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Iterative dynamics of quantum systems: non-classical properties, applications for modelling physical systems and optical realizations. A We plan to investigate the fundamental properties of iterated quantum dynamical systems: quantum walks and iterated quantum circuits. Quantum walks, on one hand, play an essential role in quantum information theory, where they are used for designing algorithms. On the other hand, quantum walks may serve as a model for more complex physical systems. Far from being just a theoretical construction, they have recently been experimentally realized on ultracold atoms trapped in optical lattices and on photons in quantum optical circuits.

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Iterative dynamics of quantum systems: non-classical properties, applications for modelling physical systems and optical realizations. A We plan to investigate the fundamental properties of iterated quantum dynamical systems: quantum walks and iterated quantum circuits. Quantum walks, on one hand, play an essential role in quantum information theory, where they are used for designing algorithms.

On the other hand, quantum walks may serve as a model for more complex physical systems. Far from being just a theoretical construction, they have recently been experimentally realized on ultracold atoms trapped in optical lattices and on photons in quantum optical circuits.

The other iterated system we focus on, iterated quantum circuits, are periodic arrangements of quantum gates and measurements applied on an ensemble of one or more qubits.

Examples of these setups are the ones used in entanglement purification protocols. According to our previous discovery, truly chaotic dynamics complex chaos may occur in iterated quantum circuits, which is also visible in the evolution of entanglement.

We aim in this project to investigate analytically and numerically these iterated systems. We will concentrate on the following areas: Topological effects in quantum walks, modeling and generalizing topological effects known in solid state physics. Percolated quantum walks, including both of the two main classes, the continuous and discrete time cases, their asymptotic behavior, mixing properties. Issues related to quantum optical realizations and applications of quantum walks in quantum information theory.

Mathematical properties of complex chaos: the fractal structure of the set of initial states leading to special asymptotic properties, e. What is the major research question? Describe here briefly the problem to be solved by the research, the starting hypothesis, and the questions addressed by the experiments.

We consider iterative dynamics of quantum systems with few dynamical components. The dynamics is prescribed by either 1 deterministic time evolution via Hamiltonians which may have periodically time dependence , and measurements, or 2 by quantum circuits of small complexity, including measurement feedback loops. Our main questions are: How can such systems be applied for modeling phenomena seen in more complicated physical systems?

What are their special properties which may enable us to employ them in quantum information protocols? Can these properties be measured in realistic settings used in current laboratory experiments? We plan to apply and further build the set of mathematical tools that we have developed over the past few years. What is the significance of the research? Describe the new perspectives opened by the results achieved, including the scientific basics of potential societal applications.

Please describe the unique strengths of your proposal in comparison to your domestic and international competitors in the given field. The quantum physics of relatively small systems has long been unaccessible experimentally, only gedanken thought experiments were discussed by physicists. Theoretical understanding of the dynamical properties of such systems is essential, on one hand, since they form the building blocks of quantum information processing.

On the other hand, these highly controllable model systems can help us understand phenomena arising in more complicated systems e. Summary and aims of the research for the public Describe here the major aims of the research for an audience with average background information. This summary is especially important for NKFI in order to inform decision-makers, media, and the taxpayers. Quantum mechanics was discovered a century ago from measurement made on billions of atoms or photons.

For a long time it was unimaginable to experiment with single quantum systems, study their interaction, measure their properties. Today, there are more and more experiments handling these individual quantum systems with ever increasing precision and control. The Nobel prize in physics in was awarded to experiments of this kind. These new experimental possibilities lead also to new questions for theoreticians, related to the time evolution of sytems composed of a small number of quantum components.

Exploiting the nonclassical properties of these systems, one could, in principle, process or transfer information in a much more effective way than with any classical computer. These simple, highly controlled setups can also model the behavior of more complicated physical systems.

In our project, we would like to theoretically study systems consisting of a small number of individual quantum components which are subjected repeatedly to the same interaction.

One of the special systems we have focused on was the quantum walk. We pointed out the role of topological invariants in various models, among others periodically driven and lossy walks, which at the same time help better understand their analog systems in solid state physics topological insulators. We could present the asymptotic dynamics of quantum walks on percolation lattices, both in the continuous- and discrete-time cases.

We pointed out that there are oscillations in the internal degree of freedom and there exist inhomogenious asymptotic solutions. Iterated quantum protocols were another focus of our research. We could show that complex chaotic behaviour, earlier discovered by us, requires exponential downscaling of the corresponding ensemble. We proposed an scheme based on the Tavis-Cummings model, in order to realize iterated chaotic dynamics. Based on this arrangement, we proposed a procedure to distinguish certain classes of quantum states.

We determined the average return time in general iterated dynamical systems with unital superoperator. Generalizing this result we proved a generalized Kac lemma for quantum systems. Tarasinski, P. Tarasinski, J. Dahlhaus: Scattering theory of topological phases in discrete-time quantum walks , Phys. A 89, , Hideaki Obuse, Janos K. B 92, , Kiss, and I. Jex: Strongly trapped two-dimensional quantum walks , Phys. A 91, , Sinkovicz, Z.

Kurucz, T. Kiss, J. Asboth: Quantized recurrence time in iterated open quantum dynamics , Phys. Asboth and J. Edge: Edge-state enhanced transport in a 2-dimensional quantum walk , Phys.

Jonathan M. Edge and Janos K. Asboth: Localization, delocalization, and topological transitions in disordered two-dimensional quantum walks Phys. B 91, — Published 5 March , Phys.

B 91, , A 94, , Sinkovicz, T. Kiss, and J. A 93, R , Tibor Rakovszky and Janos K. Asboth: Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk , Phys. A 92, , Janos K. Tibor Rakovszky, Janos K. Asboth, Andrea Alberti: Detecting topological invariants in chiral symmetric insulators via losses , Phys.

B 95, , A 95, , , Kornyik M, Michaletzky G.

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Lie-szimmetriák egy közgazdasági alkalmazása (An economic application of the Lie symmetries)

Alkalmazott matematikai lapok, In the centre of its mathematical presentation is a specic function of coordinates and velocities, i. If the integral of the Lagrangian is stationary, then the system is moving along an extremal path through the phase space, and vice versa. It can be seen, that each Lie symmetry of a Lagrangian in general corresponds to a conserved quantity, and the conservation principle is explained by a variational symmetry related to a dynamic or geometrical symmetry.

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Project Report. In our research we first gave an account of the most basic types of growth functions, and then surveyed the endeavors which seek to apply the use of growth functions to the broadest possible areas of social change. On basis of these, we targeted to reveal the signs of growth functions on some specific area of the daily newspapers. First we examined how the press reacted to the so called "Radio Tilos Affair", where one of participant declared that he "would liquidate all Christians". The evolution in time of the number of articles published reminds us of to typical S curve of the logistic growth. By using the logistic fit we could measure the state of emergency in the daily press.

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