This paper presents the geometric setting of quantum variational principles and extends it to comprise the interaction between classical and quantum degrees of freedom. Euler-Poincaré reduction theory is applied to the Schrödinger, Heisenberg, and Wigner-Moyal dynamics of pure states. This construction leads to new variational principles for the description of mixed quantum states. The corresponding momentum map properties are presented as they arise from the underlying unitary symmetries. Finally, certain semidirect-product group structures are shown to produce new variational principles for Dirac’s interaction picture and the equations of hybrid classical-quantum dynamics.
REFERENCES
1.
Y.
Aharonov
and J.
Anandan
, “Phase change during a cyclic quantum evolution
,” Phys. Rev. Lett.
58
(16
), 1593
(1987
).2.
J.
Anandan
, “Non-adiabatic non-abelian geometric phase
,” Phys. Lett. A
133
(4
), 171
–175
(1988
).3.
J.
Anandan
, “A geometric approach to quantum mechanics
,” Foundations Phys.
21
(11
), 1265
–1284
(1991
).4.
J.
Anandan
and Y.
Aharonov
, “Geometry of quantum evolution
,” Phys. Rev. Lett.
65
, 1697
–1700
(1990
).5.
O.
Andersson
and H.
Heydari
, “Dynamic distance measure on spaces of isospectral mixed quantum states
,” Entropy
15
(9
), 3688
–3697
(2013
).6.
A.
Ashtekar
and T. A.
Schilling
, “Geometry of quantum mechanics
,” AIP Conf. Proc.
342
(1
), 471
(1995
).7.
E.
Bonet-Luz
and C.
Tronci
, “Hamiltonian approach to the dynamics of Ehrenfest expectation values and Gaussian quantum states
,” preprint arXiv:1507.02607 (2015
).8.
L. J.
Boya
, J.
Cariñena
, and J.
Gracia-Bondía
, “Symplectic structure of the Aharonov-Anandan geometric phase
,” Phys. Lett. A
161
(1
), 30
–34
(1991
).9.
D. C.
Brody
and L. P.
Hughston
, “Geometric quantum mechanics
,” J. Geom. Phys.
38
(1
), 19
–53
(2001
).10.
R.
Car
and M.
Parrinello
, “Unified approach for molecular dynamics and density-functional theory
,” Phys. Rev. Lett.
55
(22
), 2471
(1985
).11.
A.
Carlini
, A.
Hosoya
, T.
Koike
, and Y.
Okudaira
, “Time-optimal quantum evolution
,” Phys. Rev. Lett.
96
, 060503
(2006
).12.
A.
Carlini
, A.
Hosoya
, T.
Koike
, and Y.
Okudaira
, “Time-optimal unitary operations
,” Phys. Rev. A
75
, 042308
(2007
).13.
H.
Cendra
, A.
Ibort
, and J.
Marsden
, “Variational principles on principal fiber bundles: A geometry theory of Clebsch potentials and Lin constraints
,” J. Geom. Phys.
4
(2
), 183
–205
(1987
).14.
H.
Cendra
and J. E.
Marsden
, “Lin constraints, Clebsch potentials and variational principles
,” Phys. D
27
(1
), 63
–89
(1987
).15.
H.
Cendra
, J. E.
Marsden
, and T. S.
Ratiu
, Lagrangian Reduction by Stages
(American Mathematical Society
, 2001
), Vol. 722
.16.
P.
Chernoff
and J.
Marsden
, “Lagrangian systems
,” in Properties of Infinite Dimensional Hamiltonian Systems
(Springer Berlin Heidelberg, Springer-Verlag
, 1974
), pp. 100
–124
.17.
E.
Chiumiento
and M.
Melgaard
, “Stiefel and Grassmann manifolds in quantum chemistry
,” J. Geom. Phys.
62
(8
), 1866
–1881
(2012
).18.
D.
Chruściński
, “Symplectic structure for the non-abelian geometric phase
,” Phys. Lett. A
186
(1
), 1
–4
(1994
).19.
J.
Clemente-Gallardo
and G.
Marmo
, “Basics of quantum mechanics, geometrization and some applications to quantum information
,” Int. J. Geom. Methods Mod. Phys.
5
, 989
(2008
).20.
D.
D’Alessandro
and M.
Dahleh
, “Optimal control of two-level quantum systems
,” IEEE Trans. Autom. Control
46
(6
), 866
–876
(2001
).21.
P. A. M.
Dirac
, “Note on exchange phenomena in the Thomas atom
,” in Mathematical Proceedings of the Cambridge Philosophical Society
(Cambridge University Press
, 1930
), Vol. 26
, pp. 376
–385
.22.
P.
Facchi
, R.
Kulkarni
, V. I.
Man’ko
, G.
Marmo
, E. C. G.
Sudarshan
, and F.
Ventriglia
, “Classical and quantum fisher information in the geometrical formulation of quantum mechanics
,” Phys. Lett. A
374
(48
), 4801
–4803
(2010
).23.
24.
F.
Gay-Balmaz
and C.
Tronci
, “Reduction theory for symmetry breaking with applications to nematic systems
,” Phys. D
239
(20
), 1929
–1947
(2010
).25.
A. N.
Grigorenko
, “Geometry of projective hilbert space
,” Phys. Rev. A
46
, 7292
–7294
(1992
).26.
U.
Güngördü
, Y.
Wan
, and M.
Nakahara
, “Non-adiabatic universal holonomic quantum gates based on abelian holonomies
,” J. Phys. Soc. Jpn.
83
(3
), 034001
(2014
).27.
E. J.
Heller
, “Time dependent variational approach to semiclassical dynamics
,” J. Chem. Phys.
64
(1
), 63
–73
(1976
).28.
D. D.
Holm
, B. A.
Kupershmidt
, and C. D.
Levermore
, “Canonical maps between poisson brackets in Eulerian and Lagrangian descriptions of continuum mechanics
,” Phys. Lett. A
98
(8
), 389
–395
(1983
).29.
D. D.
Holm
, J. E.
Marsden
, and T. S.
Ratiu
, “The Euler–Poincaré equations and semidirect products with applications to continuum theories
,” Adv. Math.
137
(1
), 1
–81
(1998
).30.
D. D.
Holm
, T.
Schmah
, C.
Stoica
, and D. C. P.
Ellis
, Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions
(Oxford University Press
, London
, 2009
), Vol. 12
.31.
N.
Khaneja
, R.
Brockett
, and S. J.
Glaser
, “Time optimal control in spin systems
,” Phys. Rev. A
63
, 032308
(2001
).32.
T. W. B.
Kibble
, “Geometrization of quantum mechanics
,” Commun. Math. Phys.
65
(2
), 189
–201
(1979
).33.
P. H.
Kramer
and M.
Saraceno
, Geometry of the Time-dependent Variational Principle in Quantum Mechanics
(Springer
, 1981
).34.
R. G.
Littlejohn
, “The semiclassical evolution of wave packets
,” Phys. Rep.
138
(4
), 193
–291
(1986
).35.
P-O
Löwdin
and P. K.
Mukherjee
, “Some comments on the time-dependent variation principle
,” Chem. Phys. Lett.
14
(1
), 1
–7
(1972
).36.
D.
Lucarelli
, “Control aspects of holonomic quantum computation
,” J. Math. Phys.
46
(5
), 052103
(2005
).37.
J. E.
Marsden
and T. S.
Ratiu
, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems
(Springer
, 1999
), Vol. 17
.38.
A.
Miyake
and M.
Wadati
, “Geometric strategy for the optimal quantum search
,” Phys. Rev. A
64
, 042317
(2001
).39.
R.
Montgomery
, “Heisenberg and isoholonomic inequalities
,” in Symplectic Geometry and Mathematical Physics
, edited by P.
Donato
, et al (Birkhäuser
, Boston
, 1991
), pp. 303
–325
.40.
J. E.
Moyal
, “Quantum mechanics as a statistical theory
,” in Mathematical Proceedings of the Cambridge Philosophical Society
(Cambridge University Press
, 1949
), Vol. 45
, pp. 99
–124
.41.
K.
Ohta
, “Time-dependent variational principle with constraints
,” Chem. Phys. Lett.
329
(3
), 248
–254
(2000
).42.
A. M.
Perelomov
, “Coherent states for arbitrary lie group
,” Commun. Math. Phys.
26
(3
), 222
–236
(1972
).43.
J. A.
Poulsen
, “A variational principle in Wigner phase-space with applications to statistical mechanics
,” J. Chem. Phys.
134
(3
), 034118
(2011
).44.
L. L.
Salcedo
, “Statistical consistency of quantum-classical hybrids
,” Phys. Rev. A
85
(2
), 022127
(2012
).45.
A.
Sawicki
, A.
Huckleberry
, and M.
Kuś
, “Symplectic geometry of entanglement
,” Commun. Math. Phys.
305
(2
), 441
–468
(2011
).46.
S.
Tanimura
, M.
Nakahara
, and D.
Hayashi
, “Exact solutions of the isoholonomic problem and the optimal control problem in holonomic quantum computation
,” J. Math. Phys.
46
, 022101
(2005
).47.
C. J.
Trahan
and R. E.
Wyatt
, Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics
, Interdisciplinary Applied Mathematics
(Springer
, 2006
).48.
A.
Uhlmann
and B.
Crell
, “Geometry of state spaces
,” in Entanglement and Decoherence
(Springer
, 2009
), pp. 1
–60
.49.
E.
Wigner
, “On the quantum correction for thermodynamic equilibrium
,” Phys. Rev.
40
(5
), 749
(1932
).50.
C.
Zachos
, D.
Fairlie
, and T.
Curtright
, Quantum Mechanics in Phase Space: An Overview with Selected Papers
(World Scientific
, 2005
), Vol. 34
.51.
L.-J.
Zhao
, Y.-S.
Li
, L.
Hao
, T.
Zhou
, and G. L.
Long
, “Geometric pictures for quantum search algorithms
,” Quantum Inf. Process.
11
(2
), 325
–340
(2012
).© 2015 AIP Publishing LLC.
2015
AIP Publishing LLC
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