) which produce the related quantum entangled systems. We finally remark that these two quantum systems, which clearly possess inequivalent entanglement properties, are locally unitarily equivalent.

DOI: 10.1142/9789814335614_0070

Related papers

Links and Quantum Entanglement ∗

A. I. Solomon, Choon-Lin Ho

We discuss the analogy between topological entanglement and quantum entanglement, particularly for tripartite quantum systems.We illustrate our approach by first discussing two clearly (topologically) inequivalent systems of three-ring links: The Borromean rings, in which the removal of any one link leaves the remaining two non-linked (or, by analogy, non-entangled); and an inequivalent system (which we call the NUS link) for which the removal of any one link leaves the remaining two linked (or, entangled in our analogy).We introduce unitary representations for the appropriate Braid Group (B3) which produce the related quantum entangled systems. We finally remark that these two quantum systems, which clearly possess inequivalent entanglement properties, are locally unitarily equivalent.We discuss the analogy between topological entanglement and quantum entanglement, particularly for tripartite quantum systems.We illustrate our approach by first discussing two clearly (topologically) inequivalent systems of three-ring links: The Borromean rings, in which the removal of any one link leaves the remaining two non-linked (or, by analogy, non-entangled); and an inequivalent system (which we call the NUS link) for which the removal of any one link leaves the remaining two linked (or, entangled in our analogy).We introduce unitary representations for the appropriate Braid Group (B3) which produce the related quantum entangled systems. We finally remark that these two quantum systems, which clearly possess inequivalent entanglement properties, are locally unitarily equivalent.

Jan 1, 20103 citations

Unitary braid matrices: bridge between topological and quantum entanglements

B. Abdesselam, Amitabha Chakrabarti

Braiding operators corresponding to the third Reidemeister move in the theory of knots and links are realized in terms of parametrized unitary matrices for all dimensions. Two distinct classes are considered. Their (non-local) unitary actions on separable pure product states of three identical subsystems (say, the spin projections of three particles) are explicitly evaluated for all dimensions. This, for our classes, is shown to generate entangled superposition of four terms in the base space. The 3-body and 2-body entanglements (in three 2-body subsystems), the 3-tangles and 2-tangles are explicitly evaluated for each class. For our matrices, these are parametrized. Varying parameters they can be made to sweep over the domain (0,1).Thus braiding operators corresponding to over- and under-crossings of three braids and, on closing ends, to topologically entangled Borromean rings are shown, in another context, to generate quantum entanglements. For higher dimensions, starting with different initial triplets one can entangle by turns, each state with all the rest. A specific coupling of three angular momenta is briefly discussed to throw more light on three body entanglements.Braiding operators corresponding to the third Reidemeister move in the theory of knots and links are realized in terms of parametrized unitary matrices for all dimensions. Two distinct classes are considered. Their (non-local) unitary actions on separable pure product states of three identical subsystems (say, the spin projections of three particles) are explicitly evaluated for all dimensions. This, for our classes, is shown to generate entangled superposition of four terms in the base space. The 3-body and 2-body entanglements (in three 2-body subsystems), the 3-tangles and 2-tangles are explicitly evaluated for each class. For our matrices, these are parametrized. Varying parameters they can be made to sweep over the domain (0,1).Thus braiding operators corresponding to over- and under-crossings of three braids and, on closing ends, to topologically entangled Borromean rings are shown, in another context, to generate quantum entanglements. For higher dimensions, starting with different initial triplets one can entangle by turns, each state with all the rest. A specific coupling of three angular momenta is briefly discussed to throw more light on three body entanglements.

arXiv:0911.0906Journal of Physical MathematicsJan 1, 20102 citations

Topological aspects of quantum entanglement

Louis Hirsch Kauffman, Eshan Mehrotra

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 $ER = EPR$ 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.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 $$ER = EPR$$ 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.

arXiv:1611.08047Quantum Information ProcessingFeb 2, 201920 citations

Unitary braid matrices: bridge between topological and quantum entanglements

B Abdesellam, A Chakrabati

Braiding operators corresponding to the third Reidemeister move in the theory of knots and links are realized in terms of parametrized unitary matrices for all dimensions.Two distinct classes are considered.Their (nonlocal) unitary actions on separable pure product states of three identical subsystems (i.e., the spin projections of three particles) are explicitly evaluated for all dimensions.This, for our classes, is shown to generate entangled superposition of four terms in the base space.The 3-body and 2-body entanglements (in three 2-body subsystems), the 3 tangles, and 2 tangles are explicitly evaluated for each class.For our matrices, these are parametrized.Varying parameters they can be made to sweep over the domain (0, 1).Thus, braiding operators corresponding to over-and undercrossings of three braids and, on closing ends, to topologically entangled Borromean rings are shown, in another context, to generate quantum entanglements.For higher dimensions, starting with different initial triplets one can entangle by turns, each state with all the rest.A specific coupling of three angular momenta is briefly discussed to throw more light on three body entanglements.Braiding operators corresponding to the third Reidemeister move in the theory of knots and links are realized in terms of parametrized unitary matrices for all dimensions.Two distinct classes are considered.Their (nonlocal) unitary actions on separable pure product states of three identical subsystems (i.e., the spin projections of three particles) are explicitly evaluated for all dimensions.This, for our classes, is shown to generate entangled superposition of four terms in the base space.The 3-body and 2-body entanglements (in three 2-body subsystems), the 3 tangles, and 2 tangles are explicitly evaluated for each class.For our matrices, these are parametrized.Varying parameters they can be made to sweep over the domain (0, 1).Thus, braiding operators corresponding to over-and undercrossings of three braids and, on closing ends, to topologically entangled Borromean rings are shown, in another context, to generate quantum entanglements.For higher dimensions, starting with different initial triplets one can entangle by turns, each state with all the rest.A specific coupling of three angular momenta is briefly discussed to throw more light on three body entanglements.

Journal of Physical MathematicsJan 1, 20100 citations

Classifying quantum entanglement through topological links

Gonçalo M. Quinta, Rui André

We propose an alternative classification scheme for quantum entanglement based on topological links. This is done by identifying a nonrigid ring to a particle, attributing the act of cutting and removing a ring to the operation of tracing out the particle, and associating linked rings to entangled particles. This analogy naturally leads us to a classification of multipartite quantum entanglement based on all possible distinct links for a given number of rings. To determine all different possibilities, we develop a formalism that associates any link to a polynomial, with each polynomial thereby defining a distinct equivalence class. To demonstrate the use of this classification scheme, we choose qubit quantum states as our example of physical system. A possible procedure to obtain qubit states from the polynomials is also introduced, providing an example state for each link class. We apply the formalism for the quantum systems of three and four qubits and demonstrate the potential of these tools in a context of qubit networks.We propose an alternative classification scheme for quantum entanglement based on topological links. This is done by identifying a nonrigid ring to a particle, attributing the act of cutting and removing a ring to the operation of tracing out the particle, and associating linked rings to entangled particles. This analogy naturally leads us to a classification of multipartite quantum entanglement based on all possible distinct links for a given number of rings. To determine all different possibilities, we develop a formalism that associates any link to a polynomial, with each polynomial thereby defining a distinct equivalence class. To demonstrate the use of this classification scheme, we choose qubit quantum states as our example of physical system. A possible procedure to obtain qubit states from the polynomials is also introduced, providing an example state for each link class. We apply the formalism for the quantum systems of three and four qubits and demonstrate the potential of these tools in a context of qubit networks.

arXiv:1803.08935Physical Review AApr 5, 201814 citations

Quantum Computing and Quantum Topology

Louis Hirsch Kauffman, Samuel J. Lomonaco

Abstract This paper is an introduction to relationships between quantum topology and quantum computing. We discuss unitary solutions to the YangBaxter equation that are universal quantum gates, quantum entanglement and topological entanglement, and we give an exposition of knot-theoretic recoupling theory, its relationship with topological quantum field theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten-Reshetikhin-Turaev invariant of three manifolds.Abstract This paper is an introduction to relationships between quantum topology and quantum computing. We discuss unitary solutions to the YangBaxter equation that are universal quantum gates, quantum entanglement and topological entanglement, and we give an exposition of knot-theoretic recoupling theory, its relationship with topological quantum field theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten-Reshetikhin-Turaev invariant of three manifolds.

Chapman & Hall/CRC eBooksSep 19, 20070 citations

B_3

) which produce the related quantum entangled systems. We finally remark that these two quantum systems, which clearly possess inequivalent entanglement properties, are locally unitarily equivalent.

ER = 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.