Bounds on changes in Ritz values for a perturbed invariant subspace of a Hermitian matrix

The Rayleigh-Ritz method is widely used for eigenvalue approximation. Given a matrix $X$ with columns that form an orthonormal basis for a subspace \X, and a Hermitian matrix $A$, the eigenvalues of $X^HAX$ are called Ritz values of $A$ with respect to \X. If the subspace \X is $A$-invariant then the Ritz values are some of the eigenvalues of $A$. If the $A$-invariant subspace \X is perturbed to give rise to another subspace \Y, then the vector of absolute values of changes in Ritz values of $A$ represents the absolute eigenvalue approximation error using \Y. We bound the error in terms of principal angles between \X and \Y. We capitalize on ideas from a recent paper [DOI: 10.1137/060649070] by A. Knyazev and M. Argentati, where the vector of absolute values of differences between Ritz values for subspaces \X and \Y was weakly (sub-)majorized by a constant times the sine of the vector of principal angles between \X and \Y, the constant being the spread of the spectrum of $A$. In that result no assumption was made on either subspace being $A$-invariant. It was conjectured there that if one of the trial subspaces is $A$-invariant then an analogous weak majorization bound should only involve terms of the order of sine squared. Here we confirm this conjecture. Specifically we prove that the absolute eigenvalue error is weakly majorized by a constant times the sine squared of the vector of principal angles between the subspaces \X and \Y, where the constant is proportional to the spread of the spectrum of $A$. For many practical cases we show that the proportionality factor is simply one, and that this bound is sharp. For the general case we can only prove the result with a slightly larger constant, which we believe is artificial.Comment: 12 pages. Accepted to SIAM Journal on Matrix Analysis and Applications (SIMAX

Entanglement of Collectively Interacting Harmonic Chains

We study the ground-state entanglement of one-dimensional harmonic chains that are coupled to each other by a collective interaction as realized e.g. in an anisotropic ion crystal. Due to the collective type of coupling, where each chain interacts with every other one in the same way,the total system shows critical behavior in the direction orthogonal to the chains while the isolated harmonic chains can be gapped and non-critical. We derive lower and most importantly upper bounds for the entanglement,quantified by the von Neumann entropy, between a compact block of oscillators and its environment. For sufficiently large size of the subsystems the bounds coincide and show that the area law for entanglement is violated by a logarithmic correction.Comment: 5 pages, 1 figur

UV/Optical Detections of Candidate Tidal Disruption Events by GALEX and CFHTLS

We present two luminous UV/optical flares from the nuclei of apparently inactive early-type galaxies at z=0.37 and 0.33 that have the radiative properties of a flare from the tidal disruption of a star. In this paper we report the second candidate tidal disruption event discovery in the UV by the GALEX Deep Imaging Survey, and present simultaneous optical light curves from the CFHTLS Deep Imaging Survey for both UV flares. The first few months of the UV/optical light curves are well fitted with the canonical t^(-5/3) power-law decay predicted for emission from the fallback of debris from a tidally disrupted star. Chandra ACIS X-ray observations during the flares detect soft X-ray sources with T_bb= (2-5) x 10^5 K or Gamma > 3 and place limits on hard X-ray emission from an underlying AGN down to L_X (2-10 keV) <~ 10^41 ergs/s. Blackbody fits to the UV/optical spectral energy distributions of the flares indicate peak flare luminosities of > 10^44-10^45 ergs/s. The temperature, luminosity, and light curves of both flares are in excellent agreement with emission from a tidally disrupted main sequence star onto a central black hole of several times 10^7 msun. The observed detection rate of our search over ~ 2.9 deg^2 of GALEX Deep Imaging Survey data spanning from 2003 to 2007 is consistent with tidal disruption rates calculated from dynamical models, and we use these models to make predictions for the detection rates of the next generation of optical synoptic surveys.Comment: 28 pages, 27 figures, 11 tables, accepted to ApJ, final corrections from proofs adde

Lower bounds on the complexity of simulating quantum gates

We give a simple proof of a formula for the minimal time required to simulate a two-qubit unitary operation using a fixed two-qubit Hamiltonian together with fast local unitaries. We also note that a related lower bound holds for arbitrary n-qubit gates.Comment: 6 page