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:

- a virtually complete finite-geometrical theory of the commutation algebra of the generalized Pauli groups associated with arbitrary multiple qudits;
- a fundamental finite-geometrical insight into the intricacies of chemical couplings and hierarchical systems of molecules;
- a deep observable-based understanding of the so-called 'black hole analogy' and a wealth of black-hole clues for understanding the quantum entanglement in geometrical terms.

**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)