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Exploring the Set of Quantum States
Stockholms universitet, Naturvetenskapliga fakulteten, Fysikum.
2007 (engelsk)Doktoravhandling, med artikler (Annet vitenskapelig)
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.

sted, utgiver, år, opplag, sider
Stockholm: Fysikum , 2007. , 82 s.
Emneord [en]
quantum mechanics, quantum states, density matrices, quantum information, geometry
HSV kategori
Forskningsprogram
teoretisk fysik
Identifikatorer
URN: urn:nbn:se:su:diva-6993ISBN: 978-91-7155-475-8 (tryckt)OAI: oai:DiVA.org:su-6993DiVA: diva2:197433
Disputas
2007-09-17, sal FB42, AlbaNova universitetscentrum, Roslagstullsbacken 21, Stockholm, 13:00
Opponent
Veileder
Tilgjengelig fra: 2007-08-15 Laget: 2007-08-15bibliografisk kontrollert
Delarbeid
1. Separability and the stella octangula
Åpne denne publikasjonen i ny fane eller vindu >>Separability and the stella octangula
2002 Inngår i: Physics Letters A, ISSN 0375-9601, Vol. 295, nr 5-6, 256-258 s.Artikkel i tidsskrift (Fagfellevurdert) Published
Identifikatorer
urn:nbn:se:su:diva-24387 (URN)
Merknad
Part of urn:nbn:se:su:diva-6993Tilgjengelig fra: 2007-08-15 Laget: 2007-08-15bibliografisk kontrollert
2. How to mix a density matrix
Åpne denne publikasjonen i ny fane eller vindu >>How to mix a density matrix
2003 Inngår i: Physical Review A, ISSN 1050-2947, Vol. 67, 012107- s.Artikkel i tidsskrift (Fagfellevurdert) Published
Identifikatorer
urn:nbn:se:su:diva-24388 (URN)
Merknad
Part of urn:nbn:se:su:diva-6993Tilgjengelig fra: 2007-08-15 Laget: 2007-08-15bibliografisk kontrollert
3. Birkhoff’s Polytope and Unistochastic Matrices, N = 3 and N = 4
Åpne denne publikasjonen i ny fane eller vindu >>Birkhoff’s Polytope and Unistochastic Matrices, N = 3 and N = 4
Vise andre…
2005 Inngår i: Communications in mathematical physics, ISSN 0010-3616, Vol. 259, nr 2, 307-324 s.Artikkel i tidsskrift (Fagfellevurdert) Published
Identifikatorer
urn:nbn:se:su:diva-24389 (URN)
Merknad
Part of urn:nbn:se:su:diva-6993Tilgjengelig fra: 2007-08-15 Laget: 2007-08-15bibliografisk kontrollert
4. Mutually unbiased bases and the complementarity polytope
Åpne denne publikasjonen i ny fane eller vindu >>Mutually unbiased bases and the complementarity polytope
2005 Inngår i: Open systems & information dynamics, ISSN 1230-1612, Vol. 12, 107-120 s.Artikkel i tidsskrift (Fagfellevurdert) Published
Identifikatorer
urn:nbn:se:su:diva-24390 (URN)
Merknad
Part of urn:nbn:se:su:diva-6993Tilgjengelig fra: 2007-08-15 Laget: 2007-08-15bibliografisk kontrollert
5. Geodesics and the best measurement for distinguishing quantum states
Åpne denne publikasjonen i ny fane eller vindu >>Geodesics and the best measurement for distinguishing quantum states
2005 Inngår i: Journal of physics. A, Mathematical and general, ISSN 0305-4470, Vol. 38, L725-L730 s.Artikkel i tidsskrift (Fagfellevurdert) Published
Identifikatorer
urn:nbn:se:su:diva-24391 (URN)
Merknad
Part of urn:nbn:se:su:diva-6993Tilgjengelig fra: 2007-08-15 Laget: 2007-08-15bibliografisk kontrollert
6. Mutually unbiased bases and Hadamard matrices of order six
Åpne denne publikasjonen i ny fane eller vindu >>Mutually unbiased bases and Hadamard matrices of order six
Vise andre…
2007 (engelsk)Inngår i: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 48, nr 5, 052106- s.Artikkel i tidsskrift (Fagfellevurdert) 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.

HSV kategori
Identifikatorer
urn:nbn:se:su:diva-24392 (URN)10.1063/1.2716990 (DOI)000246892400006 ()
Tilgjengelig fra: 2007-08-15 Laget: 2007-08-15 Sist oppdatert: 2013-05-02bibliografisk kontrollert

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