Neighbours of Einstein's Equations: Connections and Curvatures

Once the action for Einstein's equations is rewritten as a functional of an SO(3,C) connection and a conformal factor of the metric, it admits a family of ``neighbours'' having the same number of degrees of freedom and a precisely defined metric tensor. This paper analyzes the relation between the Riemann tensor of that metric and the curvature tensor of the SO(3) connection. The relation is in general very complicated. The Einstein case is distinguished by the fact that two natural SO(3) metrics on the GL(3) fibers coincide. In the general case the theory is bimetric on the fibers.Comment: 16 pages, LaTe

SL(2,R) Yang-Mills theory on a circle

The kinematics of SL(2,R) Yang-Mills theory on a circle is considered, for reasons that are spelled out. The gauge transformations exhibit hyperbolic fixed points, and this results in a physical configuration space with a non-Hausdorff "network" topology. The ambiguity encountered in canonical quantization is then much more pronounced than in the compact case, and can not be resolved through the kind of appeal made to group theory in that case.Comment: 10 pages, Goteborg ITP 94-19, Contains two files: A latex file with all figures drawn in latex and a tar archive including a slightly modified latex file (uses psfig) and nicer postscript figures+necessary macro

Degenerate Sectors of the Ashtekar Gravity

This work completes the task of solving locally the Einstein-Ashtekar equations for degenerate data. The two remaining degenerate sectors of the classical 3+1 dimensional theory are considered. First, with all densitized triad vectors linearly dependent and second, with only two independent ones. It is shown how to solve the Einstein-Ashtekar equations completely by suitable gauge fixing and choice of coordinates. Remarkably, the Hamiltonian weakly Poisson commutes with the conditions defining the sectors. The summary of degenerate solutions is given in the Appendix.Comment: 19 pages, late

Degenerate Metric Phase Boundaries

The structure of boundaries between degenerate and nondegenerate solutions of Ashtekar's canonical reformulation of Einstein's equations is studied. Several examples are given of such "phase boundaries" in which the metric is degenerate on one side of a null hypersurface and non-degenerate on the other side. These include portions of flat space, Schwarzschild, and plane wave solutions joined to degenerate regions. In the last case, the wave collides with a planar phase boundary and continues on with the same curvature but degenerate triad, while the phase boundary continues in the opposite direction. We conjecture that degenerate phase boundaries are always null.Comment: 16 pages, 2 figures; erratum included in separate file: errors in section 4, degenerate phase boundary is null without imposing field equation

A trick for passing degenerate points in Ashtekar formulation

We examine one of the advantages of Ashtekar's formulation of general relativity: a tractability of degenerate points from the point of view of following the dynamics of classical spacetime. Assuming that all dynamical variables are finite, we conclude that an essential trick for such a continuous evolution is in complexifying variables. In order to restrict the complex region locally, we propose some `reality recovering' conditions on spacetime. Using a degenerate solution derived by pull-back technique, and integrating the dynamical equations numerically, we show that this idea works in an actual dynamical problem. We also discuss some features of these applications.Comment: 9 pages by RevTeX or 16 pages by LaTeX, 3 eps figures and epsf-style file are include

Causal structure and degenerate phase boundaries

Timelike and null hypersurfaces in the degenerate space-times in the Ashtekar theory are defined in the light of the degenerate causal structure proposed by Matschull. Using the new definition of null hypersufaces, the conjecture that the "phase boundary" separating the degenerate space-time region from the non-degenerate one in Ashtekar's gravity is always null is proved under certain circumstances.Comment: 13 pages, Revte

Probability-based comparison of quantum states

We address the following state comparison problem: is it possible to design an experiment enabling us to unambiguously decide (based on the observed outcome statistics) on the sameness or difference of two unknown state preparations without revealing complete information about the states? We find that the claim "the same" can never be concluded without any doubts unless the information is complete. Moreover, we prove that a universal comparison (that perfectly distinguishes all states) also requires complete information about the states. Nevertheless, for some measurements, the probability distribution of outcomes still allows one to make an unambiguous conclusion regarding the difference between the states even in the case of incomplete information. We analyze an efficiency of such a comparison of qudit states when it is based on the SWAP-measurement. For qubit states, we consider in detail the performance of special families of two-valued measurements enabling us to successfully compare at most half of the pairs of states. Finally, we introduce almost universal comparison measurements which can distinguish almost all non-identical states (up to a set of measure zero). The explicit form of such measurements with two and more outcomes is found in any dimension.Comment: 12 pages, 6 figures, 1 table, some results are extende

The prediction of extratropical storm tracks by the ECMWF and NCEP ensemble prediction systems

The prediction of extratropical cyclones by the European Centre for Medium Range Weather Forecasts (ECMWF) and the National Centers for Environmental Prediction (NCEP) Ensemble Prediction Systems (EPS) has been investigated using an objective feature tracking methodology to identify and track the cyclones along the forecast trajectories. Overall the results show that the ECMWF EPS has a slightly higher level of skill than the NCEP EPS in the northern hemisphere (NH). However in the southern hemisphere (SH), NCEP has higher predictive skill than ECMWF for the intensity of the cyclones. The results from both EPS indicate a higher level of predictive skill for the position of extratropical cyclones than their intensity and show that there is a larger spread in intensity than position. Further analysis shows that the predicted propagation speed of cyclones is generally too slow for the ECMWF EPS and show a slight bias for the intensity of the cyclones to be overpredicted. This is also true for the NCEP EPS in the SH. For the NCEP EPS in the NH the intensity of the cyclones is underpredicted. There is small bias in both the EPS for the cyclones to be displaced towards the poles. For each ensemble forecast of each cyclone, the predictive skill of the ensemble member that best predicts the cyclones position and intensity was computed. The results are very encouraging showing that the predictive skill of the best ensemble member is significantly higher than that of the control forecast in terms of both the position and intensity of the cyclones. The prediction of cyclones before they are identified as 850 hPa vorticity centers in the analysis cycle was also considered. It is shown that an indication of extratropical cyclones can be given by at least 1 ensemble member 7 days before they are identified in the analysis. Further analysis of the ECMWF EPS shows that the ensemble mean has a higher level of skill than the control forecast, particularly for the intensity of the cyclones, 2 from day 3 of the forecast. There is a higher level of skill in the NH than the SH and the spread in the SH is correspondingly larger. The difference between the ensemble mean and spread is very small for the position of the cyclones, but the spread of the ensemble is smaller than the ensemble mean error for the intensity of the cyclones in both hemispheres. Results also show that the ECMWF control forecast has ½ to 1 day more skill than the perturbed members, for both the position and intensity of the cyclones, throughout the forecast

Collapse of the quantum correlation hierarchy links entropic uncertainty to entanglement creation

Quantum correlations have fundamental and technological interest, and hence many measures have been introduced to quantify them. Some hierarchical orderings of these measures have been established, e.g., discord is bigger than entanglement, and we present a class of bipartite states, called premeasurement states, for which several of these hierarchies collapse to a single value. Because premeasurement states are the kind of states produced when a system interacts with a measurement device, the hierarchy collapse implies that the uncertainty of an observable is quantitatively connected to the quantum correlations (entanglement, discord, etc.) produced when that observable is measured. This fascinating connection between uncertainty and quantum correlations leads to a reinterpretation of entropic formulations of the uncertainty principle, so-called entropic uncertainty relations, including ones that allow for quantum memory. These relations can be thought of as lower-bounds on the entanglement created when incompatible observables are measured. Hence, we find that entanglement creation exhibits complementarity, a concept that should encourage exploration into "entanglement complementarity relations".Comment: 19 pages, 2 figures. Added Figure 1 and various remarks to improve clarity of presentatio

A study of separability criteria for mixed three-qubit states

We study the noisy GHZ-W mixture. We demonstrate some necessary but not sufficient criteria for different classes of separability of these states. It turns out that the partial transposition criterion of Peres and the criteria of G\"uhne and Seevinck dealing with matrix elements are the strongest ones for different separability classes of this 2 parameter state. As a new result we determine a set of entangled states of positive partial transpose.Comment: 18 pages, 10 figures, PRA styl