Making Almost Commuting Matrices Commute

Suppose two Hermitian matrices $A,B$ almost commute ($\Vert [A,B] \Vert \leq \delta$). Are they close to a commuting pair of Hermitian matrices, $A',B'$, with $\Vert A-A' \Vert,\Vert B-B'\Vert \leq \epsilon$? A theorem of H. Lin shows that this is uniformly true, in that for every $\epsilon>0$ there exists a $\delta>0$, independent of the size $N$ of the matrices, for which almost commuting implies being close to a commuting pair. However, this theorem does not specify how $\delta$ depends on $\epsilon$. We give uniform bounds relating $\delta$ and $\epsilon$. We provide tighter bounds in the case of block tridiagonal and tridiagonal matrices and a fully constructive method in that case. Within the context of quantum measurement, this implies an algorithm to construct a basis in which we can make a {\it projective} measurement that approximately measures two approximately commuting operators simultaneously. Finally, we comment briefly on the case of approximately measuring three or more approximately commuting operators using POVMs (positive operator-valued measures) instead of projective measurements.Comment: 22 pages; tighter bounds; Note: fixed mistake in proof pointed out by Filonov and Kachkovski

Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems

Gapped ground states of quantum spin systems have been referred to in the physics literature as being `in the same phase' if there exists a family of Hamiltonians H(s), with finite range interactions depending continuously on $s \in [0,1]$, such that for each $s$, H(s) has a non-vanishing gap above its ground state and with the two initial states being the ground states of H(0) and H(1), respectively. In this work, we give precise conditions under which any two gapped ground states of a given quantum spin system that 'belong to the same phase' are automorphically equivalent and show that this equivalence can be implemented as a flow generated by an $s$-dependent interaction which decays faster than any power law (in fact, almost exponentially). The flow is constructed using Hastings' 'quasi-adiabatic evolution' technique, of which we give a proof extended to infinite-dimensional Hilbert spaces. In addition, we derive a general result about the locality properties of the effect of perturbations of the dynamics for quantum systems with a quasi-local structure and prove that the flow, which we call the {\em spectral flow}, connecting the gapped ground states in the same phase, satisfies a Lieb-Robinson bound. As a result, we obtain that, in the thermodynamic limit, the spectral flow converges to a co-cycle of automorphisms of the algebra of quasi-local observables of the infinite spin system. This proves that the ground state phase structure is preserved along the curve of models $H(s), 0\leq s\leq 1$.Comment: Updated acknowledgments and new email address of S

A new family of matrix product states with Dzyaloshinski-Moriya interactions

We define a new family of matrix product states which are exact ground states of spin 1/2 Hamiltonians on one dimensional lattices. This class of Hamiltonians contain both Heisenberg and Dzyaloshinskii-Moriya interactions but at specified and not arbitrary couplings. We also compute in closed forms the one and two-point functions and the explicit form of the ground state. The degeneracy structure of the ground state is also discussed.Comment: 15 pages, 1 figur

Many body physics from a quantum information perspective

The quantum information approach to many body physics has been very successful in giving new insight and novel numerical methods. In these lecture notes we take a vertical view of the subject, starting from general concepts and at each step delving into applications or consequences of a particular topic. We first review some general quantum information concepts like entanglement and entanglement measures, which leads us to entanglement area laws. We then continue with one of the most famous examples of area-law abiding states: matrix product states, and tensor product states in general. Of these, we choose one example (classical superposition states) to introduce recent developments on a novel quantum many body approach: quantum kinetic Ising models. We conclude with a brief outlook of the field.Comment: Lectures from the Les Houches School on "Modern theories of correlated electron systems". Improved version new references adde

Red Queen Coevolution on Fitness Landscapes

Species do not merely evolve, they also coevolve with other organisms. Coevolution is a major force driving interacting species to continuously evolve ex- ploring their fitness landscapes. Coevolution involves the coupling of species fit- ness landscapes, linking species genetic changes with their inter-specific ecological interactions. Here we first introduce the Red Queen hypothesis of evolution com- menting on some theoretical aspects and empirical evidences. As an introduction to the fitness landscape concept, we review key issues on evolution on simple and rugged fitness landscapes. Then we present key modeling examples of coevolution on different fitness landscapes at different scales, from RNA viruses to complex ecosystems and macroevolution.Comment: 40 pages, 12 figures. To appear in "Recent Advances in the Theory and Application of Fitness Landscapes" (H. Richter and A. Engelbrecht, eds.). Springer Series in Emergence, Complexity, and Computation, 201

Cosmological parameters from SDSS and WMAP

We measure cosmological parameters using the three-dimensional power spectrum P(k) from over 200,000 galaxies in the Sloan Digital Sky Survey (SDSS) in combination with WMAP and other data. Our results are consistent with a ``vanilla'' flat adiabatic Lambda-CDM model without tilt (n=1), running tilt, tensor modes or massive neutrinos. Adding SDSS information more than halves the WMAP-only error bars on some parameters, tightening 1 sigma constraints on the Hubble parameter from h~0.74+0.18-0.07 to h~0.70+0.04-0.03, on the matter density from Omega_m~0.25+/-0.10 to Omega_m~0.30+/-0.04 (1 sigma) and on neutrino masses from <11 eV to <0.6 eV (95%). SDSS helps even more when dropping prior assumptions about curvature, neutrinos, tensor modes and the equation of state. Our results are in substantial agreement with the joint analysis of WMAP and the 2dF Galaxy Redshift Survey, which is an impressive consistency check with independent redshift survey data and analysis techniques. In this paper, we place particular emphasis on clarifying the physical origin of the constraints, i.e., what we do and do not know when using different data sets and prior assumptions. For instance, dropping the assumption that space is perfectly flat, the WMAP-only constraint on the measured age of the Universe tightens from t0~16.3+2.3-1.8 Gyr to t0~14.1+1.0-0.9 Gyr by adding SDSS and SN Ia data. Including tensors, running tilt, neutrino mass and equation of state in the list of free parameters, many constraints are still quite weak, but future cosmological measurements from SDSS and other sources should allow these to be substantially tightened.Comment: Minor revisions to match accepted PRD version. SDSS data and ppt figures available at http://www.hep.upenn.edu/~max/sdsspars.htm

A Measurement of the Branching Fraction for the Inclusive B --> X(s) gamma Decays with the Belle Detector

We have measured the branching fraction of the inclusive radiative B meson decay B --> X(s) gamma to be Br(B->X(s)gamma)=(3.36 +/- 0.53(stat) +/- 0.42(sys) +0.50-0.54(th)) x 10^{-4}. The result is based on a sample of 6.07 x 10^6 BBbar events collected at the Upsilon(4S) resonance with the Belle detector at the KEKB asymmetric e^+e^- storage ring.Comment: 14 pages, 6 Postsript figures, uses elsart.cl

Constraining Cut-off Physics in the Cosmic Microwave Background

We investigate the ability to constrain oscillatory features in the primordial power spectrum using current and future cosmic microwave background observations. In particular, we study the observability of an oscillation arising from imprints of physics at the cut-off energy scale. We perform a likelihood analysis on the WMAP data set, and find that the current data set constrains the amplitude of the oscillations to be less than 0.77 at 2-sigma, consistent with a power spectrum without oscillations. In addition, we investigate the fundamental limitations in the measurement of oscillation parameters by studying the constraints from a cosmic variance limited experiment. We find that such an experiment is capable of constraining the amplitude of such oscillations to be below 0.005, implying that reasonable models with cut-off energy scales Lambda>200 H_infl are unobservable through the microwave background.Comment: 16 pages, 7 figures; PRD accepted versio

Disorder-assisted error correction in Majorana chains

It was recently realized that quenched disorder may enhance the reliability of topological qubits by reducing the mobility of anyons at zero temperature. Here we compute storage times with and without disorder for quantum chains with unpaired Majorana fermions - the simplest toy model of a quantum memory. Disorder takes the form of a random site-dependent chemical potential. The corresponding one-particle problem is a one-dimensional Anderson model with disorder in the hopping amplitudes. We focus on the zero-temperature storage of a qubit encoded in the ground state of the Majorana chain. Storage and retrieval are modeled by a unitary evolution under the memory Hamiltonian with an unknown weak perturbation followed by an error-correction step. Assuming dynamical localization of the one-particle problem, we show that the storage time grows exponentially with the system size. We give supporting evidence for the required localization property by estimating Lyapunov exponents of the one-particle eigenfunctions. We also simulate the storage process for chains with a few hundred sites. Our numerical results indicate that in the absence of disorder, the storage time grows only as a logarithm of the system size. We provide numerical evidence for the beneficial effect of disorder on storage times and show that suitably chosen pseudorandom potentials can outperform random ones.Comment: 50 pages, 7 figure

Search for Lorentz and CPT violation using sidereal time dependence of neutrino flavor transitions over a short baseline

A class of extensions of the Standard Model allows Lorentz and CPT violations, which can be identified by the observation of sidereal modulations in the neutrino interaction rate. A search for such modulations was performed using the T2K on-axis near detector. Two complementary methods were used in this study, both of which resulted in no evidence of a signal. Limits on associated Lorentz and CPT-violating terms from the Standard Model extension have been derived by taking into account their correlations in this model for the first time. These results imply such symmetry violations are suppressed by a factor of more than 10 20 at the GeV scale