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 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.
Highlights
The third Reidemeister move in the theory of knots and links imposes equivalence between two specific sequences of over- and undercrossing of three braids
The equality sign imposes the essential Reidemeister constraint. This will be the link of our matrices to topological entanglement
We have presented before two quite distinct classes of unitary N 2 × N 2 braid matrices [2, 3], one real and for even N and the other complex, for all N
Summary
The superposition can be shown to imply entanglement On assuming it can be expressed as a product ( xi|xi ) ⊗ ( yi|yi ) ⊗ ( zi|zi ), one runs into contradictions. We will obtain explicitly the intrinsic 3-body entanglement (3 tangles) and the 2-body entanglements (2 tangles) of the three subsystems They will be expressed in terms of (f0, f1, f2, f3) of (2.6). It was noted [2, 3] that for all these parameters pure imaginary, this class corresponds to unitary braid matrices. Repeated actions of B with different parameters will modify the coefficients as follows:. For simplicity, we will restrict our study (Section 3) to the two sets of coefficients (2.6) and (2.12)
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