Differential Geometry and Geometric Structures

Main Page | Research Areas | Geometry and Quantum Theory

Geometry and Quantum Theory

Presently, we exhibit finite geometries which describe the commutation relations between generalised Pauli operators and their Kronecker products.

Astronomical Institute

Astronomical Institute, Slovak Academy of Sciences

Finite Geometries Behind Hilbert Spaces

Slovak head of project: Metod Saniga
Astronomický ústav
Slovenská akadémia vied
Tatranská Lomnica
Slovakia

Austrian head of project: Hans Havlicek

Action Slovakia-Austria

Supported by the Slovak and Austrian Governments
(Aktion Slowakei - Österreich), project number 58s2, 2007-2008.

Finite Projective Ring Geometries:

An Intriguing Emerging Link Between Quantum Information Theory, Black-Hole Physics, and Chemistry of Coupling

Cooperation group at the Centre for Interdisciplinary Research, University of Bielefeld (Germany)

August 1st to October 31st, 2009.

Principal Organiser: Hans Havlicek, Vienna University of Technology, Austria
Co-Organiser: Metod Saniga, Astronomical Institute of the Slovak Academy of Sciences, Tatranská Lomnica, Slovakia

Further Information

ZiF Cooperation Group 2009

Finite Ring Geometries: Where Qudits meet Black Holes

Slovak head of project: Metod Saniga, Astronomický ústav, Slovenská akadémia vied, Tatranská Lomnica, Slovakia

Austrian participants: Hans Havlicek (head) and Boris Odehnal.

Supported under the Slovak-Austrian Science and Technology Cooperation Agreement under grants SK 07-2009 (Austrian side) and SK-AT-0001-08 (Slovak side), 2009-2010.

Doily

The smallest generalised quadrangle

Two-qubits and the smallest generalised quadrangle

M. Saniga and M. Planat discovered in 2006 that completely new vistas open up if, instead of dealing with a given Hilbert space itself, one considers the associated space of generalised Pauli operators/matrices. They demonstrated that the operators' space characterising the simplest non-trivial systems, so-called two-qubits, is isomorphic to the generalised quadrangle of order two.

The the generalised quadrangle of order two is a point-line geometry with 15 points and 15 lines. Each point is on 3 lines and, dually, also each line is incident with 3 points. Collinear points correspond to commuting operators.

Generalisations

Some steps to generalise the above Saniga-Planat Theorem have already been taken, thereby establishing a relationship to symplectic polar spaces (K. Thas).

In our ongoing research we aim at further generalisations of this Theorem to systems of N-qudits for arbitrary numbers N and d. Here the projective line over the ring of integers modulo d plays a prominent role.

Publications

H. Havlicek and K. Svozil: Density conditions for quantum propositions, J. Math. Phys. 37 (11) (1996), 5337-5341.
Preprint (PDF)
H. Havlicek, G. Krenn, J. Summhammer, and K. Svozil: Colouring the rational quantum sphere and the Kochen-Specker theorem, J. Phys. A 34 (2001), 3071-3077.
Preprint (PDF)
M. Saniga, M. Planat, P. Pracna, and H. Havlicek: The Veldkamp-space of two-qubits, SIGMA Symmetry Integrability Geom. Methods Appl. 3 (2007), paper 075, 7 pp. (electronic).
Preprint (PDF)
H. Havlicek and M. Saniga: Projective ring line of a specific qudit, J. Phys. A 40 (2007), F943-F952.
Preprint (PDF)
H. Havlicek and M. Saniga: Projective ring line of an arbitrary single qudit, J. Phys. A 41 (2008), Article ID 015302, 12 pp.
Preprint (PDF)
M. Saniga, H. Havlicek, M. Planat, and P. Pracna: "Twin Fano-Snowflakes" over the smallest ring of ternions, SIGMA Symmetry Integrability Geom. Methods Appl. 4 (2008), paper 050, 7 pp. (electronic).
Preprint (PDF)
H. Havlicek and M. Saniga: Vectors, cyclic submodules, and projective spaces linked with ternions, J. Geom. 92 (2009), 79-90.
Preprint (PDF)
H. Havlicek, B. Odehnal, and M. Saniga: Factor-group-generated polar spaces and (multi-)qudits, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), paper 096, 15 pp. (electronic).
Preprint (PDF) | Article (PDF, external link)
H. Havlicek, B. Odehnal, and M. Saniga: Möbius pairs of simplices and commuting Pauli operators, Math. Pannonica 21 (2010), 115-128.
Preprint (PDF)