## Journal of Physical Mathematics

### Unitary braid matrices: bridge between topological and quantum entanglements

#### Abstract

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.

#### Article information

Source
J. Phys. Math., Volume 2 (2010), Article ID P100804, 14 pages.

Dates
First available in Project Euclid: 22 September 2011

https://projecteuclid.org/euclid.jpm/1316724049

Digital Object Identifier
doi:10.4303/jpm/P100804

Zentralblatt MATH identifier
1264.81238

Subjects
Primary: 86-08: Computational methods

#### Citation

Abdesselam, B.; Chakrabarti, A. Unitary braid matrices: bridge between topological and quantum entanglements. J. Phys. Math. 2 (2010), Article ID P100804, 14 pages. doi:10.4303/jpm/P100804. https://projecteuclid.org/euclid.jpm/1316724049

#### References

• B. Abdesselam and A. Chakrabarti. Multiparameter statistical models from $N^2\times N^2$ braid matrices: Explicit eigenvalues of transfer matrices ${\bf T}^{(r)}$, spin chains, factorizable scatterings for all $N$. Submitted to J. Phys. A: Math. Theor..
• B. Abdesselam, A. Chakrabarti, V. K. Dobrev, and S. G. Mihov. Higher dimensional multiparameter unitary and nonunitary braid matrices: even dimensions. J. Math. Phys., 48 (2007), 103505.
• –––. Higher dimensional unitary braid matrices: Construction, associated structures, and entanglements. J. Math. Phys., 48 (2007), 053508.
• H. A. Cartret and A. Sudbery. Local symmetry properties of pure three-qubit states. J. Phys. A: Math. Gen., 33 (2000), 4981–5002.
• A. Chakrabarti. Entangled states, Lorentz transformations and spin precession in magnetic fields. J. Phys. A: Math. Theor., 42 (2009), 245205.
• –––. On the coupling of 3 angular momenta. Ann. Inst. H. Poincaré, 1 (1964), 301–327.
• V. Coffman, J. Kundu, and W. K. Woothers. Distributed entanglement. Phys. Rev. A, 61 (2000), 052306.
• D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger. Bell's theorem without inequalities. Ann. J. Phys., 58 (1990), 1131–1143.
• E. Jung, M. Hwang, and D. Park. Three-tangle for rank-three mixed states: Mixture of Greenberger-Horne-Zeilinger, $W$, and flipped $W$-states. Phys. Rev. A, 79 (2009), 024306.
• L. H. Kauffman and S. J. Lomonaco Jr. Braiding operators are universal quantum gates. New J. Phys., 6 (2004), 134.
• J. M. Lévy-Leblond and M. Nahas. Symmetrical coupling of three angular momenta. J. Math. Phys., 6 (1965), 1372.
• Y. M. Zhang and M.-L. Ge. GHZ states, Almost-complex structure of Yang-Baxter equation. Quant. Inf. Proc., 6 (2007), 363–379. }