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arXiv:1602.02671

Qudit Isotopy

Arthur M. Jaffe, Zhengwei Liu, Alex Wozniakowski

Published Feb 8, 2016• 4 citations
arXivHTMLPDF

Abstract

We explore a general diagrammatic framework to understand qudits and their braiding, especially in its relation to entanglement. This involves understanding the role of isotopy in interpreting diagrams that implement entangling gates as well as some standard quantum information protocols. We give qudit Pauli operators $X,Y,Z$ and comment on their structure, both from an algebraic and from a diagrammatic point of view. We explain alternative models for diagrammatic interpretations of qudits and their transformations. We use our diagrammatic approach to define an entanglement-relay protocol for long-distance entanglement. Our approach rests on algebraic and topological relations discovered in the study of planar para algebras. In summary, this work provides bridges between the new theory of planar para algebras and quantum information, especially in questions involving entanglement.

DOI
10.48550/arxiv.1602.02671

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This paper explores the intricate connections between various algebraic structures—Banach algebras, C*-algebras, and von Neumann algebras—and their profound implications for the theory of quantum entanglement and quantum information. We establish a comprehensive framework that bridges the abstract concepts of functional analysis with the concrete demands of quantum mechanics. Beginning with the foundational properties of Banach algebras, we progressively introduce C*-algebras as a natural setting for quantum observables and states, followed by von Neumann algebras, which are essential for describing quantum measurement and infinite-dimensional quantum systems. A central focus is to delineate how quantum entanglement, a cornerstone resource in quantum information science, can be rigorously characterized and quantified within this hierarchical algebraic framework. We integrate recent game-theoretic approaches to quantum information, providing new perspectives on quantum relative entropy, mutual information, and channel capacity. By demonstrating how the rich structure of C*-algebras acts as a pivotal bridge, we illuminate the mathematical coherence underlying the transition from general algebraic concepts to the specialized domain of quantum information processing, offering a unified view of quantum phenomena. The work also delves into the operational justifications for these algebraic descriptions, particularly in the context of infinite quantum systems.
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