A dynamical point of view of Quantum Information: entropy and pressure

Quantum Information is a new area of research which has been growing rapidly since last decade. This topic is very close to potential applications to the so called Quantum Computer. In our point of view it makes sense to develop a more "dynamical point of view" of this theory. We want to consider the concepts of entropy and pressure for "stationary systems" acting on density matrices which generalize the usual ones in Ergodic Theory (in the sense of the Thermodynamic Formalism of R. Bowen, Y. Sinai and D. Ruelle). We consider the operator $\mathcal{L}$ acting on density matrices $\rho\in \mathcal{M}_N$ over a finite $N$-dimensional complex Hilbert space $\mathcal{L}(\rho):=\sum_{i=1}^k tr(W_i\rho W_i^*)V_i\rho V_i^*,$ where $W_i$ and $V_i$, $i=1,2,...k$ are operators in this Hilbert space. $\mathcal{L}$ is not a linear operator. In some sense this operator is a version of an Iterated Function System (IFS). Namely, the $V_i\,(.)\,V_i^*=:F_i(.)$, $i=1,2,...,k$, play the role of the inverse branches (acting on the configuration space of density matrices $\rho$) and the $W_i$ play the role of the weights one can consider on the IFS. We suppose that for all $\rho$ we have that $\sum_{i=1}^k tr(W_i\rho W_i^*)=1$. A family $W:=\{W_i\}_{i=1,..., k}$ determines a Quantum Iterated Function System (QIFS) $\mathcal{F}_{W}$, $\mathcal{F}_W=\{\mathcal{M}_N,F_i,W_i\}_{i=1,..., k}.

Asteroseismic Theory of Rapidly Oscillating Ap Stars

This paper reviews some of the important advances made over the last decade concerning theory of roAp stars.Comment: 9 pages, 5 figure

A dynamical point of view of Quantum Information: Wigner measures

We analyze a known version of the discrete Wigner function and some connections with Quantum Iterated Funcion Systems. This paper is a follow up of "A dynamical point of view of Quantum Information: entropy and pressure" by the same authors

A Thermodynamic Formalism for density matrices in Quantum Information

We consider new concepts of entropy and pressure for stationary systems acting on density matrices which generalize the usual ones in Ergodic Theory. Part of our work is to justify why the definitions and results we describe here are natural generalizations of the classical concepts of Thermodynamic Formalism (in the sense of R. Bowen, Y. Sinai and D. Ruelle). It is well-known that the concept of density operator should replace the concept of measure for the cases in which we consider a quantum formalism. We consider the operator $\Lambda$ acting on the space of density matrices $\mathcal{M}_N$ over a finite $N$-dimensional complex Hilbert space $$\Lambda(\rho):=\sum_{i=1}^k tr(W_i\rho W_i^*)\frac{V_i\rho V_i^*}{tr(V_i\rho V_i^*)},$$ where $W_i$ and $V_i$, $i=1,2,..., k$ are linear operators in this Hilbert space. In some sense this operator is a version of an Iterated Function System (IFS). Namely, the $V_i\,(.)\,V_i^*=:F_i(.)$, $i=1,2,...,k$, play the role of the inverse branches (i.e., the dynamics on the configuration space of density matrices) and the $W_i$ play the role of the weights one can consider on the IFS. In this way a family $W:=\{W_i\}_{i=1,..., k}$ determines a Quantum Iterated Function System (QIFS). We also present some estimates related to the Holevo bound

Asymptotic Entanglement Dynamics and Geometry of Quantum States

A given dynamics for a composite quantum system can exhibit several distinct properties for the asymptotic entanglement behavior, like entanglement sudden death, asymptotic death of entanglement, sudden birth of entanglement, etc. A classification of the possible situations was given in [M. O. Terra Cunha, {\emph{New J. Phys}} {\bf{9}}, 237 (2007)] but for some classes there were no known examples. In this work we give a better classification for the possibile relaxing dynamics at the light of the geometry of their set of asymptotic states and give explicit examples for all the classes. Although the classification is completely general, in the search of examples it is sufficient to use two qubits with dynamics given by differential equations in Lindblad form (some of them non-autonomous). We also investigate, in each case, the probabilities to find each possible behavior for random initial states.Comment: 9 pages, 2 figures; revised version accepted for publication in J. Phys. A: Math. Theo

Quantum Stochastic Processes, Quantum Iterated Function Systems and Entropy

We describe some basic results for Quantum Stochastic Processes and present some new results about a certain class of processes which are associated to Quantum Iterated Function Systems (QIFS). We discuss questions related to the Markov property and we present a de nition of entropy which is induced by a QIFS. This definition is a natural generalization of the Shannon-Kolmogorov entropy from Ergodic Theory

KINEMATIC AND ELECTROMYOGRAPHIC BEHAVIOUR OF BASKETBALL PLAYERS’ INJURED AND HEALTHY ANKLES DURING THE JUMP ONTO AN UNSTABLE BOARD

The purpose of this study was to identify kinematic and electromyographic differences of jumps performed by basketball players with healthy and previously sprained ankles. 25 elite basketball players with healthy (n=17) and already sprained ankles (n=28) jumped five times in unipodal support from a stable surface onto a round Freeman board. During the jump the flight phase of those athletes with already sprained ankles was shorter which may indicate less preparation time for the moment of contact with the surface and for the respective load. When landing, they also positioned their ankle in a more plantar flexion and generally, the contraction of their foot musles was stronger than that of the healthy athletes. The groups’ differing movement behaviour of the lower leg possibly explains resulting ankle injuries. These results indicate that it might be necessary to train athletes to jump in “safe positions” in order to prevent ankle sprains

Assessment of fibre orientation and distribution in steel fibre reinforced self-compacting concrete panels

The benefits of adding fibres to concrete lie, mostly, in improving the post-cracking behaviour, since its ability to transfer stresses across cracked sections is substantially increased. The post-cracking strength is dependent not only on the fibre geometry, mechanical performance and fibre/matrix interface properties, but also on the fibre orientation and distribution. Previous works have shown that in self-compacting concrete matrices, there is a preferential fibre alignment according to the concrete’s flow in the fresh state. Having in mind that fibres are more efficient if they are oriented according the principal tensile stresses, a preferential fibre alignment on a certain direction could either enhance or diminish the material and the structural performance of this composite. In this paper, it is investigated the influence of the fibre orientation and distribution on the post-cracking behaviour of the steel fibre reinforced self-compacting concrete (SFRSCC). To perform this evaluation, SFRSCC panels were casted from their centre point. Two self-compacting mixtures were prepared using the same base mix proportions. For each SFRSCC panel cylindrical specimens were extracted and the post-cracking behaviour was assessed from a crack width controlled splitting tensile test