktop application for building string graphs and graphical theories, as well as performing automated graph rewriting visually. The second is QuantoCoSy, which performs fully automated, model-driven theory creation using a procedure called conjecture synthesis.

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