Quantum Integrals of Motion for the Heisenberg Spin Chain

An explicit expression for all the quantum integrals of motion for the isotropic Heisenberg $s=1/2$ spin chain is presented. The conserved quantities are expressed in terms of a sum over simple polynomials in spin variables. This construction is direct and independent of the transfer matrix formalism. Continuum limits of these integrals in both ferrromagnetic and antiferromagnetic sectors are briefly discussed.Comment: 10 pages Report #: LAVAL-PHY-94-2

On the relation between states and maps in infinite dimensions

Relations between states and maps, which are known for quantum systems in finite-dimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators $\mathcal{L}_2(\mathcal{H}_2,\mathcal{H}_1)$ and the corresponding tensor products $\mathcal{H}_1\otimes\mathcal{H}_2^*$ of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map $\mathcal{C}:\mathcal{L}_1(\mathcal{L}_2(\mathcal{H}_2,\mathcal{H}_1))\to \mathcal{L}_\infty(\mathcal{L}(\mathcal{H}_2),\mathcal{L}_1(\mathcal{H}_1))$ from trace-class operators on $\mathcal{L}_2(\mathcal{H}_2,\mathcal{H}_1)$ (with the nuclear norm) into compact operators mapping the space of all bounded operators on $\mathcal{H}_2$ into trace class operators on $\mathcal{H}_1$ (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.Comment: 19 page

Symmetries, group actions, and entanglement

We address several problems concerning the geometry of the space of Hermitian operators on a finite-dimensional Hilbert space, in particular the geometry of the space of density states and canonical group actions on it. For quantum composite systems we discuss and give examples of measures of entanglement.Comment: 21 page

Segre maps and entanglement for multipartite systems of indistinguishable particles

We elaborate the concept of entanglement for multipartite system with bosonic and fermionic constituents and its generalization to systems with arbitrary parastatistics. The entanglement is characterized in terms of generalized Segre maps, supplementing thus an algebraic approach to the problem by a more geometric point of view.Comment: 16 pages, the version to appear in J. Phys. A. arXiv admin note: text overlap with arXiv:1012.075

Geometry of quantum dynamics in infinite dimension

We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple. Our approach is genuinely infinite-dimensional and including a Lagrangian formalism in which self-adjoint (Schroedinger) operators are obtained as Lagrangian submanifolds associated with the Lagrangian. As a byproduct we obtain also results concerning coadjoint orbits of the unitary group in infinite dimension, embedding of the Hilbert projective space of pure states in the unitary group, and an approach to self-adjoint extensions of symmetric relations.Comment: 32 page

Convex bodies of states and maps

We give a general solution to the question when the convex hulls of orbits of quantum states on a finite-dimensional Hilbert space under unitary actions of a compact group have a non-empty interior in the surrounding space of all density states. The same approach can be applied to study convex combinations of quantum channels. The importance of both problems stems from the fact that, usually, only sets with non-vanishing volumes in the embedding spaces of all states or channels are of practical importance. For the group of local transformations on a bipartite system we characterize maximally entangled states by properties of a convex hull of orbits through them. We also compare two partial characteristics of convex bodies in terms of largest balls and maximum volume ellipsoids contained in them and show that, in general, they do not coincide. Separable states, mixed-unitary channels and k-entangled states are also considered as examples of our techniques.Comment: 18 pages, 1 figur

Load balancing of communication channels with the use of routing protocols

In the article the authors propose a method for load-balancing of network resources forthe case which uses a routing protocols. In the first part of the article the authors present currentlyused algorithms for load balancing and possibilities of their modification. Through the introductionof additional hardware components for each node: the agent and the probe; it is possible to monitorand control the current system performance. The whole analyzed network is treated as a complexsystem. This allows to eliminate overloading of route nodes (through ongoing analysis of the optimaloperating point for a given node). Load balancing can be achieved using a modified mechanism ofECMP. The proposed approach allows for dynamic adjustment of load to network resources and thuseffectively to balance network traffic

Queuing in terms of complex systems

Limited resources are a natural feature of most real systems, both artificial and naturalones. This causes the need for effective management of access to existing resources. In this area,queuing systems are of special application. However, they are treated as simple systems for whichtwo states are characteristic: work underload and work on the border of thermodynamic equilibrium.This approach is reflected in existing queue management mechanisms, that need to keep them in oneof two mentioned states. On the other hand, they should be considered from the point of complexsystems view, for which the third operation states: overload state is natural as well. In order to becloser to this issue, in this paper the authors consider queues performance from the perspective ofcomplex systems