TY - JOUR

T1 - Permutation and its partial transpose

AU - Zhang, Yong

AU - Kauffman, Louis H.

AU - Werner, Reinhard F.

N1 - Funding information: For L. H. Kauffman, most of this effort was sponsored by the Defense Advanced Research Projects Agency (DARPA) and Air Force Research Laboratory, Air Force Material Command, USAF, under agreement F30602-01-2-05022. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright annotations thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Defense Advanced Research Projects Agency, the Air Force Research Laboratory, or the U.S. Government. (Copyright 2006.) It gives L. H. Kauffman great pleasure to acknowledge support from NSF Grant DMS-0245588. Y. Zhang thanks X. Y. Li and X. Q. Li-Jost for encouragements and supports, M. L. Ge for fruitful collaborations and stimulating discussions, and the Mathematisches Forschungsinstitut Oberwolfach for the hospitality during his stay. This work is in part supported by NSFC–10447134 and SRF for ROCS, SEM.

PY - 2007

Y1 - 2007

N2 - Permutation and its partial transpose play important roles in quantum information theory. The Werner state is recognized as a rational solution of the Yang-Baxter equation, and the isotropic state with an adjustable parameter is found to form a braid representation. The set of permutation's partial transposes is an algebra called the PPT algebra, which guides the construction of multipartite symmetric states. The virtual knot theory, having permutation as a virtual crossing, provides a topological language describing quantum computation as having permutation as a swap gate. In this paper, permutation's partial transpose is identified with an idempotent of the Temperley-Lieb algebra. The algebra generated by permutation and its partial transpose is found to be the Brauer algebra. The linear combinations of identity, permutation and its partial transpose can form various projectors describing tangles; braid representations; virtual braid representations underlying common solutions of the braid relation and Yang-Baxter equations; and virtual Temperley-Lieb algebra which is articulated from the graphical viewpoint. They lead to our drawing a picture called the ABPK diagram describing knot theory in terms of its corresponding algebra, braid group and polynomial invariant. The paper also identifies non-trivial unitary braid representations with universal quantum gates, derives a Hamiltonian to determine the evolution of a universal quantum gate, and further computes the Markov trace in terms of a universal quantum gate for a link invariant to detect linking numbers.

AB - Permutation and its partial transpose play important roles in quantum information theory. The Werner state is recognized as a rational solution of the Yang-Baxter equation, and the isotropic state with an adjustable parameter is found to form a braid representation. The set of permutation's partial transposes is an algebra called the PPT algebra, which guides the construction of multipartite symmetric states. The virtual knot theory, having permutation as a virtual crossing, provides a topological language describing quantum computation as having permutation as a swap gate. In this paper, permutation's partial transpose is identified with an idempotent of the Temperley-Lieb algebra. The algebra generated by permutation and its partial transpose is found to be the Brauer algebra. The linear combinations of identity, permutation and its partial transpose can form various projectors describing tangles; braid representations; virtual braid representations underlying common solutions of the braid relation and Yang-Baxter equations; and virtual Temperley-Lieb algebra which is articulated from the graphical viewpoint. They lead to our drawing a picture called the ABPK diagram describing knot theory in terms of its corresponding algebra, braid group and polynomial invariant. The paper also identifies non-trivial unitary braid representations with universal quantum gates, derives a Hamiltonian to determine the evolution of a universal quantum gate, and further computes the Markov trace in terms of a universal quantum gate for a link invariant to detect linking numbers.

U2 - 10.1142/S021974990700302X

DO - 10.1142/S021974990700302X

M3 - Article

VL - 5

SP - 469

EP - 507

JO - Int. J. Quant. Inf.

JF - Int. J. Quant. Inf.

IS - 4

ER -