3,665 research outputs found
On inequalities for normalized Schur functions
We prove a conjecture of Cuttler et al.~[2011] [A. Cuttler, C. Greene, and M.
Skandera; \emph{Inequalities for symmetric means}. European J. Combinatorics,
32(2011), 745--761] on the monotonicity of \emph{normalized Schur functions}
under the usual (dominance) partial-order on partitions. We believe that our
proof technique may be helpful in obtaining similar inequalities for other
symmetric functions.Comment: This version fixes the error of the previous on
Exact inequalities for sums of asymmetric random variables, with applications
Let \BS_1,...,\BS_n be independent identically distributed random variables
each having the standardized Bernoulli distribution with parameter .
Let if and
if . Let . Let be such a function that and
are nondecreasing and convex. Then it is proved that for all nonnegative
numbers one has the inequality \E f(c_1\BS_1+...+c_n\BS_n)\le\E
f(s^{(m)}\cdot(\BS_1+...+\BS_n)), where . The lower bound on is exact for each
. Moreover, \E f(c_1\BS_1+...+c_n\BS_n) is Schur-concave in
. A number of related results are presented, including
ones for the ``symmetric'' case.
A number of corollaries are obtained, including upper bounds on generalized
moments and tail probabilities of (super)martingales with differences of
bounded asymmetry, and also upper bounds on the maximal function of such
(super)martingales. It is shown that these results may be important in certain
statistical applications.Comment: 41 pages; a minor inaccuracy in Remark 1.4 is corrected; a few
references and short comments are adde
Positive maps, majorization, entropic inequalities, and detection of entanglement
In this paper, we discuss some general connections between the notions of
positive map, weak majorization and entropic inequalities in the context of
detection of entanglement among bipartite quantum systems. First, basing on the
fact that any positive map can
be written as the difference between two completely positive maps
, we propose a possible way to generalize the
Nielsen--Kempe majorization criterion. Then we present two methods of
derivation of some general classes of entropic inequalities useful for the
detection of entanglement. While the first one follows from the aforementioned
generalized majorization relation and the concept of the Schur--concave
decreasing functions, the second is based on some functional inequalities. What
is important is that, contrary to the Nielsen--Kempe majorization criterion and
entropic inequalities, our criteria allow for the detection of entangled states
with positive partial transposition when using indecomposable positive maps. We
also point out that if a state with at least one maximally mixed subsystem is
detected by some necessary criterion based on the positive map , then
there exist entropic inequalities derived from (by both procedures)
that also detect this state. In this sense, they are equivalent to the
necessary criterion [I\ot\Lambda](\varrho_{AB})\geq 0. Moreover, our
inequalities provide a way of constructing multi--copy entanglement witnesses
and therefore are promising from the experimental point of view. Finally, we
discuss some of the derived inequalities in the context of recently introduced
protocol of state merging and possibility of approximating the mean value of a
linear entanglement witness.Comment: the published version, 25 pages in NJP format, 6 figure
On the monotonicity of scalar curvature in classical and quantum information geometry
We study the statistical monotonicity of the scalar curvature for the
alpha-geometries on the simplex of probability vectors. From the results
obtained and from numerical data we are led to some conjectures about quantum
alpha-geometries and Wigner-Yanase-Dyson information. Finally we show that this
last conjecture implies the truth of the Petz conjecture about the monotonicity
of the scalar curvature of the Bogoliubov-Kubo-Mori monotone metric.Comment: 20 pages, 2 .eps figures; (v2) section 2 rewritten, typos correcte
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