We study quantum entanglements induced on product states by the action of 8-vertex braid matrices, rendered unitary with purely imaginary spectral parameters (rapidity).The unitarity is displayed via the 'canonical factorization' of the coefficients of the projectors spanning the basis.This adds one more new facet to the famous and fascinating features of the 8vertex model.The double periodicity and the analytic properties of the elliptic functions involved lead to a rich structure of the 3-tangle quantifying the entanglement.We thus explore the complex relationship between topological and quantum entanglement.
We study quantum entanglements induced on product states by the action of 8-vertex braid matrices, rendered unitary with purely imaginary spectral parameters (rapidity).The unitarity is displayed via the 'canonical factorization' of the coefficients of the projectors spanning the basis.This adds one more new facet to the famous and fascinating features of the 8vertex model.The double periodicity and the analytic properties of the elliptic functions involved lead to a rich structure of the 3-tangle quantifying the entanglement.We thus explore the complex relationship between topological and quantum entanglement.
We construct (2n)2×(2n)2 unitary braid matrices R̂ for n⩾2 generalizing the class known for n=1. A set of (2n)×(2n) matrices (I,J,K,L) is defined. R̂ is expressed in terms of their tensor products (such as K⊗J), leading to a canonical formulation for all n. Complex projectors P± provide a basis for our real, unitary R̂. Baxterization is obtained. Diagonalizations and block diagonalizations are presented. The loss of braid property when R̂ (n>1) is block diagonalized in terms of R̂ (n=1) is pointed out and explained. For odd dimension (2n+1)2×(2n+1)2, a previously constructed braid matrix is complexified to obtain unitarity. R̂LL and R̂TT algebras, chain Hamiltonians, potentials for factorizable S matrices, and complex noncommutative spaces are all studied briefly in the context of our unitary braid matrices. Turaev construction of link invariants is formulated for our case. We conclude with comments concerning 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.
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.
Braid matrices R̂(θ), corresponding to vector representations, are spectrally decomposed obtaining a ratio fi(θ)/fi(−θ) for the coefficient of each projector Pi appearing in the decomposition. This directly yields a factorization (F(−θ))−1F(θ) for the braid matrix, implying also the relation R̂(−θ)R̂(θ)=I. This is achieved for GLq(n), SOq(2n+1), SOq(2n), Spq(2n) for all n and also for various other interesting cases including the eight-vertex matrix. We explain how the limits θ→±∞ can be interpreted to provide factorizations of the standard (non-Baxterized) braid matrices. A systematic approach to diagonalization of projectors and hence of braid matrices is presented with explicit constructions for GLq(2), GLq(3), SOq(3), SOq(4), Spq(4) and various other cases such as the eight-vertex one. For a specific nested sequence of projectors diagonalization is obtained for all dimensions. The canonical property implemented in the diagonalizers is mutual orthogonality of the rows. In each factor F(θ) our diagonalizer again factors out all dependence on the spectral parameter θ as a diagonal matrix. Applications of our formalism to the construction of L-operators and transfer matrices are indicated. In an Appendix our type of factorization is compared to another one proposed by other authors.
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.
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.