General correlation functions of the Clauser-Horne-Shimony-Holt inequality for arbitrarily high-dimensional systems

We generalize the correlation functions of the Clauser-Horne-Shimony-Holt (CHSH) inequality to arbitrarily high-dimensional systems. Based on this generalization, we construct the general CHSH inequality for bipartite quantum systems of arbitrarily high dimensionality, which takes the same simple form as CHSH inequality for two-dimension. This inequality is optimal in the same sense as the CHSH inequality for two dimensional systems, namely, the maximal amount by which the inequality is violated consists with the maximal resistance to noise. We also discuss the physical meaning and general definition of the correlation functions. Furthermore, by giving another specific set of the correlation functions with the same physical meaning, we realize the inequality presented in [Phys. Rev. Lett. {\bf 88,}040404 (2002)].Comment: 4 pages, accepted by Phys. Rev. Let

The Complexity of Testing Monomials in Multivariate Polynomials

The work in this paper is to initiate a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by certain economically compact structure has a multilinear monomial in its sum-product expansion. The complexity aspects of this problem and its variants are investigated with two folds of objectives. One is to understand how this problem relates to critical problems in complexity, and if so to what extent. The other is to exploit possibilities of applying algebraic properties of polynomials to the study of those problems. A series of results about $\Pi\Sigma\Pi$ and $\Pi\Sigma$ polynomials are obtained in this paper, laying a basis for further study along this line

Dependence of Temporal Properties on Energy in Long-Lag, Wide-Pulse Gamma-Ray Bursts

We employed a sample compiled by Norris et al. (2005, ApJ, 625, 324) to study the dependence of the pulse temporal properties on energy in long-lag, wide-pulse gamma-ray bursts. Our analysis shows that the pulse peak time, rise time scale and decay time scale are power law functions of energy, which is a preliminary report on the relationships between the three quantities and energy. The power law indexes associated with the pulse width, rise time scale and decay time scale are correlated and the correlation between the indexes associated with the pulse width and the decay time scale is more obvious. In addition, we have found that the pulse peak lag is strongly correlated with the CCF lag, but the centroid lag is less correlated with the peak lag and CCF lag. Based on these results and some previous investigations, we tend to believe that all energy-dependent pulse temporal properties may come from the joint contribution of both the hydrodynamic processes of the outflows and the curvature effect, where the energy-dependent spectral lag may be mainly dominated by the dynamic process and the energy-dependent pulse width may be mainly determined by the curvature effect.Comment: 20 pages, 7 figures, added references, matched to published version, accepted for publication in PAS

Quantum phase transition in a three-level atom-molecule system

We adopt a three-level bosonic model to investigate the quantum phase transition in an ultracold atom-molecule conversion system which includes one atomic mode and two molecular modes. Through thoroughly exploring the properties of energy level structure, fidelity, and adiabatical geometric phase, we confirm that the system exists a second-order phase transition from an atommolecule mixture phase to a pure molecule phase. We give the explicit expression of the critical point and obtain two scaling laws to characterize this transition. In particular we find that both the critical exponents and the behaviors of ground-state geometric phase change obviously in contrast to a similar two-level model. Our analytical calculations show that the ground-state geometric phase jumps from zero to ?pi/3 at the critical point. This discontinuous behavior has been checked by numerical simulations and it can be used to identify the phase transition in the system.Comment: 8 pages,8 figure