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Non-Abelian generalization of off-diagonal geometric phases
Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Quantum Chemistry. (Quantum Information Theory)
Centre for Quantum Computation, DAMTP, Univ. of Cambridge, UK.
Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Quantum Chemistry. (Quantum information theory)
2007 (English)In: Europhysics letters, ISSN 0295-5075, E-ISSN 1286-4854, Vol. 78, no 6, 60004- p.Article in journal (Refereed) Published
Abstract [en]

If a quantum system evolves in a noncyclic fashion the corresponding geometric phase or holonomy may not be fully defined. Off-diagonal geometric phases have been developed to deal with such cases. Here, we generalize these phases to the non-Abelian case, by introducing off-diagonal holonomies that involve evolution of more than one subspace of the underlying Hilbert space. Physical realizations of the off-diagonal holonomies in adiabatic evolution and interferometry are put forward.

Place, publisher, year, edition, pages
2007. Vol. 78, no 6, 60004- p.
National Category
Physical Sciences
Research subject
URN: urn:nbn:se:uu:diva-96150DOI: 10.1209/0295-5075/78/60004ISI: 000248593700004OAI: oai:DiVA.org:uu-96150DiVA: diva2:170627
Available from: 2007-09-03 Created: 2007-09-03 Last updated: 2011-10-06
In thesis
1. Quantum Holonomies: Concepts and Applications to Quantum Computing and Interferometry
Open this publication in new window or tab >>Quantum Holonomies: Concepts and Applications to Quantum Computing and Interferometry
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Quantum holonomies are investigated in different contexts.

A geometric phase is proposed for decomposition dependent evolution, where each component of a given decomposition of a mixed state evolves independently. It is shown that this geometric phase only depends on the path traversed in the space of decompositions.

A holonomy is associated to general paths of subspaces of a Hilbert space, both discrete and continuous. This opens up the possibility of constructing quantum holonomic gates in the open path setting. In the discrete case it is shown that it is possible to associate two distinct holonomies to a given path. Interferometric setups for measuring both holonomies are

provided. It is further shown that there are cases when the holonomy is only partially defined. This has no counterpart in the Abelian setting.

An operational interpretation of amplitudes of density operators is provided. This allows for a direct interferometric realization of Uhlmann's parallelity condition, and the possibility of measuring the Uhlmann holonomy for sequences of density operators.

Off-diagonal geometric phases are generalized to the non-Abelian case. These off-diagonal holonomies are undefined for cyclic evolution, but must contain members of non-zero rank if all standard holonomies are undefined. Experimental setups for measuring the off-diagonal holonomies are proposed.

The concept of nodal free geometric phases is introduced. These are constructed from gauge invariant quantities, but do not share the nodal point structure of geometric phases and off-diagonal geometric phases. An interferometric setup for measuring nodal free geometric phases is provided, and it is shown that these phases could be useful in geometric quantum computation.

A holonomy associated to a sequence of quantum maps is introduced. It is shown that this holonomy is related to the Uhlmann holonomy. Explicit examples are provided to illustrate the general idea.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2007. 66 p.
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 335
Physics, Quantum Holonomy, Geometric Phase, Quantum Computation, Completely Positive Map, Mixed State, Interferometry, Fysik
urn:nbn:se:uu:diva-8185 (URN)978-91-554-6955-9 (ISBN)
Public defence
2007-09-25, Häggsalen, Ångströmslaboratoriet, Lägerhyddsvägen 1, Uppsala, 10:00
Available from: 2007-09-03 Created: 2007-09-03Bibliographically approved

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Kult, DavidSjöqvist, Erik
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