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

Ram pressure stripping of the cool core of the Ophiuchus Cluster

(abridged) We report results from a Chandra study of the central regions of the nearby, X-ray bright, Ophiuchus Cluster (z = 0.03), the second-brightest cluster in the sky. Our study reveals a dramatic, close-up view of the stripping and potential destruction of a cool core within a rich cluster. The X-ray emission from the Ophiuchus Cluster core exhibits a comet-like morphology extending to the north, driven by merging activity, indicative of ram-pressure stripping caused by rapid motion through the ambient cluster gas. A cold front at the southern edge implies a velocity of 1000$\pm$200 km/s (M~0.6). The X-ray emission from the cluster core is sharply peaked. As previously noted, the peak is offset by 4 arcsec (~2 kpc) from the optical center of the associated cD galaxy, indicating that ram pressure has slowed the core, allowing the relatively collisionless stars and dark matter to carry on ahead. The cluster exhibits the strongest central temperature gradient of any massive cluster observed to date: the temperature rises from 0.7 keV within 1 kpc of the brightness peak, to 10 keV by 30 kpc. A strong metallicity gradient is also observed within the same region. This supports a picture in which the outer parts of the cool core have been stripped by ram-pressure due to its rapid motion. The cooling time of the innermost gas is very short, ~5$\times10^7$ yrs. Within the central 10 kpc radius, multiple small-scale fronts and a complex thermodynamic structure are observed, indicating significant motions. Beyond the central 50 kpc, and out to a radius ~150 kpc, the cluster appears relatively isothermal and has near constant metallicity. The exception is a large, coherent ridge of enhanced metallicity observed to trail the cool core, and which is likely to have been stripped from it.Comment: Accepted to MNRAS. 11 pages, 9 figure

Entanglement measure for general pure multipartite quantum states

We propose an explicit formula for an entanglement measure of pure multipartite quantum states, then study a general pure tripartite state in detail, and at end we give some simple but illustrative examples on four-qubits and m-qubits states.Comment: 5 page

Restoration of degraded grasslands, but not invasion by Prosopis juliflora, avoids trade-offs between climate change mitigation and other ecosystem services

Grassland degradation and the concomitant loss of soil organic carbon is widespread in tropical arid and semi-arid regions of the world. Afforestation of degraded grassland, sometimes by using invasive alien trees, has been put forward as a legitimate climate change mitigation strategy. However, even in cases where tree encroachment of degraded grasslands leads to increased soil organic carbon, it may come at a high cost since the restoration of grassland-characteristic biodiversity and ecosystem services will be blocked. We assessed how invasion by Prosopis juliflora and restoration of degraded grasslands in a semi-arid region in Baringo, Kenya affected soil organic carbon, biodiversity and fodder availability. Thirty years of grassland restoration replenished soil organic carbon to 1 m depth at a rate of 1.4% per year and restored herbaceous biomass to levels of pristine grasslands, while plant biodiversity remained low. Invasion of degraded grasslands by P. juliflora increased soil organic carbon primarily in the upper 30 cm and suppressed herbaceous vegetation. We argue that, in contrast to encroachment by invasive alien trees, restoration of grasslands in tropical semi-arid regions can both serve as a measure for climate change mitigation and help restore key ecosystem services important for pastoralists and agro-pastoralist communities

Separability criteria and bounds for entanglement measures

Employing a recently proposed separability criterion we develop analytical lower bounds for the concurrence and for the entanglement of formation of bipartite quantum systems. The separability criterion is based on a nondecomposable positive map which operates on state spaces with even dimension N >= 4, and leads to a class of nondecomposable optimal entanglement witnesses. It is shown that the bounds derived here complement and improve the existing bounds obtained from the criterion of positive partial transposition and from the realignment criterion.Comment: 8 pages, 2 figure

Constraining Gas Motions in the Intra-Cluster Medium

The detailed velocity structure of the diffuse X-ray emitting intra-cluster medium (ICM) remains one of the last missing key ingredients in understanding the microphysical properties of these hot baryons and constraining our models of the growth and evolution of structure on the largest scales in the Universe. Direct measurements of the gas velocities from the widths and shifts of X-ray emission lines were recently provided for the central region of the Perseus Cluster of galaxies by Hitomi, and upcoming high-resolution X-ray microcalorimeters onboard XRISM and Athena are expected to extend these studies to many more systems. In the mean time, several other direct and indirect methods have been proposed for estimating the velocity structure in the ICM, ranging from resonant scattering to X-ray surface brightness fluctuation analysis, the kinematic Sunyaev-Zeldovich effect, or using optical line emitting nebulae in the brightest cluster galaxies as tracers of the motions of the ambient plasma. Here, we review and compare the existing estimates of the velocities of the hot baryons, as well as the various overlapping physical processes that drive motions in the ICM, and discuss the implications of these measurements for constraining the viscosity and identifying the source of turbulence in clusters of galaxies

Entanglement and the Kondo effect in double quantum dots

We investigate entanglement between electrons in serially coupled double quantum dots attached to non interacting leads. In addition to local repulsion we consider the influence of capacitive inter-dot interaction. We show how the competition between extended Kondo and local singlet phases determines the ground state and thereby the entanglement.Comment: EPJ Special Topics 200

Searching for extremal PPT entangled states

We study extremality in various sets of states that have positive partial transposes. One of the tools we use for this purpose is the recently formulated criterion allowing to judge if a given state is extremal in the set of PPT states. First we investigate qubit--ququart states and show that the only candidates for extremal PPT entangled states (PPTES) have ranks of the state and its partial transposition (5,5) or (5,6) (equivalently (6,5)). Then, examples of extremal states of (5,5) type and the so--called edge states of type (5,6) are provided. We also make an attempt to explore the set of PPT states with ranks (5,6). Finally, we discuss what are the possible configurations of ranks of density matrices and their respective partial transposition in general three-qubit and four-qubit symmetric states for which there may exist extremal entangled PPT states. For instance in the first case we show that the only possibilities are (4,4,4) and (4,4,5).Comment: 12 pages, 2 figures, revised version due to the partial overlap with results of arXiv:0704.3348, some new results on extremality in multi-qubit systems added, contribution to the special issue of Optics Communications in memory of Krzysztof Wodkiewic