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arXiv:1404.4897 · Europhysics Letters (EPL)

Multipartite d-level GHZ bases associated with generalized braid matrices

Gangcheng Wang, Chunfang Sun, Chunfeng Wu, Бо Лю, Yan Zhang, Kang Xue

Published Sep 23, 2014• 10 citations
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Abstract

We investigate the generalized braid relation ($d-$level $N-$body braid relation) and its application to quantum entanglement. By means of finite-dimensional representations of Heisenberg-Weyl algebra, a set of $d^{N}\times d^{N}$ unitary matrix representations satisfying the generalized braid relation can be constructed. Such generalized braid matrices can entangle $d-$level $N-$partite quantum states. Acting the generalized braid matrices on the standard basis, one can obtain a set of maximally entangled basis. Further study shows that such entangled basis can be viewed as the $d-$level $N-$partite Greenberger-Horne-Zeilinger (GHZ) basis.

DOI
10.1209/0295-5075/108/10001

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We investigate the generalized braid relation (d−d-d−level N−N-N−body braid relation) and its application to quantum entanglement. By means of finite-dimensional representations of Heisenberg-Weyl algebra, a set of dN×dNd^{N}\times d^{N}dN×dN unitary matrix representations satisfying the generalized braid relation can be constructed. Such generalized braid matrices can entangle d−d-d−level N−N-N−partite quantum states. Acting the generalized braid matrices on the standard basis, one can obtain a set of maximally entangled basis. Further study shows that such entangled basis can be viewed as the d−d-d−level N−N-N−partite Greenberger-Horne-Zeilinger (GHZ) basis.
Important developments in fault-tolerant quantum computation using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauffman and Lomonaco that a specific braiding operator from the solution of the Yang-Baxter equation, namely the Bell matrix, is universal implies that in principle all quantum gates can be constructed from braiding operators together with single qubit gates. In this paper we present a new class of braiding operators from the Temperley-Lieb algebra that generalizes the Bell matrix to multi-qubit systems, thus unifying the Hadamard and Bell matrices within the same framework. Unlike previous braiding operators, these new operators generate directly , from separable basis states, important entangled states such as the generalized Greenberger-Horne-Zeilinger states, cluster-like states, and other states with varying degrees of entanglement.
Using a braid group representation based on the Temperley-Lieb algebra, we construct braid quantum gates that could generate entangled n -partite D -level qudit states. D different sets of D n × D n unitary representation of the braid group generators are presented. With these generators the desired braid quantum gates are obtained. We show that the generalized GHZ states, which are maximally entangled states, can be obtained directly from these braid quantum gates without resorting to further local unitary transformations. We also point out an interesting observation, namely for a general multi-qudit state there exists a unitary braid quantum gate based on the Temperley-Lieb algebra that connects it from one of its component basis states, if the coefficient of the component state is such that the square of its norm is no less than 1/4.
Important developments in fault-tolerant quantum computation using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauffman and Lomonaco that a specific braiding operator from the solution of the Yang-Baxter equation, namely the Bell matrix, is universal implies that in principle all quantum gates can be constructed from braiding operators together with single qubit gates. In this paper we present a new class of braiding operators from the Temperley-Lieb algebra that generalizes the Bell matrix to multi-qubit systems, thus unifying the Hadamard and Bell matrices within the same framework. Unlike previous braiding operators, these new operators generate directly from separable basis states important entangled states such as the generalized Greenberger-Horne-Zeilinger states, cluster-like states, and other states with varying degrees of entanglement.
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.
We find an algebraic formula for the NNN-partite concurrence of NNN qubits in an XXX matrix. XXX matrices are density matrices whose only nonzero elements are diagonal or antidiagonal when written in an orthonormal basis. We use our formula to study the dynamics of the NNN-partite entanglement of NNN remote qubits in generalized NNN-party Greenberger-Horne-Zeilinger (GHZ) states. We study the case in which each qubit interacts with a local amplitude damping channel. It is shown that only one type of GHZ state loses its entanglement in finite time; for the rest, NNN-partite entanglement dies out asymptotically. Algebraic formulas for the entanglement dynamics are given in both cases. We directly confirm that the half-life of the entanglement is proportional to the inverse of NNN. When entanglement vanishes in finite time, the time at which entanglement vanishes can decrease or increase with NNN depending on the initial state. In the macroscopic limit, this time is independent of the initial 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.