Olivier Giraud

Université Paris-Sud 11, Orsay, Île-de-France, France

Are you Olivier Giraud?

Claim your profile

Publications (43)155.21 Total impact

  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We propose a generalisation of the Bloch sphere representation for mixed spin states based on covariant matrices introduced by Weinberg in 1964 in the context of quantum field theory. We show that these matrices form a tight frame, enabling a compact and transparent representation of density matrices of arbitrary spin in terms of tensors that share the most important properties of Bloch vectors. We investigate the properties of this representation, and give various applications, notably a novel characterization of anti-coherent spin states.
    09/2014;
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We expose two scenarios for the breakdown of quantum multifractality under the effect of perturbations. In the first scenario, multifractality survives below a certain scale of the quantum fluctuations. In the other one, the fluctuations of the wave functions are changed at every scale and each multifractal dimension smoothly goes to the ergodic value. We use as generic examples a one-dimensional dynamical system and the three-dimensional Anderson model at the metal-insulator transition. Our results suggest that quantum multifractality breakdown is universal and follows one of these two scenarios depending on the perturbation. We also discuss the experimental implications.
    Physical Review Letters 06/2014; 112:234101. · 7.73 Impact Factor
  • Source
    V. Kandiah, B. Georgeot, O. Giraud
    [Show abstract] [Hide abstract]
    ABSTRACT: We analyze the game of go from the point of view of complex networks. We construct three different directed networks of increasing complexity, defining nodes as local patterns on plaquettes of increasing sizes, and links as actual successions of these patterns in databases of real games. We discuss the peculiarities of these networks compared to other types of networks. We explore the ranking vectors and community structure of the networks and show that this approach enables to extract groups of moves with common strategic properties. We also investigate different networks built from games with players of different levels or from different phases of the game. We discuss how the study of the community structure of these networks may help to improve the computer simulations of the game. More generally, we believe such studies may help to improve the understanding of human decision process.
    05/2014;
  • Source
    E Bogomolny, O Giraud
    [Show abstract] [Hide abstract]
    ABSTRACT: For random matrices with treelike structure there exists a recursive relation for the local Green functions whose solution permits us to find directly many important quantities in the limit of infinite matrix dimensions. The purpose of this article is to investigate and compare expressions for the spectral density of random regular graphs, based on easy approximations for real solutions of the recursive relation valid for trees with large coordination number. The obtained formulas are in a good agreement with the results of numerical calculations even for small coordination number.
    Physical Review E 12/2013; 88(6-1):062811. · 2.31 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We investigate the set of quantum channels acting on a single qubit. We provide a compact generalization of the Fujiwara-Algoet conditions for complete positivity to non-unital qubit channels, which we then use to characterize the possible geometric forms of the pure output of the channel. We provide universal sets of quantum channels for all unital qubit channels as well as for all extremal (not necessarily unital) qubit channels, in the sense that all qubit channels in these sets can be obtained by concatenation of channels in the corresponding universal set. We also show that our universal sets are essentially minimal.
    06/2013;
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We calculate analytically, for finite-size matrices, joint probability densities of ratios of level spacings in ensembles of random matrices characterized by their associated confining potential. We focus on the ratios of two spacings between three consecutive real eigenvalues, as well as certain generalizations such as the overlapping ratios. The resulting formulas are further analyzed in detail in two specific cases: the beta-Hermite and the beta-Laguerre cases, for which we offer explicit calculations for small N. The analytical results are in excellent agreement with numerical simulations of usual random matrix ensembles, and with the level statistics of a quantum many-body lattice model and zeros of the Riemann zeta function.
    Journal of Physics A Mathematical and Theoretical 05/2013; 46(35). · 1.77 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We derive expressions for the probability distribution of the ratio of two consecutive level spacings for the classical ensembles of random matrices. This ratio distribution was recently introduced to study spectral properties of many-body problems, as, contrary to the standard level spacing distributions, it does not depend on the local density of states. Our Wigner-like surmises are shown to be very accurate when compared to numerics and exact calculations in the large matrix size limit. Quantitative improvements are found through a polynomial expansion. Examples from a quantum many-body lattice model and from zeros of the Riemann zeta function are presented.
    Physical Review Letters 02/2013; 110(8):084101. · 7.73 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We study a version of the mathematical Ruijsenaars-Schneider model and reinterpret it physically in order to describe the spreading with time of quantum wave packets in a system where multifractality can be tuned by varying a parameter. We compare different methods to measure the multifractality of wave packets and identify the best one. We find the multifractality to decrease with time until it reaches an asymptotic limit, which is different from the multifractality of eigenvectors but related to it, as is the rate of the decrease. Our results could guide the study of experimental situations where multifractality is present in quantum systems.
    Physical Review E 11/2012; 86(5-2):056215. · 2.31 Impact Factor
  • Source
    E Bogomolny, O Giraud
    [Show abstract] [Hide abstract]
    ABSTRACT: We construct perturbation series for the qth moment of eigenfunctions of various critical random-matrix ensembles in the strong multifractality regime close to localization. Contrary to previous investigations, our results are valid in the region q<1/2. Our findings allow one to verify, at first leading orders in the strong multifractality limit, the symmetry relation for anomalous fractal dimensions Δ(q)=Δ(1-q), recently conjectured for critical models where an analog of the metal-insulator transition takes place. It is known that this relation is verified at leading order in the weak multifractality regime. Our results thus indicate that this symmetry holds in both limits of small and large coupling constant. For general values of the coupling constant we present careful numerical verifications of this symmetry relation for different critical random-matrix ensembles. We also present an example of a system closely related to one of these critical ensembles, but where the symmetry relation, at least numerically, is not fulfilled.
    Physical Review E 04/2012; 85(4 Pt 2):046208. · 2.31 Impact Factor
  • Source
    Olivier Giraud, Petr Braun, Daniel Braun
    [Show abstract] [Hide abstract]
    ABSTRACT: We give an explicit parametrization of the set of mixed quantum states and of the set of mixed classical states for a spin--1. Classical states are defined as states with a positive Glauber-Sudarshan P-function. They are at the same time the separable symmetric states of two qubits. We explore the geometry of this set, and show that its boundary consists of a two-parameter family of ellipsoids. The boundary does not contain any facets, but includes straight-lines corresponding to mixtures of pure classical states.
    Physical Review A 10/2011; · 3.04 Impact Factor
  • Source
    E Bogomolny, O Giraud
    [Show abstract] [Hide abstract]
    ABSTRACT: Fractal dimensions of eigenfunctions for various critical random matrix ensembles are investigated in perturbation series in the regimes of strong and weak multifractality. In both regimes, we obtain expressions similar to those of the critical banded random matrix ensemble extensively discussed in the literature. For certain ensembles, the leading-order term for weak multifractality can be calculated within standard perturbation theory. For other models, such a direct approach requires modifications, which are briefly discussed. Our analytical formulas are in good agreement with numerical calculations.
    Physical Review E 09/2011; 84(3 Pt 2):036212. · 2.31 Impact Factor
  • Source
    Bertrand Georgeot, Olivier Giraud
    [Show abstract] [Hide abstract]
    ABSTRACT: We study the game of go from a complex network perspective. We construct a directed network using a suitable definition of tactical moves including local patterns, and study this network for different datasets of professional tournaments and amateur games. The move distribution follows Zipf's law and the network is scale free, with statistical peculiarities different from other real directed networks, such as e. g. the World Wide Web. These specificities reflect in the outcome of ranking algorithms applied to it. The fine study of the eigenvalues and eigenvectors of matrices used by the ranking algorithms singles out certain strategic situations. Our results should pave the way to a better modelization of board games and other types of human strategic scheming.
    EPL (Europhysics Letters) 05/2011; · 2.26 Impact Factor
  • Source
    E. Bogomolny, O. Giraud, C. Schmit
    [Show abstract] [Hide abstract]
    ABSTRACT: We propose new classes of random matrix ensembles whose statistical properties are intermediate between statistics of Wigner-Dyson random matrices and Poisson statistics. The construction is based on integrable N-body classical systems with a random distribution of momenta and coordinates of the particles. The Lax matrices of these systems yield random matrix ensembles whose joint distribution of eigenvalues can be calculated analytically thanks to integrability of the underlying system. Formulas for spacing distributions and level compressibility are obtained for various instances of such ensembles.
    Nonlinearity 04/2011; 24(11). · 1.60 Impact Factor
  • Source
    O. Giraud, K. Thas
    [Show abstract] [Hide abstract]
    ABSTRACT: In a celebrated paper '"Can one hear the shape of a drum?"' M. Kac [Amer. Math. Monthly 73, 1 (1966)] asked his famous question about the existence of nonisometric billiards having the same spectrum of the Laplacian. This question was eventually answered positively in 1992 by the construction of noncongruent planar isospectral pairs. This review highlights mathematical and physical aspects of isospectrality.
    Review of Modern Physics 01/2011; 82(3). · 44.98 Impact Factor
  • Source
    E Bogomolny, O Giraud
    [Show abstract] [Hide abstract]
    ABSTRACT: Based on numerical and perturbation series arguments we conjecture that for certain critical random matrix models the information dimension of eigenfunctions D(1) and the spectral compressibility χ are related by the simple equation χ+D(1)/d=1, where d is system dimensionality.
    Physical Review Letters 01/2011; 106(4):044101. · 7.73 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We study numerically multifractal properties of two models of one-dimensional quantum maps: a map with pseudointegrable dynamics and intermediate spectral statistics and a map with an Anderson-like transition recently implemented with cold atoms. Using extensive numerical simulations, we compute the multifractal exponents of quantum wave functions and study their properties, with the help of two different numerical methods used for classical multifractal systems (box-counting and wavelet methods). We compare the results of the two methods over a wide range of values. We show that the wave functions of the Anderson map display a multifractal behavior similar to eigenfunctions of the three-dimensional Anderson transition but of a weaker type. Wave functions of the intermediate map share some common properties with eigenfunctions at the Anderson transition (two sets of multifractal exponents, with similar asymptotic behavior), but other properties are markedly different (large linear regime for multifractal exponents even for strong multifractality, different distributions of moments of wave functions, and absence of symmetry of the exponents). Our results thus indicate that the intermediate map presents original properties, different from certain characteristics of the Anderson transition derived from the nonlinear sigma model. We also discuss the importance of finite-size effects.
    Physical Review E 10/2010; 82(4 Pt 2):046206. · 2.31 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We build a quantum algorithm which uses the Grover quantum search procedure in order to sample the exact equilibrium distribution of a wide range of classical statistical mechanics systems. The algorithm is based on recently developed exact Monte Carlo sampling methods, and yields a polynomial gain compared to classical procedures.
    Physical Review Letters 06/2010; 104(25):250502. · 7.73 Impact Factor
  • Source
    Olivier Giraud, Petr Braun, Daniel Braun
    [Show abstract] [Hide abstract]
    ABSTRACT: We introduce a measure of 'quantumness' for any quantum state in a finite-dimensional Hilbert space, based on the distance between the state and the convex set of classical states. The latter are defined as states that can be written as a convex sum of projectors onto coherent states. We derive the general properties of this measure of non-classicality and use it to identify, for a given dimension of Hilbert space, the 'Queen of Quantum' (QQ) states, i.e. the most non-classical quantum states. In three dimensions, we obtain the QQ state analytically and show that it is unique up to rotations. In up to 11-dimensional Hilbert spaces, we find the QQ states numerically, and show that in terms of their Majorana representation they are highly symmetric bodies, which for dimensions 5 and 7 correspond to Platonic bodies.
    New Journal of Physics 06/2010; 12(6):063005. · 4.06 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We study numerically the spectrum and eigenstate properties of the Google matrix of various examples of directed networks such as vocabulary networks of dictionaries and university World Wide Web networks. The spectra have gapless structure in the vicinity of the maximal eigenvalue for Google damping parameter α equal to unity. The vocabulary networks have relatively homogeneous spectral density, while university networks have pronounced spectral structures which change from one university to another, reflecting specific properties of the networks. We also determine specific properties of eigenstates of the Google matrix, including the PageRank. The fidelity of the PageRank is proposed as a characterization of its stability.
    Physical Review E 05/2010; 81(5 Pt 2):056109. · 2.31 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We propose a detailed study of the geometric entanglement properties of pure symmetric N-qubit states, focusing more particularly on the identification of symmetric states with a high geometric entanglement and how their entanglement behaves asymptotically for large N. We show that much higher geometric entanglement with improved asymptotical behavior can be obtained in comparison with the highly entangled balanced Dicke states studied previously. We also derive an upper bound for the geometric measure of entanglement of symmetric states. The connection with the quantumness of a state is discussed.
    Physical Review A 03/2010; · 3.04 Impact Factor

Publication Stats

332 Citations
155.21 Total Impact Points

Institutions

  • 2009–2014
    • Université Paris-Sud 11
      • Laboratoire de Physique Théorique et Modèles Statistiques
      Orsay, Île-de-France, France
  • 2007
    • University of Toulouse
      Tolosa de Llenguadoc, Midi-Pyrénées, France
  • 2005
    • Paul Sabatier University - Toulouse III
      • Laboratoire de Physique Théorique - UMR 5152 - LPT
      Toulouse, Midi-Pyrenees, France
  • 2004
    • University of Bristol
      • School of Mathematics
      Bristol, England, United Kingdom