Cluster and group synchronization in delay-coupled networks

We investigate the stability of synchronized states in delay-coupled networks where synchronization takes place in groups of different local dynamics or in cluster states in networks with identical local dynamics. Using a master stability approach, we find that the master stability function shows a discrete rotational symmetry depending on the number of groups. The coupling matrices that permit solutions on group or cluster synchronization manifolds show a very similar symmetry in their eigenvalue spectrum, which helps to simplify the evaluation of the master stability function. Our theory allows for the characterization of stability of different patterns of synchronized dynamics in networks with multiple delay times, multiple coupling functions, but also with multiple kinds of local dynamics in the networks' nodes. We illustrate our results by calculating stability in the example of delay-coupled semiconductor lasers and in a model for neuronal spiking dynamics.Comment: 11 pages, 7 figure

Hierarchies of Geometric Entanglement

We introduce a class of generalized geometric measures of entanglement. For pure quantum states of $N$ elementary subsystems, they are defined as the distances from the sets of $K$-separable states ($K=2,...,N$). The entire set of generalized geometric measures provides a quantification and hierarchical ordering of the different bipartite and multipartite components of the global geometric entanglement, and allows to discriminate among the different contributions. The extended measures are applied to the study of entanglement in different classes of $N$-qubit pure states. These classes include $W$ and $GHZ$ states, and their symmetric superpositions; symmetric multi-magnon states; cluster states; and, finally, asymmetric generalized $W$-like superposition states. We discuss in detail a general method for the explicit evaluation of the multipartite components of geometric entanglement, and we show that the entire set of geometric measures establishes an ordering among the different types of bipartite and multipartite entanglement. In particular, it determines a consistent hierarchy between $GHZ$ and $W$ states, clarifying the original result of Wei and Goldbart that $W$ states possess a larger global entanglement than $GHZ$ states. Furthermore, we show that all multipartite components of geometric entanglement in symmetric states obey a property of self-similarity and scale invariance with the total number of qubits and the number of qubits per party.Comment: 16 pages, 7 figures. Final version, to appear in Phys. Rev.

Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence

We propose a family of exactly solvable toy models for the AdS/CFT correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an isometry from the bulk Hilbert space to the boundary Hilbert space. The entire tensor network is an encoder for a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindler-wedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed by Almheiri et. al in arXiv:1411.7041.Comment: 40 Pages + 25 Pages of Appendices. 38 figures. Typos and bibliographic amendments and minor correction

Entanglement quantification by local unitaries

Invariance under local unitary operations is a fundamental property that must be obeyed by every proper measure of quantum entanglement. However, this is not the only aspect of entanglement theory where local unitaries play a relevant role. In the present work we show that the application of suitable local unitary operations defines a family of bipartite entanglement monotones, collectively referred to as "mirror entanglement". They are constructed by first considering the (squared) Hilbert-Schmidt distance of the state from the set of states obtained by applying to it a given local unitary. To the action of each different local unitary there corresponds a different distance. We then minimize these distances over the sets of local unitaries with different spectra, obtaining an entire family of different entanglement monotones. We show that these mirror entanglement monotones are organized in a hierarchical structure, and we establish the conditions that need to be imposed on the spectrum of a local unitary for the associated mirror entanglement to be faithful, i.e. to vanish on and only on separable pure states. We analyze in detail the properties of one particularly relevant member of the family, the "stellar mirror entanglement" associated to traceless local unitaries with nondegenerate spectrum and equispaced eigenvalues in the complex plane. This particular measure generalizes the original analysis of [Giampaolo and Illuminati, Phys. Rev. A 76, 042301 (2007)], valid for qubits and qutrits. We prove that the stellar entanglement is a faithful bipartite entanglement monotone in any dimension, and that it is bounded from below by a function proportional to the linear entropy and from above by the linear entropy itself, coinciding with it in two- and three-dimensional spaces.Comment: 13 pages, 3 figures. Improved and generalized proof of monotonicity of the mirror and stellar entanglemen

Policy forums: Why do they exist and what are they used for?

Policy forums are issue-based intermediary organizations where diverse types of political and societal actors repeatedly interact. Policy forums are important elements of modern governance systems as they allow actors to learn, negotiate, or build trust. They can vary in composition, size, membership logic, and other distinct features. This article lays the foundation of a theory of policy forums based on three interrelated elements: First, it discusses conditions for the formation of a forum and describes the logic of these organizations as one of an asymmetric multipartite exchange. Second, it enumerates the potential set of goals and motivations of participating actors that are fed into this exchange. Third, it proposes eight different dimensions on which policy forums differ and which affect the exchange mechanisms among actors. We claim that empirical work on policy forums should systematically take these elements into account and propose elements of a research agenda

Entanglement in continuous variable systems: Recent advances and current perspectives

We review the theory of continuous-variable entanglement with special emphasis on foundational aspects, conceptual structures, and mathematical methods. Much attention is devoted to the discussion of separability criteria and entanglement properties of Gaussian states, for their great practical relevance in applications to quantum optics and quantum information, as well as for the very clean framework that they allow for the study of the structure of nonlocal correlations. We give a self-contained introduction to phase-space and symplectic methods in the study of Gaussian states of infinite-dimensional bosonic systems. We review the most important results on the separability and distillability of Gaussian states and discuss the main properties of bipartite entanglement. These include the extremal entanglement, minimal and maximal, of two-mode mixed Gaussian states, the ordering of two-mode Gaussian states according to different measures of entanglement, the unitary (reversible) localization, and the scaling of bipartite entanglement in multimode Gaussian states. We then discuss recent advances in the understanding of entanglement sharing in multimode Gaussian states, including the proof of the monogamy inequality of distributed entanglement for all Gaussian states, and its consequences for the characterization of multipartite entanglement. We finally review recent advances and discuss possible perspectives on the qualification and quantification of entanglement in non Gaussian states, a field of research that is to a large extent yet to be explored.Comment: 61 pages, 7 figures, 3 tables; Published as Topical Review in J. Phys. A, Special Issue on Quantum Information, Communication, Computation and Cryptography (v3: few typos corrected

The Rainbow Prim Algorithm for Selecting Putative Orthologous Protein Sequences

We present a selection method designed for eliminating species redundancy in clusters of putative orthologous sequences, to be applied as a post-processing procedure to pre-clustered data obtained from other methods. The algorithm can always zero-out the cluster redundancy while preserving the number of species of the original cluster