Variation of discrete spectra for non-selfadjoint perturbations of selfadjoint operators

Let B=A+K where A is a bounded selfadjoint operator and K is an element of the von Neumann-Schatten ideal S_p with p>1. Let {\lambda_n} denote an enumeration of the discrete spectrum of B. We show that \sum_n \dist(\lambda_n, \sigma(A))^p is bounded from above by a constant multiple of |K|_p^p. We also derive a unitary analog of this estimate and apply it to obtain new estimates on zero-sets of Cauchy transforms.Comment: Differences to previous version: Extended Introduction, new Section 5, additional references. To appear in Int. Eq. Op. Theor

Quantum Correlations in Large-Dimensional States of High Symmetry

In this article, we investigate how quantum correlations behave for the so-called Werner and pseudo-pure families of states. The latter refers to states formed by mixing any pure state with the totally mixed state. We derive closed expressions for the Quantum Discord (QD) and the Relative Entropy of Quantumness (REQ) for these families of states. For Werner states, the classical correlations are seen to vanish in high dimensions while the amount of quantum correlations remain bounded and become independent of whether or not the the state is entangled. For pseudo-pure states, nearly the opposite effect is observed with both the quantum and classical correlations growing without bound as the dimension increases and only as the system becomes more entangled. Finally, we verify that pseudo-pure states satisfy the conjecture of [\textit{Phys. Rev. A} \textbf{84}, 052110 (2011)] which says that the Geometric Measure of Discord (GD) always upper bounds the squared Negativity of the state

Generating qudits with d=3,4 encoded on two-photon states

We present an experimental method to engineer arbitrary pure states of qudits with d=3,4 using linear optics and a single nonlinear crystal.Comment: 4 pages, 1 eps figure. Minor changes. The title has been changed for publication on Physical Review

Extensions of Lieb's concavity theorem

The operator function (A,B)\to\tr f(A,B)(K^*)K, defined on pairs of bounded self-adjoint operators in the domain of a function f of two real variables, is convex for every Hilbert Schmidt operator K, if and only if f is operator convex. As a special case we obtain a new proof of Lieb's concavity theorem for the function (A,B)\to\tr A^pK^*B^{q}K, where p and q are non-negative numbers with sum p+q\le 1. In addition, we prove concavity of the operator function (A,B)\to \tr(A(A+\mu_1)^{-1}K^* B(B+\mu_2)^{-1}K) on its natural domain D_2(\mu_1,\mu_2), cf. Definition 4.1Comment: The format of one reference is changed such that CiteBase can identify i

Second-order Democratic Aggregation

Aggregated second-order features extracted from deep convolutional networks have been shown to be effective for texture generation, fine-grained recognition, material classification, and scene understanding. In this paper, we study a class of orderless aggregation functions designed to minimize interference or equalize contributions in the context of second-order features and we show that they can be computed just as efficiently as their first-order counterparts and they have favorable properties over aggregation by summation. Another line of work has shown that matrix power normalization after aggregation can significantly improve the generalization of second-order representations. We show that matrix power normalization implicitly equalizes contributions during aggregation thus establishing a connection between matrix normalization techniques and prior work on minimizing interference. Based on the analysis we present {\gamma}-democratic aggregators that interpolate between sum ({\gamma}=1) and democratic pooling ({\gamma}=0) outperforming both on several classification tasks. Moreover, unlike power normalization, the {\gamma}-democratic aggregations can be computed in a low dimensional space by sketching that allows the use of very high-dimensional second-order features. This results in a state-of-the-art performance on several datasets

The ground state of a class of noncritical 1D quantum spin systems can be approximated efficiently

We study families H_n of 1D quantum spin systems, where n is the number of spins, which have a spectral gap \Delta E between the ground-state and first-excited state energy that scales, asymptotically, as a constant in n. We show that if the ground state |\Omega_m> of the hamiltonian H_m on m spins, where m is an O(1) constant, is locally the same as the ground state |\Omega_n>, for arbitrarily large n, then an arbitrarily good approximation to the ground state of H_n can be stored efficiently for all n. We formulate a conjecture that, if true, would imply our result applies to all noncritical 1D spin systems. We also include an appendix on quasi-adiabatic evolutions.Comment: 9 pages, 1 eps figure, minor change

UV and X-ray Spectral Lines of FeXXIII Ion for Plasma Diagnostics

We have calculated X-ray and UV spectra of Be-like Fe (FeXXIII) ion in collisional-radiative model including all fine-structure transitions among the 2s^2, 2s2p, 2p^2, 2snl, and 2pnl levels where n=3 and 4, adopting data for the collision strengths by Zhang & Sampson (1992) and by Sampson, Goett, & Clark (1984). Some line intensity ratios can be used for the temperature diagnostics. We show 5 ratios in UV region and 9 ratios in X-ray region as a function of electron temperature and density at 0.3keV < T_e < 10keV and $n_e = 1 - 10^{25} cm^{-3}$. The effect of cascade in these line ratios and in the level population densities are discussed.Comment: LaTeX, 18 pages, 10 Postscript figures. To appear in Physica Script

Information theoretic treatment of tripartite systems and quantum channels

A Holevo measure is used to discuss how much information about a given POVM on system $a$ is present in another system $b$, and how this influences the presence or absence of information about a different POVM on $a$ in a third system $c$. The main goal is to extend information theorems for mutually unbiased bases or general bases to arbitrary POVMs, and especially to generalize "all-or-nothing" theorems about information located in tripartite systems to the case of \emph{partial information}, in the form of quantitative inequalities. Some of the inequalities can be viewed as entropic uncertainty relations that apply in the presence of quantum side information, as in recent work by Berta et al. [Nature Physics 6, 659 (2010)]. All of the results also apply to quantum channels: e.g., if \EC accurately transmits certain POVMs, the complementary channel \FC will necessarily be noisy for certain other POVMs. While the inequalities are valid for mixed states of tripartite systems, restricting to pure states leads to the basis-invariance of the difference between the information about $a$ contained in $b$ and $c$.Comment: 21 pages. An earlier version of this paper attempted to prove our main uncertainty relation, Theorem 5, using the achievability of the Holevo quantity in a coding task, an approach that ultimately failed because it did not account for locking of classical correlations, e.g. see [DiVincenzo et al. PRL. 92, 067902 (2004)]. In the latest version, we use a very different approach to prove Theorem

Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators

Let $t\mapsto A(t)$ for $t\in T$ be a $C^M$-mapping with values unbounded operators with compact resolvents and common domain of definition which are self-adjoint or normal. Here $C^M$ stands for C^\om (real analytic), a quasianalytic or non-quasianalytic Denjoy-Carleman class, $C^\infty$, or a H\"older continuity class C^{0,\al}. The parameter domain $T$ is either $\mathbb R$ or $\mathbb R^n$ or an infinite dimensional convenient vector space. We prove and review results on $C^M$-dependence on $t$ of the eigenvalues and eigenvectors of $A(t)$.Comment: 8 page