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arXiv:math/0702188 · Journal of Mathematical Physics

Higher dimensional unitary braid matrices: Construction, associated structures, and entanglements

B. Abdesselam, Amitabha Chakrabarti, V. K. Dobrev, S. G. Mihov

Published May 1, 2007• 21 citations
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Abstract

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.

DOI
10.1063/1.2737266

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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.
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.
In this paper, we present reducible representation of the n2 braid group representation which is constructed on the tensor product of n-dimensional spaces. Specifically, it is shown that via a combining method, we can construct more n2 dimensional braiding S-matrices which satisfy the braid relations. By Yang–Baxterization approach, we derive a 9 × 9 unitary [Formula: see text]-matrix according to a 9 × 9 braiding S-matrix we have constructed. The entanglement properties of [Formula: see text]-matrix is investigated, and the arbitrary degree of entanglement for two-qutrit entangled states can be generated via [Formula: see text]-matrix acting on the standard basis.
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 (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.
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.
A basis of N2 projectors, each an N2×N2 matrix with constant elements, is implemented to construct a class of braid matrices R̂(θ), θ being the spectral parameter. Only odd values of N are considered here. Our ansatz for the projectors Pα appearing in the spectral decomposition of R̂(θ) leads to exponentials exp(mαθ) as the coefficient of Pα. The sums and differences of such exponentials on the diagonal and the antidiagonal, respectively, provide the (2N2−1) nonzero elements of R̂(θ). One element at the center is normalized to unity. A class of supplementary constraints imposed by the braid equation leaves 12(N+3)(N−1) free parameters mα. The diagonalizer of R̂(θ) is presented for all N. Transfer matrices t(θ) and L(θ) operators corresponding to our R̂(θ) are studied. Our diagonalizer signals specific combinations of the components of the operators that lead to a quadratic algebra of N2 constant N×N matrices. The θ dependence factors out for such combinations. R̂(θ) is developed in a power series in θ. The basic difference arising for even dimensions is made explicit. Some special features of our R̂(θ) are discussed in a concluding section.