2,697 research outputs found
Generating random density matrices
We study various methods to generate ensembles of random density matrices of
a fixed size N, obtained by partial trace of pure states on composite systems.
Structured ensembles of random pure states, invariant with respect to local
unitary transformations are introduced. To analyze statistical properties of
quantum entanglement in bi-partite systems we analyze the distribution of
Schmidt coefficients of random pure states. Such a distribution is derived in
the case of a superposition of k random maximally entangled states. For another
ensemble, obtained by performing selective measurements in a maximally
entangled basis on a multi--partite system, we show that this distribution is
given by the Fuss-Catalan law and find the average entanglement entropy. A more
general class of structured ensembles proposed, containing also the case of
Bures, forms an extension of the standard ensemble of structureless random pure
states, described asymptotically, as N \to \infty, by the Marchenko-Pastur
distribution.Comment: 13 pages in latex with 8 figures include
On the structure of the body of states with positive partial transpose
We show that the convex set of separable mixed states of the 2 x 2 system is
a body of constant height. This fact is used to prove that the probability to
find a random state to be separable equals 2 times the probability to find a
random boundary state to be separable, provided the random states are generated
uniformly with respect to the Hilbert-Schmidt (Euclidean) distance. An
analogous property holds for the set of positive-partial-transpose states for
an arbitrary bipartite system.Comment: 10 pages, 1 figure; ver. 2 - minor changes, new proof of lemma
Spectral density of generalized Wishart matrices and free multiplicative convolution
We investigate the level density for several ensembles of positive random
matrices of a Wishart--like structure, , where stands for a
nonhermitian random matrix. In particular, making use of the Cauchy transform,
we study free multiplicative powers of the Marchenko-Pastur (MP) distribution,
, which for an integer yield Fuss-Catalan
distributions corresponding to a product of independent square random
matrices, . New formulae for the level densities are derived
for and . Moreover, the level density corresponding to the
generalized Bures distribution, given by the free convolution of arcsine and MP
distributions is obtained. We also explain the reason of such a curious
convolution. The technique proposed here allows for the derivation of the level
densities for several other cases.Comment: 10 latex pages including 4 figures, Ver 4, minor improvements and
references updat
Extremal spacings between eigenphases of random unitary matrices and their tensor products
Extremal spacings between eigenvalues of random unitary matrices of size N
pertaining to circular ensembles are investigated. Explicit probability
distributions for the minimal spacing for various ensembles are derived for N =
4. We study ensembles of tensor product of k random unitary matrices of size n
which describe independent evolution of a composite quantum system consisting
of k subsystems. In the asymptotic case, as the total dimension N = n^k becomes
large, the nearest neighbor distribution P(s) becomes Poissonian, but
statistics of extreme spacings P(s_min) and P(s_max) reveal certain deviations
from the Poissonian behavior
A Better Definition of the Kilogram
This article reviews several recent proposed redefinitions of the kilogram,
and compares them with respect to practical realizations, uncertainties
(estimated standard deviations), and educational aspects.Comment: 10 pages, no figure
Structural Properties, Order-Disorder Phenomena and Phase Stability of Orotic Acid Crystal Forms
Orotic acid (OTA) is reported to exist in the anhydrous (AH), monohydrate (Hy1) and dimethylsulfoxide monosolvate (SDMSO) forms. In this study we investigate the (de)hydration/desolvation behavior, aiming at an understanding of the elusive structural features of anhydrous OTA by a combination of experimental and computational techniques, namely, thermal analytical methods, gravimetric moisture (de)sorption studies, water activity measurements, X-ray powder diffraction, spectroscopy (vibrational, solid-state NMR), crystal energy landscape and chemical shift calculations. The Hy1 is a highly stable hydrate, which dissociates above 135°C and loses only a small part of the water when stored over desiccants (25°C) for more than one year. In Hy1, orotic acid and water molecules are linked by strong hydrogen bonds in nearly perfectly planar arranged stacked layers. The layers are spaced by 3.1 Å and not linked via hydrogen-bonds. Upon dehydration the X-ray powder diffraction and solid-state NMR peaks become broader indicating some disorder in the anhydrous form. The Hy1 stacking reflection (122) is maintained, suggesting that the OTA molecules are still arranged in stacked layers in the dehydration product. Desolvation of SDMSO, a non-layer structure, results in the same AH phase as observed upon dehydrating Hy1. Depending on the desolvation conditions different levels of order-disorder of layers present in anhydrous OTA are observed, which is also suggested by the computed low energy crystal structures. These structures provide models for stacking faults as intergrowth of different layers is possible. The variability in anhydrate crystals is of practical concern as it affects the moisture dependent stability of AH with respect to hydration
Coarse-grained entanglement classification through orthogonal arrays
Classification of entanglement in multipartite quantum systems is an open
problem solved so far only for bipartite systems and for systems composed of
three and four qubits. We propose here a coarse-grained classification of
entanglement in systems consisting of subsystems with an arbitrary number
of internal levels each, based on properties of orthogonal arrays with
columns. In particular, we investigate in detail a subset of highly entangled
pure states which contains all states defining maximum distance separable
codes. To illustrate the methods presented, we analyze systems of four and five
qubits, as well as heterogeneous tripartite systems consisting of two qubits
and one qutrit or one qubit and two qutrits.Comment: 38 pages, 1 figur
Invariant sets for discontinuous parabolic area-preserving torus maps
We analyze a class of piecewise linear parabolic maps on the torus, namely
those obtained by considering a linear map with double eigenvalue one and
taking modulo one in each component. We show that within this two parameter
family of maps, the set of noninvertible maps is open and dense. For cases
where the entries in the matrix are rational we show that the maximal invariant
set has positive Lebesgue measure and we give bounds on the measure. For
several examples we find expressions for the measure of the invariant set but
we leave open the question as to whether there are parameters for which this
measure is zero.Comment: 19 pages in Latex (with epsfig,amssymb,graphics) with 5 figures in
eps; revised version: section 2 rewritten, new example and picture adde
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