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Exploring the Set of Quantum States
Stockholm University, Faculty of Science, Department of Physics.
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Quantum mechanical properties of finite dimensional quantum systems are used within the field of quantum information. In this thesis the set of states (density matrices) for such systems is studied and described, largely in geometrical terms. The introductory part also acquaints the reader with relevant background about majorization, bistochastic matrices, mutually unbiased bases, Hadamard matrices and entanglement, with the aim to make the papers attached easier to read.

Paper I considers Peres' criterion for separability, for two qubit states. Paper II deals with the problem of how density matrices can be mixed from pure states, especially what probability distributions over pure states that are possible. In Paper III the set of bistochastic matrices–Birkhoff's polytope–and the subset of unistochastic matrices is studied, with a detailed description in dimensions 3 and 4. In Paper IV it is seen how the states of a complete set of mutually unbiased bases form a polytope in the set of density matrices, with certain combinatorial properties. A search for mutually unbiased bases in dimension 6 is presented in Paper VI, which includes a thorough discussion on 6 by 6 Hadamard matrices. Paper V presents a result about geodesics in the set of quantum states with respect to the curved Bures-Uhlmann geometry.

Place, publisher, year, edition, pages
Stockholm: Fysikum , 2007. , 82 p.
Keyword [en]
quantum mechanics, quantum states, density matrices, quantum information, geometry
National Category
Physical Sciences
Research subject
Theoretical Physics
Identifiers
URN: urn:nbn:se:su:diva-6993ISBN: 978-91-7155-475-8 (print)OAI: oai:DiVA.org:su-6993DiVA: diva2:197433
Public defence
2007-09-17, sal FB42, AlbaNova universitetscentrum, Roslagstullsbacken 21, Stockholm, 13:00
Opponent
Supervisors
Available from: 2007-08-15 Created: 2007-08-15Bibliographically approved
List of papers
1. Separability and the stella octangula
Open this publication in new window or tab >>Separability and the stella octangula
2002 In: Physics Letters A, ISSN 0375-9601, Vol. 295, no 5-6, 256-258 p.Article in journal (Refereed) Published
Identifiers
urn:nbn:se:su:diva-24387 (URN)
Note
Part of urn:nbn:se:su:diva-6993Available from: 2007-08-15 Created: 2007-08-15Bibliographically approved
2. How to mix a density matrix
Open this publication in new window or tab >>How to mix a density matrix
2003 In: Physical Review A, ISSN 1050-2947, Vol. 67, 012107- p.Article in journal (Refereed) Published
Identifiers
urn:nbn:se:su:diva-24388 (URN)
Note
Part of urn:nbn:se:su:diva-6993Available from: 2007-08-15 Created: 2007-08-15Bibliographically approved
3. Birkhoff’s Polytope and Unistochastic Matrices, N = 3 and N = 4
Open this publication in new window or tab >>Birkhoff’s Polytope and Unistochastic Matrices, N = 3 and N = 4
Show others...
2005 In: Communications in mathematical physics, ISSN 0010-3616, Vol. 259, no 2, 307-324 p.Article in journal (Refereed) Published
Identifiers
urn:nbn:se:su:diva-24389 (URN)
Note
Part of urn:nbn:se:su:diva-6993Available from: 2007-08-15 Created: 2007-08-15Bibliographically approved
4. Mutually unbiased bases and the complementarity polytope
Open this publication in new window or tab >>Mutually unbiased bases and the complementarity polytope
2005 In: Open systems & information dynamics, ISSN 1230-1612, Vol. 12, 107-120 p.Article in journal (Refereed) Published
Identifiers
urn:nbn:se:su:diva-24390 (URN)
Note
Part of urn:nbn:se:su:diva-6993Available from: 2007-08-15 Created: 2007-08-15Bibliographically approved
5. Geodesics and the best measurement for distinguishing quantum states
Open this publication in new window or tab >>Geodesics and the best measurement for distinguishing quantum states
2005 In: Journal of physics. A, Mathematical and general, ISSN 0305-4470, Vol. 38, L725-L730 p.Article in journal (Refereed) Published
Identifiers
urn:nbn:se:su:diva-24391 (URN)
Note
Part of urn:nbn:se:su:diva-6993Available from: 2007-08-15 Created: 2007-08-15Bibliographically approved
6. Mutually unbiased bases and Hadamard matrices of order six
Open this publication in new window or tab >>Mutually unbiased bases and Hadamard matrices of order six
Show others...
2007 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 48, no 5, 052106- p.Article in journal (Refereed) Published
Abstract [en]

We report on a search for mutually unbiased bases (MUBs) in six dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organize our results. Finally, we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.

National Category
Physical Sciences
Identifiers
urn:nbn:se:su:diva-24392 (URN)10.1063/1.2716990 (DOI)000246892400006 ()
Available from: 2007-08-15 Created: 2007-08-15 Last updated: 2013-05-02Bibliographically approved

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