Physikalische Mathematik

Unitary braid matrices: bridge between topological and quantum entanglements

Abstract

B. Abdesellam and A. Chakrabati

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 en-tanglements (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 correspond-ing to over- and undercrossings of three braids and, on closing ends, to topologically entangled Borromean rings are shown, in another context, to generate quantum entan-glements. For higher dimensions, starting with di erent initial triplets one can entangle by turns, each state with all the rest. A speci c coupling of three angular momenta is brie y discussed to throw more light on three body entanglements.