Sufficient dimension reduction based on an ensemble of minimum average variance estimators

We introduce a class of dimension reduction estimators based on an ensemble of the minimum average variance estimates of functions that characterize the central subspace, such as the characteristic functions, the Box--Cox transformations and wavelet basis. The ensemble estimators exhaustively estimate the central subspace without imposing restrictive conditions on the predictors, and have the same convergence rate as the minimum average variance estimates. They are flexible and easy to implement, and allow repeated use of the available sample, which enhances accuracy. They are applicable to both univariate and multivariate responses in a unified form. We establish the consistency and convergence rate of these estimators, and the consistency of a cross validation criterion for order determination. We compare the ensemble estimators with other estimators in a wide variety of models, and establish their competent performance.Comment: Published in at http://dx.doi.org/10.1214/11-AOS950 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

An entanglement measure for n-qubits

Recently, Coffman, Kundu, and Wootters introduced the residual entanglement for three qubits to quantify the three-qubit entanglement in Phys. Rev. A 61, 052306 (2000). In Phys. Rev. A 65, 032304 (2007), we defined the residual entanglement for $n$ qubits, whose values are between 0 and 1. In this paper, we want to show that the residual entanglement for $n$ qubits is a natural measure of entanglement by demonstrating the following properties. (1). It is SL-invariant, especially LU-invariant. (2). It is an entanglement monotone. (3). It is invariant under permutations of the qubits. (4). It vanishes or is multiplicative for product states.Comment: 16 pages, no figure

Is hadronic flow produced in p--Pb collisions at the Large Hadron Collider?

Using the Ultra-relativistic Quantum Molecular Dynamics ({\tt UrQMD}) model, we investigate the azimuthal correlations in p--Pb collisions at $\sqrt{s_{_{\rm NN}}}=5.02$ TeV. It is shown that the simulated hadronic p--Pb system can not generate the collective flow signatures, but mainly behaves as a non-flow dominant system. However, the characteristic $v_{2}(p_{\rm T})$ mass-ordering of pions, kaons and protons is observed in {\tt UrQMD} simulations, which is the consequence of hadronic interactions and not necessarily associated with strong fluid-like expansions.Comment: 4 pages, 4 figures, proceedings for the 12th International Conference on Nucleus-Nucleus Collisions (21-26 June 2015, Catania

Method for classifying multiqubit states via the rank of the coefficient matrix and its application to four-qubit states

We construct coefficient matrices of size 2^l by 2^{n-l} associated with pure n-qubit states and prove the invariance of the ranks of the coefficient matrices under stochastic local operations and classical communication (SLOCC). The ranks give rise to a simple way of partitioning pure n-qubit states into inequivalent families and distinguishing degenerate families from one another under SLOCC. Moreover, the classification scheme via the ranks of coefficient matrices can be combined with other schemes to build a more refined classification scheme. To exemplify we classify the nine families of four qubits introduced by Verstraete et al. [Phys. Rev. A 65, 052112 (2002)] further into inequivalent subfamilies via the ranks of coefficient matrices, and as a result, we find 28 genuinely entangled families and all the degenerate classes can be distinguished up to permutations of the four qubits. We also discuss the completeness of the classification of four qubits into nine families