TY - JOUR

T1 - On Uncertainty, Braiding and Entanglement in Geometric Quantum Mechanics

AU - Benvegnu', Alberto

AU - Spera, Mauro

PY - 2006

Y1 - 2006

N2 - 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.

AB - 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.

KW - Artin’s braid group

KW - Geometric quantum mechanics

KW - Jacobi fields

KW - Segre and Veronese maps

KW - elliptic curves

KW - links

KW - quantum entanglement

KW - Artin’s braid group

KW - Geometric quantum mechanics

KW - Jacobi fields

KW - Segre and Veronese maps

KW - elliptic curves

KW - links

KW - quantum entanglement

UR - http://hdl.handle.net/10807/35684

M3 - Article

SN - 0129-055X

VL - 18

SP - 1075

EP - 1102

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

ER -