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Holonomy for quantum channels
Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Quantum Chemistry.
Centre for Quantum Computation, DAMTP, Cambridge, UK.
Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Quantum Chemistry. (Quantum information theory)
2008 (English)In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 77, no 1, 012114- p.Article in journal (Refereed) Published
Abstract [en]

A quantum holonomy reflects the curvature of some underlying structure of quantum-mechanical systems, such as that associated with quantum states. Here, we extend the notion of holonomy to families of quantum channels, i.e., trace-preserving completely positive maps. By the use of the Jamiolkowski isomorphism, we show that the proposed channel holonomy is related to the Uhlmann holonomy. The general theory is illustrated for specific examples. We put forward a physical realization of the channel holonomy in terms of interferometry. This enables us to identify a gauge-invariant physical object that directly relates to the channel holonomy. Parallel transport condition and concomitant gauge structure are delineated in the case of smoothly parametrized families of channels. Finally, we point out that interferometer tests that have been carried out in the past to confirm the 4 pi rotation symmetry of the neutron spin can be viewed as early experimental realizations of the channel holonomy.

Place, publisher, year, edition, pages
2008. Vol. 77, no 1, 012114- p.
Keyword [en]
Geometric phase, quantum holonomy, completely positive maps, quantum channels, Uhlmann holonomy, mixed quantum states, quantum interferometry, Jamiolkowski isomorphism
National Category
Physical Sciences
Research subject
Physics
Identifiers
URN: urn:nbn:se:uu:diva-96152DOI: 10.1103/PhysRevA.77.012114ISI: 000252862000029OAI: oai:DiVA.org:uu-96152DiVA: diva2:170629
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.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 335
Keyword
Physics, Quantum Holonomy, Geometric Phase, Quantum Computation, Completely Positive Map, Mixed State, Interferometry, Fysik
Identifiers
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
Opponent
Supervisors
Available from: 2007-09-03 Created: 2007-09-03Bibliographically approved

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Sjöqvist, Erik

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