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 -