An Intensive Mini-Workshop on

FINITE PROJECTIVE GEOMETRIES IN QUANTUM THEORY

August 1 – 4, 2007 / Tatranská Lomnica / Slovakia

(UPDATED 23/07/2007)

August 1 (Wed):

10:00–12:00: Quantum Entanglement

(M. Ziman, RCQI Bratislava, SK)

The concept and basic properties of quantum entanglement will be reviewed. The mathematical tools (such as positive maps, entanglement witnesses, Jamiolkowski isomorphism) useful in the analysis of entanglement will be introduced. We will focus on operational meaning of bipartite entanglement and compare its meaning with correlations and nonlocality. In the last part we will introduce several measures used for quantification of the entanglement content of quantum states.

14:00–15:45: How to Implement Quantum Computing from the Tensorial Product of Two Pauli Matrices

(A.-C. Baboin, FEMTO-ST/CNRS Besançon, FR)

Quantum-teleportation-based-model of quantum computing was announced by Michael Nielsen in 2001. Over the past few years, both the concept and the minimal amount of resources needed have been clarified by Debbie Leung and others. Recently, it has been shown by Simon Perdrix that quantum teleportation can be replaced by a state transfer so that the new model of quantum computing only needs a single two-qubit observable, i.e the tensorial product of two Pauli matrices Z*X, a few one-qubit observables, e.g. X and Z, and a single auxiliary qubit. A summary of these developments will be given.

15:45–16:15: Coffee & Refreshments

16:15–18:00: Variation on a Theme of Q-uons: From Supersymmetry to Quantum Information

(M. Kibler, IPN/UCBL, Lyon, FR)

A first part of this talk is devoted to the description and some applications of q-uons or k-fermions, with q = exp(2i\pi /k) for k = 2, 3, …, which are operators interpolating between fermion operators (k = 2) and boson operators (k \rightarrow \infty). We show how such operators can be used for (i) a realization of a generalized Weyl-Heisenberg algebra Wk connected to k-fractional supersymmetric quantum mechanics and (ii) a polar decomposition of the Lie algebra of the group SU(2). A second part deals with a unified approach of mutually unbiased bases (MUBs) and positive operator valued measures (POVMs) used in quantum information. In particular the polar decomposition of SU(2) is employed for generating MUBs in a Hilbert space the dimension of which is a prime.

August 2 (Thu):

10:00–12:00: Projective Ring Lines and Finite Generalized Quadrangles

(M. Saniga, AsU SAV, Tatranská Lomnica, SK)

A detailed account of basic properties of projective lines defined over finite commutative rings and finite generalized quadrangles will be given. As per projective ring lines, after introducing the definition and basic concepts of neighbour and distant we shall give a number of illustrative examples; these will include lines defined over all rings of order four as well as the “exceptional” line defined over the full two-by-two matrix ring with entries in GF(2). Concerning quadrangles, we will highlight the most important properties and illustrate these on the smallest finite generalized quadrangle, the “doily:” various representations of the latter will be presented and its relation with specific projective ring lines will be shown.

14:00–15:45: Geometry of Finite-Dimensional Hilbert Spaces

(M. Planat, FEMTO-ST/CNRS, Besançon, FR)

The commutation relations between the generalized Pauli operators, and their maximal sets of commuting operators, follow an identifiable graph theoretical/geometrical pattern. Two-qubit observables have the geometry of the symplectic generalized quadrangle of order two W(2), and higher order $N$-qudit observables (N a power of a prime number) are related to polar spaces of the corresponding order. A new recognized feature is the multi-line property of the geometry of observables, starting with the smallest composite dimension six. An overview of recent progress will be given.

15:45–16:15: Coffee & Refreshments

16:15–18:00: A Mathematician's Insight into the Saniga-Planat Theorem

(H. Havlicek, TUW, Vienna, AT)

Only recently it was demonstrated by Saniga and Planat that the space of operators characterising two-qubits is isomorphic to the generalised quadrangle of order two. They conjectured also a more general result linking  $N$-qubits ($N>2$) and symplectic polar spaces of rank $N$ over the field with two elements. A very short and elegant proof of this fundamental theorem is due to Koen Thas. We aim at providing a deeper insight to this result and its proof. This should be useful in our ongoing study of Finite Geometries behind Hilbert Spaces.

August 3 (Fri):

10:00–12:00: Implications for a Spatially Discrete Transition Amplitude in the Twin-Slit Experiment

(M. Stuckey, Elizabethtown College (PA), US)

A discrete path integral formalism is used to obtain the transition amplitude between “sources” (slits and detector) in the twin-slit experiment of quantum mechanics. This method explicates the normally tacit construct of dynamic entities with temporal duration. The resulting amplitude is compared to that of standard (spacetime embedded) quantum mechanics in order to relate “source” dynamics and spatial separation. The implied metric embodies non-separability, in stark contrast to the metric of general relativity. Thus, this approach may have implications for quantum gravity.

14:00–15:45: Naissance of a System

(J. Drahorád, TEIN, Prague, CZ)

In the talk a development of concepts related to existence and basic mechanisms of living systems will be presented. For their description, it was necessary to develop a tool, which enabled an exact representation of the boundary between living and lifeless systems. It is based on a hierarchical structure of sets of elements and is used to study the origin of such hierarchy and its implications. This tool and its efficiency are demonstrated in several applications, covering various levels of complexity of living systems all the way to lifeless systems.

15:45–16:15: Coffee & Refreshments

16:15–18:00: The Duality Principle as the Basic (A)Symmetry in the General Description of Nature

(P. Pracna, JH-Inst AV CR, Prague, CZ)

The reflections of principles of existence and evolution of living systems will be confronted with the still unresolved issues of behavior (description) of complex systems, which are attempted to be described by physical laws. It will be discussed that the problem with representing the evolution (arrow of time) is closely related to the hierarchical structure of systems at all levels of complexity. The hierarchical principle is shown to be also linked to the assumption that every system exists not as an isolated entity, but as an object in a certain environment. Application of the hierarchical approach may allow to circumvent the necessity of describing the system with its environment only by statistical tools. The possible relation of the hierarchical principle with issues of finite geometry will be discussed.

August 4 (Sat):

Full-day sightseeing trip to Cracow (Poland), together with accompanying persons