Recent interest in aspects common to quantum information and condensed matter has prompted a flurry of activity at the border of these disciplines that were far distant until a few years ago. Numerous interesting questions have been addressed so far. Here an important part of this field, the properties of the entanglement in many-body systems, are reviewed. The zero and finite temperature properties of entanglement in interacting spin, fermion, and boson model systems are discussed. Both bipartite and multipartite entanglement will be considered. In equilibrium entanglement is shown tightly connected to the characteristics of the phase diagram. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium entangled states are generated and manipulated by means of many-body Hamiltonians.

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Entanglement in many-body systems

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Lanthanide single-molecule magnets.

Simulating quantum mechanics is known to be a difficult computational problem, especially when dealing with large systems However, this difficulty may be overcome by using some controllable quantum system to study another less controllable or accessible quantum system, ie, quantum simulation Quantum simulation promises to have applications in the study of many problems in, eg, condensed-matter physics, high-energy physics, atomic physics, quantum chemistry and cosmology Quantum simulation could be implemented using quantum computers, but also with simpler, analog devices that would require less control, and therefore, would be easier to construct A number of quantum systems such as neutral atoms, ions, polar molecules, electrons in semiconductors, superconducting circuits, nuclear spins and photons have been proposed as quantum simulators This review outlines the main theoretical and experimental aspects of quantum simulation and emphasizes some of the challenges and promises of this fast-growing field

Quantum Simulation

Random-matrix theories in quantum physics : common concepts

Journal of Statistical Mechanics: theory and experiment

We investigate the asymptotic properties of inertial modes confined in a spherical shell when viscosity tends to zero. We first consider the mapping made by the characteristics of the hyperbolic equation (Poincare's equation) satisfied by inviscid solutions. Charac- teristics are straight lines in a meridional section of the shell, and the mapping shows that, generically, these lines converge towards a periodic orbit which acts like an attrac- tor (the associated Lyapunov exponent is always negative or zero). We show that these attractors exist in bands of frequencies the size of which decreases with the number of reflection points of the attractor. At the bounding frequencies the associated Lyapunov exponent is generically either zero or minus infinity. We further show that for a given frequency the number of coexisting attractors is finite. We then examine the relation between this characteristic path and eigensolutions of the inviscid problem and show that in a purely two-dimensional problem, convergence towards an attractor means that the associated velocity field is not square-integrable. We give arguments which generalize this result to three dimensions. Then, using a sphere immersed in a fluid filling the whole space, we study the critical latitude singularity and show that the velocity field diverges as 1/ √ d, d being the distance to the characteristic grazing the inner sphere. We then consider the viscous problem and show how viscosity transforms singularities into internal shear layers which in general betray an attractor expected at the eigenfre- quency of the mode. Investigating the structure of these shear layers, we find that they are nested layers, the thinnest and most internal layer scaling with E 1/3 -scale, E being the Ekman number; for this latter layer, we give its analytical form and show its simi- larity to vertical 1 -shear layers of steady flows. Using an inertial wave packet traveling around an attractor, we give a lower bound on the thickness of shear layers and show how eigenfrequencies can be computed in principle. Finally, we show that as viscosity decreases, eigenfrequencies tend towards a set of values which is not dense in (0,2), contrary to the case of the full sphere ( is the angular velocity of the system). Hence, our geometrical approach opens the possibility of describing the eigenmodes and eigenvalues for astrophysical/geophysical Ekman numbers (10 −10 − 10 −20 ), which are out of reach numerically, and this for a wide class of containers.

Inertial waves in a rotating spherical shell : attractors and asymptotic spectrum

We study a generic model of quantum computer, composed of many qubits coupled by short-range interaction. Above a critical interqubit coupling strength, quantum chaos sets in, leading to quantum ergodicity of the computer eigenstates. In this regime the noninteracting qubit structure disappears, the eigenstates become complex, and the operability of the computer is destroyed. Despite the fact that the spacing between multiqubit states drops exponentially with the number of qubits n, we show that the quantum chaos border decreases only linearly with n. This opens a broad parameter region where the efficient operation of a quantum computer remains possible.

https://www.quantware.ups-tlse.fr/dima/myrefs/my107c.pdf

Quantum chaos border for quantum computing

We investigate the asymptotic properties of inertial modes confined in a spherical shell when viscosity tends to zero. We first consider the mapping made by the characteristics of the hyperbolic equation (Poincare's equation) satisfied by inviscid solutions. Characteristics are straight lines in a meridional section of the shell, and the mapping shows that, generically, these lines converge towards a periodic orbit which acts like an attractor. 
We then examine the relation between this characteristic path and eigensolutions of the inviscid problem and show that in a purely two-dimensional problem, convergence towards an attractor means that the associated velocity field is not square-integrable. We give arguments which generalize this result to three dimensions. We then consider the viscous problem and show how viscosity transforms singularities into internal shear layers which in general betray an attractor expected at the eigenfrequency of the mode. We find that there are nested layers, the thinnest and most internal layer scaling with $E^{1/3}$-scale, $E$ being the Ekman number. Using an inertial wave packet traveling around an attractor, we give a lower bound on the thickness of shear layers and show how eigenfrequencies can be computed in principle. Finally, we show that as viscosity decreases, eigenfrequencies tend towards a set of values which is not dense in $[0,2\Omega]$, contrary to the case of the full sphere ($\Omega$ is the angular velocity of the system). Hence, our geometrical approach opens the possibility of describing the eigenmodes and eigenvalues for astrophysical/geophysical Ekman numbers ($10^{-10}-10^{-20}$), which are out of reach numerically, and this for a wide class of containers.

Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum

In nonlinear dynamics, basins of attraction link a given set of initial conditions to its corresponding final states. This notion appears in a broad range of applications where several outcomes are possible, which is a common situation in neuroscience, economy, astronomy, ecology and many other disciplines. Depending on the nature of the basins, prediction can be difficult even in systems that evolve under deterministic rules. From this respect, a proper classification of this unpredictability is clearly required. To address this issue, we introduce the basin entropy, a measure to quantify this uncertainty. Its application is illustrated with several paradigmatic examples that allow us to identify the ingredients that hinder the prediction of the final state. The basin entropy provides an efficient method to probe the behavior of a system when different parameters are varied. Additionally, we provide a sufficient condition for the existence of fractal basin boundaries: when the basin entropy of the boundaries is larger than log2, the basin is fractal.

Basin entropy: a new tool to analyze uncertainty in dynamical systems.

We study the standard generic quantum computer model, which describes a realistic isolated quantum computer with fluctuations in individual qubit energies and residual short-range interqubit couplings. It is shown that in the limit where the fluctuations and couplings are small compared to the one-qubit energy spacing, the spectrum has a band structure, and a renormalized Hamiltonian is obtained which describes the eigenstate properties inside one band. Studies are concentrated on the central band of the computer (``core'') with the highest density of states. We show that above a critical interqubit coupling strength, quantum chaos sets in, leading to a quantum ergodicity of the computer eigenstates. In this regime the ideal qubit structure disappears, the eigenstates become complex, and the operability of the computer is quickly destroyed. We confirm that the quantum chaos border decreases only linearly with the number of qubits n, although the spacing between multiqubit states drops exponentially with n. The investigation of time evolution in the quantum computer shows that in the quantum chaos regime, an ideal (noninteracting) state quickly disappears, and exponentially many states become mixed after a short chaotic time scale for which the dependence on system parameters is determined. Below the quantum chaos border an ideal state can survive for long times, and an be used for computation. The results show that a broad parameter region does exist where the efficient operation of a quantum computer is possible.

https://www.quantware.ups-tlse.fr/dima/myrefs/my112c.pdf

Bertrand Georgeot

Papers

Inertial waves in a rotating spherical shell : attractors and asymptotic spectrum

Quantum chaos border for quantum computing

Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum

Basin entropy: a new tool to analyze uncertainty in dynamical systems.

Emergence of quantum chaos in the quantum computer core and how to manage it