The existence of quantum correlations that allow one party to steer the quantum state of another party is a counterintuitive quantum effect that was described at the beginning of the past century. ...Steering occurs if entanglement can be proven even though the description of the measurements on one party is not known, while the other side is characterized. We introduce the concept of steering maps, which allow us to unlock sophisticated techniques that were developed in regular entanglement detection and to use them for certifying steerability. As an application, we show that this allows us to go beyond even the canonical steering scenario; it enables a generalized dimension-bounded steering where one only assumes the Hilbert space dimension on the characterized side, with no description of the measurements. Surprisingly, this does not weaken the detection strength of very symmetric scenarios that have recently been carried out in experiments.
We introduce a new, semi-supervised classification method that extensively exploits knowledge. The method has three steps. First, the manifold regularization mechanism, adapted from the Laplacian ...support vector machine (LapSVM), is adopted to mine the manifold structure embedded in all training data, especially in numerous label-unknown data. Meanwhile, by converting the labels into pairwise constraints, the pairwise constraint regularization formula (PCRF) is designed to compensate for the few but valuable labelled data. Second, by further combining the PCRF with the manifold regularization, the precise manifold and pairwise constraint jointly regularized formula (MPCJRF) is achieved. Third, by incorporating the MPCJRF into the framework of the conventional SVM, our approach, referred to as semi-supervised classification with extensive knowledge exploitation (SSC-EKE), is developed. The significance of our research is fourfold: 1) The MPCJRF is an underlying adjustment, with respect to the pairwise constraints, to the graph Laplacian enlisted for approximating the potential data manifold. This type of adjustment plays the correction role, as an unbiased estimation of the data manifold is difficult to obtain, whereas the pairwise constraints, converted from the given labels, have an overall high confidence level. 2) By transforming the values of the two terms in the MPCJRF such that they have the same range, with a trade-off factor varying within the invariant interval 0, 1), the appropriate impact of the pairwise constraints to the graph Laplacian can be self-adaptively determined. 3) The implication regarding extensive knowledge exploitation is embodied in SSC-EKE. That is, the labelled examples are used not only to control the empirical risk but also to constitute the MPCJRF. Moreover, all data, both labelled and unlabelled, are recruited for the model smoothness and manifold regularization. 4) The complete framework of SSC-EKE organically incorporates multiple theories, such as joint manifold and pairwise constraint-based regularization, smoothness in the reproducing kernel Hilbert space, empirical risk minimization, and spectral methods, which facilitates the preferable classification accuracy as well as the generalizability of SSC-EKE.
In this work, we introduce self-adaptive methods for solving variational inequalities with Lipschitz continuous and quasimonotone mapping(or Lipschitz continuous mapping without monotonicity) in real ...Hilbert space. Under suitable assumptions, the convergence of algorithms are established without the knowledge of the Lipschitz constant of the mapping. The results obtained in this paper extend some recent results in the literature. Some preliminary numerical experiments and comparisons are reported.
In this paper, we introduce two different kinds of iterative algorithms, which are based on the inertial Tseng's method and the viscosity method. They are intended to solve the variational inequality ...problems governed by the mappings of pseudo-monotone type. Strong convergence theorems are established in Hilbert spaces. Practical examples in fuzzy environment are given to show the applicability and effectiveness of the proposed algorithms.
Motivated by some recent twaddles on Mazur rotations problem, we study the “dynamics” of the semigroup of contractive automorphisms of Banach spaces, mostly in finite-dimensional spaces. We focus on ...the metric aspects of the “action” of such semigroups, the size of the orbits and semitransitivity properties, and their impact on the geometry of the unit ball of the underlying space.
Summary
The performance of adaptive estimators that employ embedding in reproducing kernel Hilbert spaces (RKHS) depends on the choice of the location of basis kernel centers. Parameter convergence ...and error approximation rates depend on where and how the kernel centers are distributed in the state‐space. In this article, we develop the theory that relates parameter convergence and approximation rates to the position of kernel centers. We develop criteria for choosing kernel centers in a specific class of systems by exploiting the fact that the state trajectory regularly visits the neighborhood of the positive limit set. Two algorithms, based on centroidal Voronoi tessellations and Kohonen self‐organizing maps, are derived to choose kernel centers in the RKHS embedding method. Finally, we implement these methods on two practical examples and test their effectiveness.
The Rényi entanglement entropy (REE) of the states excited by local operators in two-dimensional irrational conformal field theories (CFTs), especially in Liouville field theory (LFT) and N = 1 ...super-Liouville field theory (SLFT), has been investigated. In particular, the excited states obtained by acting on the vacuum with primary operators were considered. We start from evaluating the second REE in a compact c = 1 free boson field theory at generic radius, which is an irrational CFT. Then we focus on the two special irrational CFTs, e.g., LFT and SLFT. In these theories, the second REE of such local excited states becomes divergent in early and late time limits. For simplicity, we study the memory effect of REE for the two classes of the local excited states in LFT and SLFT. In order to restore the quasiparticles picture, we define the difference of REE between target and reference states, which belong to the same class. The variation of the difference of REE between early and late time limits always coincides with the log of the ratio of the fusion matrix elements between target and reference states. Furthermore, the locally excited states by acting generic descendent operators on the vacuum have been also investigated. The variation of the difference of REE is the summation of the log of the ratio of the fusion matrix elements between the target and reference states and an additional normalization factor. Since the identity operator (or vacuum state) does not live in the Hilbert space of LFT and SLFT and no discrete terms contribute to REE in the intermediate channel, the variation of the difference of REE between target and reference states is no longer the log of the quantum dimension which is shown in the 1 + 1 -dimensional rational CFTs (RCFTs).
In this paper, we investigate the uniform exponential stability of a semi-discrete scheme for a Schrödinger equation under boundary stabilizing feedback control in the natural state space ...<inline-formula><tex-math notation="LaTeX">L^{2}(0,1)</tex-math></inline-formula>. This study is significant since a time domain energy multiplier that allows proving the exponential stability of this continuous Schrödinger system has not yet found, thus leading to a major mathematical challenge to the uniform exponential stability of the corresponding semi-discretization systems, which is an open problem for a long time. Although the powerful frequency domain energy multiplier approach has been used in proving exponential stability for PDEs since 1980s, its use to the uniform exponential stability of the semi-discrete scheme for PDEs has not been reported yet. The difficulty associated with the uniformity is that due to the parameter of the step size, it involves infinitely many matrices in different state spaces that need to be considered simultaneously. Based on the Huang-Prüss frequency domain criterion for uniform exponential stability of a family of <inline-formula><tex-math notation="LaTeX">C_{0}</tex-math></inline-formula>-semigroups in Hilbert spaces, we solve this problem for the first time by proving the uniform boundedness for all the resolvents of these matrices on the imaginary axis. The proof almost exactly follows the procedure for the exponential stability of the continuous counterpart, highlighting the advantage of this discretization method.
Investigating many-body localization (MBL) using exact numerical methods is limited by the exponential growth of the Hilbert space. However, localized eigenstates display multifractality and only ...extend over a vanishing fraction of the Hilbert space. Here, building on this remarkable property, we develop a simple yet efficient decimation scheme to discard the irrelevant parts of the Hilbert space of the random-field Heisenberg chain. This leads to a Hilbert space fragmentation in small clusters, allowing to access larger systems at strong disorder. The MBL transition is quantitatively predicted, together with a geometrical interpretation of MBL multifractality as a shattering of the Hilbert space.
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