On Uncertainty, Braiding and Entanglement in Geometric Quantum Mechanics

Acting within the framework of geometric quantum mechanics, an interpretation of quantum uncertainty is discussed in terms of Jacobi fields, and a connection with the theory of elliptic curves is outlined, via classical integrability of Schr¨odinger’s dynamics and the cross-ratio interpretation of quantum transition probabilities. Furthermore, a thoroughly geometrical construction of all special unitary representations of the 3-strand braid group on the quantum 1-qubit space is given, and the connection of one of them with elliptic curves admitting complex multiplication automorphisms — the physically relevant one corresponding to the anharmonic ratio — is shown. Also, contact is made with the Temperley–Lieb algebra theoretic constructions of Kauffman and Lomonaco, and it is shown that the standard trace relative to one of the above representations computes the Jones polynomial for particular values of the parameter, for knots arising as closures of 3-strand braids. Subsequently, a geometric entanglement criterion (in terms of Segre embeddings) is discussed, together with a projective geometrical portrait for quantum 2-gates. Finally, Aravind’s idea of describing quantum states via knot theory is critically analyzed, and a geometrical picture — involving a blend of SU(2)- representation theory, classical projective geometry, binary trees and Brunnian and Hopf links — is set up in order to describe successive measurements made upon generalized GHZ states, close in spirit to the quantum knot picture again devised by Kauffman and Lomonaco.