17,430 research outputs found
Bounds of restricted isometry constants in extreme asymptotics: formulae for Gaussian matrices
Restricted Isometry Constants (RICs) provide a measure of how far from an
isometry a matrix can be when acting on sparse vectors. This, and related
quantities, provide a mechanism by which standard eigen-analysis can be applied
to topics relying on sparsity. RIC bounds have been presented for a variety of
random matrices and matrix dimension and sparsity ranges. We provide explicitly
formulae for RIC bounds, of n by N Gaussian matrices with sparsity k, in three
settings: a) n/N fixed and k/n approaching zero, b) k/n fixed and n/N
approaching zero, and c) n/N approaching zero with k/n decaying inverse
logrithmically in N/n; in these three settings the RICs a) decay to zero, b)
become unbounded (or approach inherent bounds), and c) approach a non-zero
constant. Implications of these results for RIC based analysis of compressed
sensing algorithms are presented.Comment: 40 pages, 5 figure
The out-equilibrium 2D Ising spin glass: almost, but not quite, a free-field theory
We consider the spatial correlation function of the two-dimensional Ising
spin glass under out-equilibrium conditions. We pay special attention to the
scaling limit reached upon approaching zero temperature. The field-theory of a
non-interacting field makes a surprisingly good job at describing the spatial
shape of the correlation function of the out-equilibrium Edwards-Anderson Ising
model in two dimensions.Comment: 20 pages + 5 Figure
Minimum output entropy of a non-Gaussian quantum channel
We introduce a model of non-Gaussian quantum channel that stems from the
combination of two physically relevant processes occurring in open quantum
systems, namely amplitude damping and dephasing. For it we find input states
approaching zero output entropy, while respecting the input energy constraint.
These states fully exploit the infinite dimensionality of the Hilbert space.
Upon truncation of the latter, the minimum output entropy remains finite and
optimal input states for such a case are conjectured thanks to numerical
evidences
Nonclassical Radiation from Thermal Cavities in the Ultrastrong Coupling Regime
Thermal or chaotic light sources emit radiation characterized by a slightly
enhanced probability of emitting photons in bunches, described by a zero-delay
second-order correlation function . Here we explore
photon-coincidence counting statistics of thermal cavities in the ultrastrong
coupling regime, where the atom-cavity coupling rate becomes comparable to the
cavity resonance frequency. We find that, depending on the system temperature
and coupling rate, thermal photons escaping the cavity can display very
different statistical behaviors, characterised by second-order correlation
functions approaching zero or greatly exceeding two.Comment: results on frequency resolved photon correlations added, to appear in
Phys. Rev. Let
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