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ZiF Cooperation Group

Finite Projective Ring Geometries: An Intriguing Emerging Link Between Quantum Information Theory, Black-Hole Physics and Chemistry of Coupling

August - October 2009

Convenors: Hans Havlicek (Wien, AUT), Metod Saniga (Tatranská Lomnica, SVK)

Michel Planat, Metod Saniga, Thomas Honold, Petr Pracna, Andrea Blunck, Hans Havlicek
(Photo: Thomas Balls-Thies)

The project focuses on finite geometries (in particular geometries over rings) and their role as a link and unifying platform between at first sight rather different domains of physics and chemistry.
Quantum information theory:
Here we focus our attention at two goals: On the one hand, we aim at extending our finite-ring-geometrical theory of the generalized Pauli group of a single qudit to the case of multiple qudits. On the other hand, we plan to ascertain which geometrical and combinatorial aspects of a projective ring geometry directly relate to entanglement in a given quantum system, and to which extent these entities can provide a quantitative measure for entanglement.
Chemistry of coupling:
We intend to put forward first rudimentary discrete models of chemical bonds that are fully devoid of divergences and degenerations exhibited by continuous models. As a subsequent step, we plan to explore the finest traits of hierarchical properties of chemically bonded systems, with a particular focus on a gradual disappearance of the boundary between individual building blocks, and, hence, of the dichotomy 'the object and its surroundings'.
Stringy black holes:
We would like to inspect which finite geometries underlie the (already known and yet to be unveiled) relations between black-hole entropies and entanglement invariants characterizing multi-qubit/qudit systems. A particular attention will be paid to the role of the Fano plane and its recently discovered non-unimodular 'Snowflake' generalizations, generalized polygons of small order, and biplanes of small order.

The main outcome of the project should be:

Fellows of the Cooperation Group:
Andrea Blunck (Hamburg), Péter Lévay (Budapest), Michel Planat (Besançon), Petr Pracna (Prag)

Associated Members:
Andrea Sanigová (Tatranská Lomnica), Michael Duff (Imperial College London, United Kingdom), Thomas Honold (Hangzhou, China), Alexander Kreuzer (University of Hamburg), Apostol Vourdas (University of Bradford, United Kingdom)