Multipartite quantum correlations: symplectic and algebraic geometry approach

We review a geometric approach to classification and examination of quantum correlations in composite systems. Since quantum information tasks are usually achieved by manipulating spin and alike systems or, in general, systems with a finite number of energy levels, classification problems are usually treated in frames of linear algebra. We proposed to shift the attention to a geometric description. Treating consistently quantum states as points of a projective space rather than as vectors in a Hilbert space we were able to apply powerful methods of differential, symplectic and algebraic geometry to attack the problem of equivalence of states with respect to the strength of correlations, or, in other words, to classify them from this point of view. Such classifications are interpreted as identification of states with `the same correlations properties' i.e. ones that can be used for the same information purposes, or, from yet another point of view, states that can be mutually transformed one to another by specific, experimentally accessible operations. It is clear that the latter characterization answers the fundamental question `what can be transformed into what \textit{via} available means?'. Exactly such an interpretations, i.e, in terms of mutual transformability can be clearly formulated in terms of actions of specific groups on the space of states and is the starting point for the proposed methods.Comment: 29 pages, 9 figures, 2 tables, final form submitted to the journa

Global entangling properties of the coupled kicked tops

We study global entangling properties of the system of coupled kicked tops testing various hypotheses and predictions concerning entanglement in quantum chaotic systems. In order to analyze the averaged initial entanglement production rate and the averaged asymptotic entanglement different ensembles of initial product states are evolved. Two different ensembles with natural probability distribution are considered: product states of independent spin-coherent states and product states of arbitrary states. It appears that the choice of either of these ensembles results in significantly different averaged entanglement behavior. We investigate also a relation between the averaged asymptotic entanglement and the mean entanglement of the eigenvectors of an evolution operator. Lower bound on the averaged asymptotic entanglement is derived, expressed in terms of the eigenvector entanglement.Comment: 11 pages, 7 figures, RevTe

Symplectic geometry of entanglement

We present a description of entanglement in composite quantum systems in terms of symplectic geometry. We provide a symplectic characterization of sets of equally entangled states as orbits of group actions in the space of states. In particular, using Kostant-Sternberg theorem, we show that separable states form a unique Kaehler orbit, whereas orbits of entanglement states are characterized by different degrees of degeneracy of the canonical symplectic form on the complex projective space. The degree of degeneracy may be thus used as a new geometric measure of entanglement and we show how to calculate it for various multiparticle systems providing also simple criteria of separability. The presented method is general and can be applied also under different additional symmetry conditions stemming, eg. from the indistinguishability of particles.Comment: LaTex, 31 pages, typos correcte

The Role of Biological Diversity in Agroecosystems and Organic Farming

Ecosystems are the basis of life and all human activities. Conservation of biological diversity is very important for the proper functioning of the ecosystem and for delivering ecosystem services. Maintaining high biodiversity in agroecosystems makes agricultural production more sustainable and economically viable. Agricultural biodiversity ensures, for example, pollination of crops, biological crop protection, maintenance of proper structure and fertility of soils, protection of soils against erosion, nutrient cycling, and control of water flow and distribution. The effects of the loss of biodiversity may not be immediately apparent, but they may increase the sensitivity of the ecosystems to various abiotic and biotic stresses. The combination of biodiversity conservation with profitable food production is one of the tasks of modern sustainable agriculture that faces the necessity of reconciling the productive, environmental, and social goals. As further intensification of production and increase in the use of chemical pesticides, fertilizers, and water to increase yields are increasingly criticized, global agriculture is looking for other biological and agrotechnical methods in order to meet the requirements of global food production

Information Infrastructure for Cooperative Research in Neuroscience

The paper describes a framework for efficient sharing of knowledge between research groups, which have been working for several years without flaws. The obstacles in cooperation are connected primarily with the lack of platforms for effective exchange of experimental data, models, and algorithms. The solution to these problems is proposed by construction of the platform (EEG.pl) with the semantic aware search scheme between portals. The above approach implanted in the international cooperative projects like NEUROMATH may bring the significant progress in designing efficient methods for neuroscience research

Dynamics of quantum entanglement

A model of discrete dynamics of entanglement of bipartite quantum state is considered. It involves a global unitary dynamics of the system and periodic actions of local bistochastic or decaying channel. For initially pure states the decay of entanglement is accompanied with an increase of von Neumann entropy of the system. We observe and discuss revivals of entanglement due to unitary interaction of both subsystems. For some mixed states having different marginal entropies of both subsystems (one of them larger than the global entropy and the other one one smaller) we find an asymmetry in speed of entanglement decay. The entanglement of these states decreases faster, if the depolarizing channel acts on the "classical" subsystem, characterized by smaller marginal entropy.Comment: 10 pages, Revtex, 10 figures, refined versio

Geometry of entangled states

Geometric properties of the set of quantum entangled states are investigated. We propose an explicit method to compute the dimension of local orbits for any mixed state of the general K x M problem and characterize the set of effectively different states (which cannot be related by local transformations). Thus we generalize earlier results obtained for the simplest 2 x 2 system, which lead to a stratification of the 6D set of N=4 pure states. We define the concept of absolutely separable states, for which all globally equivalent states are separable.Comment: 16 latex pages, 4 figures in epsf, minor corrections, references updated, to appear in Phys. Rev.

Concurrence classes for general pure multipartite states

We propose concurrence classes for general pure multipartite states based on an orthogonal complement of a positive operator valued measure on quantum phase. In particular, we construct $W^{m}$ class, $GHZ^{m}$, and $GHZ^{m-1}$ class concurrences for general pure $m$-partite states. We give explicit expressions for $W^{3}$ and $GHZ^{3}$ class concurrences for general pure three-partite states and for $W^{4}$, $GHZ^{4}$, and $GHZ^{3}$ class concurrences for general pure four-partite states.Comment: 14 page

Quantum Iterated Function Systems

Iterated functions system (IFS) is defined by specifying a set of functions in a classical phase space, which act randomly on an initial point. In an analogous way, we define a quantum iterated functions system (QIFS), where functions act randomly with prescribed probabilities in the Hilbert space. In a more general setting a QIFS consists of completely positive maps acting in the space of density operators. We present exemplary classical IFSs, the invariant measure of which exhibits fractal structure, and study properties of the corresponding QIFSs and their invariant states.Comment: 12 pages, 1 figure include

Concurrence classes for an arbitrary multi-qubit state based on positive operator valued measure

In this paper, we propose concurrence classes for an arbitrary multi-qubit state based on orthogonal complement of a positive operator valued measure, or POVM in short, on quantum phase. In particular, we construct concurrence for an arbitrary two-qubit state and concurrence classes for the three- and four-qubit states. And finally, we construct $W^{m}$ and $GHZ^{m}$ class concurrences for multi-qubit states. The unique structure of our POVM enables us to distinguish different concurrence classes for multi-qubit states.Comment: 8 page