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arXiv:2503.08846

A Tutorial on Knots and Quantum Mechanics

Dmitry Melnikov

Published Mar 11, 2025• 0 citations
arXivHTMLPDF

Abstract

These notes review a description of quantum mechanics in terms of the topology of spaces, basing on the axioms of Topological Quantum Field Theory and path integral formalism. In this description quantum states and operators are encoded by the topology of spaces that are used as modules to build the quantum mechanical model, while expectation values and probabilities are given by topological invariants of spaces, knots and links. The notes focus on the specific way the topology encodes quantum mechanical features, or, equivalently, on how these features can be controlled through the topology. A topological classification of entanglement is discussed, as well as properties of entanglement entropy and basic quantum protocols. The primary aim is to build a less conventional diagrammatic intuition about quantum mechanics, expanding the paradigm of ``Quantum Picturalism".

DOI
10.48550/arxiv.2503.08846

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These notes review a description of quantum mechanics in terms of the topology of spaces, basing on the axioms of Topological Quantum Field Theory and path integral formalism. In this description quantum states and operators are encoded by the topology of spaces that are used as modules to build the quantum mechanical model, while expectation values and probabilities are given by topological invariants of spaces, knots and links. The notes focus on the specific way the topology encodes quantum mechanical features, or, equivalently, on how these features can be controlled through the topology. A topological classification of entanglement is discussed, as well as properties of entanglement entropy and basic quantum protocols. The primary aim is to build a less conventional diagrammatic intuition about quantum mechanics, expanding the paradigm of ``Quantum Picturalism".
Abstract Topological quantum field theories (TQFT) encode properties of quantum states in the topological features of abstract manifolds. One can use the topological avatars of quantum states to develop intuition about different concepts and phenomena of quantum mechanics. In this paper we focus on the class of simplest topologies provided by a specific TQFT and investigate what the corresponding states teach us about entanglement. These “planar connectome” states are defined by graphs of simplest topology for a given adjacency matrix. In the case of bipartite systems the connectomes classify different types of entanglement matching the classification of stochastic local operations and classical communication (SLOCC). The topological realization makes explicit the nature of entanglement as a resource and makes apparent a number of its properties, including monogamy and characteristic inequalities for the entanglement entropy. It also provides tools and hints to engineer new measures of entanglement and other applications. Here the approach is used to construct purely topological versions of the dense coding and quantum teleportation protocols, giving diagrammatic interpretation of the role of entanglement in quantum computation and communication. Finally, the topological concepts of entanglement and quantum teleportation are employed in a simple model of information retrieval from a causally disconnected region, similar to the interior of an evaporating black hole.
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It has been conjectured that quantum entanglement operators can be lifted to braiding operators by the way of the topological quantum field theory axioms set forth by Witten and Atiyah. Moreover, it can be readily shown that quantum link invariants need entanglement to construct topological invariants. Given these results and the already dense mathematical framework underlying topology and quantum field theory, we propose that, through the usage of quantum algebra and bracket models, we can identify a significant area of overlap where entangling R-matrix solutions to the Yang-Baxter equation can be used to construct invariants of knots and links. Such a connection illuminates the relationship between quantum and topological entanglement, and allows us to draw novel insights about properties of entangling gates, for example the question of what it is that allows some entangling R-matrices to generate topological invariants when others do not. In conclusion, by studying the boundary between topological and quantum entanglement we construct novel topological invariants that can have significant impact on the study of quantum computing and can help make progress towards the problem of representing the Artin braid group as unitary matrices.
This paper investigates the interplay between quantum entanglement and topological field theory, proposing a framework for quantum entanglement that leverages the principles of topological geometry to reveal the fundamental nature of entanglement and explore its intricate relationship with topological structures. We aim to provide a novel approach to understanding entanglement, moving beyond traditional interpretations and exploring potential connections within the realm of topological physics. The core of this work focuses on a rigorous mathematical formulation of entanglement through the lens of topological invariants, ultimately seeking to uncover deeper insights into the essence of quantum entanglement. The proposed mechanism involves a deep integration of quantum mechanics and topological geometry, offering a potential paradigm shift in our understanding of this fascinating phenomenon. The paper concludes with a discussion of the implications of this approach for both quantum information theory and the broader field of topological physics.
Abstract Entanglement is a special feature of the quantum world that reflects the existence of subtle, often non-local, correlations between local degrees of freedom. In topological theories such non-local correlations can be given a very intuitive interpretation: quantum entanglement of subsystems means that there are “strings” connecting them. More generally, an entangled state, or similarly, the density matrix of a mixed state, can be represented by cobordisms of topological spaces. Using a formal mathematical definition of TQFT we construct basic examples of entangled states and compute their von Neumann entropy.
This paper introduces a novel approach to characterizing and analyzing quantum state environments, termed "Quantum State Topologies." It proposes a probabilistic topological framework that leverages principles from probability theory and topology to reveal the intrinsic topological structure inherent in quantum state landscapes. The core mechanism involves studying the evolution of quantum state environments through probabilistic methods, identifying topological invariants, and constructing a probability-based topological model. This framework offers a new perspective for understanding and manipulating complex quantum systems, potentially leading to advancements in quantum information processing, materials science, and fundamental quantum mechanics. The methodology presented here aims to bridge the gap between the descriptive power of topological methods and the probabilistic nature of quantum mechanics, providing a robust and flexible tool for analyzing quantum environments. The key contributions lie in the development of a mathematically rigorous framework for quantifying and classifying topological features within quantum state environments, moving beyond traditional approaches that often rely on simplified models or specific physical scenarios.