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Symmetry

A Haptic Model of Entanglement, Gauge Symmetries and Minimal Interaction Based on Knot Theory

Stefan Heusler, Malte S. Ubben

Published Nov 12, 2019• 8 citations
Publisher

Abstract

The Heegaard splitting of S U ( 2 ) is a particularly useful representation for quantum phases of spin j-representation arising in the mapping S 1 → S 3, which can be related to ( 2 j , 2 ) torus knots in Hilbert space. We show that transitions to homotopically equivalent knots can be associated with gauge invariance, and that the same mechanism is at the heart of quantum entanglement. In other words, (minimal) interaction causes entanglement. Particle creation is related to cuts in the knot structure. We show that inner twists can be associated with operations with the quaternions ( I , J , K ), which are crucial to understand the Hopf mapping S 3 → S 2. We discuss the relationship between observables on the Bloch sphere S 2, and knots with inner twists in Hilbert space. As applications, we discuss selection rules in atomic physics, and the status of virtual particles arising in Feynman diagrams. Using a simple paper strip model revealing the knot structure of quantum phases in Hilbert space including inner twists, a h a p t i c model of entanglement and gauge symmetries is proposed, which may also be valid for physics education.

DOI
10.3390/sym11111399

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A generalization of the famous Dirac belt trick opens up the way to a haptic model for quantum phases of fermions and bosons in Hilbert space based on knot theory. We introduce a simple paper strip model as an aid for visualization of the quantum phases before and after Hopf-mapping, which can be extended to arbitrary spin states with almost no mathematical formalism. Knot theory arises naturally, leading to the Jones polynomials derived from Artin’s braid group for fermionic knots and for bosonic links. The paper strip model explicitly illuminates the relation between these knots and links within the S U ( 2 ) -representation of spin-jstates in C 2 j + 1 before Hopf-mapping and the number p = 2 j of nodes in the stellar representation in C P 1 after Hopf mapping.

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Modelling spin

Stefan Heusler, Malte S. Ubben

Abstract Based on a geometric view on the famous Dirac belt trick in terms of a Möbius strip, we propose a simple haptic model for spin states as standing waves in non-trivial geometry. We compare two representations of the model: the spinor representation in SU (2), which geometrically is described by ‘Dirac belt’ states on S 3 , and the well-known Bloch-sphere representation on S 2 . We show that after Hopf-mapping S 3 → S 2 , the position and number of nodal points are sufficient to describe the spin state and make the relation between the representations on S 2 and on S 3 explicit in simple haptic models. Our approach is well-suited for graduate physics courses, and provides a thorough understanding of the complex geometry of spin states beyond the usual Bloch-sphere visualisation. It turns out that an even number 2 l of nodal points can be related to nodal lines of spherical harmonics Y lm , an odd number can be traced back to spin states. In this sense, our model provides a unified view for orbital and spin states based on a simple picture of standing waves.

European Journal of Physics•Sep 24, 2018•5 citations
The Heegaard splitting of S U ( 2 ) is a particularly useful representation for quantum phases of spin j-representation arising in the mapping S 1 → S 3, which can be related to ( 2 j , 2 ) torus knots in Hilbert space. We show that transitions to homotopically equivalent knots can be associated with gauge invariance, and that the same mechanism is at the heart of quantum entanglement. In other words, (minimal) interaction causes entanglement. Particle creation is related to cuts in the knot structure. We show that inner twists can be associated with operations with the quaternions ( I , J , K ), which are crucial to understand the Hopf mapping S 3 → S 2. We discuss the relationship between observables on the Bloch sphere S 2, and knots with inner twists in Hilbert space. As applications, we discuss selection rules in atomic physics, and the status of virtual particles arising in Feynman diagrams. Using a simple paper strip model revealing the knot structure of quantum phases in Hilbert space including inner twists, a h a p t i c model of entanglement and gauge symmetries is proposed, which may also be valid for physics education.
A generalization of the famous Dirac belt trick opens up the way to a haptic model for quantum phases of fermions and bosons in Hilbert space based on knot theory. We introduce a simple paper strip model as an aid for visualization of the quantum phases before and after Hopf-mapping, which can be extended to arbitrary spin states with almost no mathematical formalism. Knot theory arises naturally, leading to the Jones polynomials derived from Artin’s braid group for fermionic knots and for bosonic links. The paper strip model explicitly illuminates the relation between these knots and links within the S U ( 2 ) -representation of spin-jstates in C 2 j + 1 before Hopf-mapping and the number p = 2 j of nodes in the stellar representation in C P 1 after Hopf mapping.
We argue that the usual Bloch sphere is insufficient in various aspects for the representation of qubits in quantum information theory. For example, spin flip operations with the quaternions I J K = e 2 π i 2 = − 1 and J I K = + 1 cannot be distinguished on the Bloch sphere. We show that a simple knot theoretic extension of the Bloch sphere representation is sufficient to track all unitary operations for single qubits. Next, we extend the Bloch sphere representation to entangled states using knot theory. As applications, we first discuss contextuality in quantum physics—in particular the Kochen-Specker theorem. Finally, we discuss some arguments against many-worlds theories within our knot theoretic model of entanglement. The key ingredients of our approach are symmetries and geometric properties of the unitary group.
Kauffman and Lomonaco (New J Phys 4:73.1–73.18, 2002. arXiv:quant-ph/0401090 , New J Phys 6:134.1–134.40, 2004) explored the idea of understanding quantum entanglement (the non-local correlation of certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In Alagic et al. (Yang–Baxter operators need quantum entanglement to distinguish knots, 2015. arXiv:1507.05979v1 ), it is shown that entanglement is a necessary condition for forming non-trivial invariants of knots from braid closures via solutions to the Yang–Baxter equation. We show that the arguments used by Alagic et al. (2015) generalize to essentially the same results for quantum invariant state summation models of knots. In one case (the unoriented swap case) we give an example of a Yang–Baxter operator, and associated quantum invariant, that can detect the Hopf link. Again this is analogous to the results of Alagic et al. (2015). We also give a class of R matrices that are entangling and are weak invariants of classical knots and links yet strong invariants of virtual knots and links. We also give an example of an SU(2) representation of the three-strand braid group that models the Jones polynomial for closures of three-strand braids. This invariant is a quantum model for the Jones polynomial restricted to three-strand braids, and it does not involve quantum entanglement. These relationships between topological braiding and quantum entanglement can be used as a framework for future work in understanding the properties of entangling gates in topological quantum computing. The paper ends with a discussion of the Aravind hypothesis about the direct relationship of knots and quantum entanglement and the ER=EPRER = EPRER=EPR E R = E P R hypothesis about the relationship of quantum entanglement with the connectivity of space. We describe how, given a background space and a quantum tensor network, to construct a new topological space that welds the network and the background space together. This construction embodies the principle that quantum entanglement and topological connectivity are intimately related.
In 2+1 dimensions, gravity is an SU(2) topological gauge theory that can be written as BF theory. In the condensed matter literature, the Hamiltonian realization of BF theory for finite groups is known as the Kitaev model. The corresponding Hamiltonian yields magnetic and electric point-like excitations, both supported by punctures. In this context, the cylinder plays a special role since the gluing of two cylinders results in another cylinder, hence defining an algebra on the Hilbert space of states, referred to as Ocneanu's tube algebra. By choosing specific graph-states on the cylinder, we can confirm explicitly that this algebra is equivalent to the Drinfel'd double of the gauge group. The representation theory of the Drinfel'd double can then be used to define a basis of excited states associated with any punctured Riemann surface. This result can be adapted so as to define a new basis for the gauge invariant Hilbert space of lattice gauge theories, and a fortiori loop quantum gravity, replacing the well-known spin network basis. In doing so, we naturally shift the focus from the underlying lattice to the excitations themselves, this notion of excitations being understood with respect to a given vacuum state, namely the so-called BF vacuum. This basis diagonalizes so-called ribbon operators that provide Dirac observables. Furthermore, it has an inherent hierarchical organization allowing to design states with a multi-scale structure. This turns out to be extremely precious for studying the large scale structure of the theory. It also provides a new notion of subsystems for gauge theories. Being solely based on excitations, this leads to a completely relational way of defining regions. This can be used to define a new notion of entanglement entropy for lattice gauge theories and (2+1)d gravity coupled to point particles. These techniques can also be generalized so as to define excitation bases for (3+1)d gauge models of topological phases. Two approaches are considered in this thesis. The first one consists in generalizing Ocneanu's tube algebra by replacing the cylinder with the manifold obtained by cutting open a three-torus along one direction. This defines an algebraic structure extending the Drinfel'd double that can be used in a similar fashion as in (2+1)d. The other approach relies on Heegaard splittings of three-manifolds which perform a decomposition into handle-bodies along so-called Heegaard surfaces. We use this technique to encode the Hilbert space of flat connections with curvature excitations of a three-manifold into the Hilbert space of flat connections on a 2d Heegaard surface, hence making the results derived in (2+1)d available to the study of the (3+1)d case.
We consider Chern-Simons theory for gauge group G at level k on 3-manifolds M n with boundary consisting of n topologically linked tori. The Euclidean path integral on M n defines a quantum state on the boundary, in the n-fold tensor product of the torus Hilbert space. We focus on the case where M n is the link-complement of some n-component link inside the three-sphere S 3. The entanglement entropies of the resulting states define framing-independent link invariants which are sensitive to the topology of the chosen link. For the Abelian theory at level k (G = U(1) k ) we give a general formula for the entanglement entropy associated to an arbitrary (m|n − m) partition of a generic n-component link into sub-links. The formula involves the number of solutions to certain Diophantine equations with coefficients related to the Gauss linking numbers (mod k) between the two sublinks. This formula connects simple concepts in quantum information theory, knot theory, and number theory, and shows that entanglement entropy between sublinks vanishes if and only if they have zero Gauss linking (mod k). For G = SU(2) k , we study various two and three component links. We show that the 2-component Hopf link is maximally entangled, and hence analogous to a Bell pair, and that the Whitehead link, which has zero Gauss linking, nevertheless has entanglement entropy. Finally, we show that the Borromean rings have a “W-like” entanglement structure (i.e., tracing out one torus does not lead to a separable state), and give examples of other 3-component links which have “GHZ-like” entanglement (i.e., tracing out one torus does lead to a separable state).
Abstract Based on a geometric view on the famous Dirac belt trick in terms of a Möbius strip, we propose a simple haptic model for spin states as standing waves in non-trivial geometry. We compare two representations of the model: the spinor representation in SU (2), which geometrically is described by ‘Dirac belt’ states on S 3 , and the well-known Bloch-sphere representation on S 2 . We show that after Hopf-mapping S 3 → S 2 , the position and number of nodal points are sufficient to describe the spin state and make the relation between the representations on S 2 and on S 3 explicit in simple haptic models. Our approach is well-suited for graduate physics courses, and provides a thorough understanding of the complex geometry of spin states beyond the usual Bloch-sphere visualisation. It turns out that an even number 2 l of nodal points can be related to nodal lines of spherical harmonics Y lm , an odd number can be traced back to spin states. In this sense, our model provides a unified view for orbital and spin states based on a simple picture of standing waves.