4,502 research outputs found
The Upper Bound of Frobenius Related Length Functions
In this paper, we study the asymptotic behavior of lengths of \tor modules
of homologies of complexes under the iterations of the Frobenius functor in
positive characteristic. We first give upper bounds to this type of length
functions in lower dimensional cases and then construct a counterexample to the
general situation. The motivation of studying such length functions arose
initially from an asymptotic length criterion given in [D4] which is a
sufficient condition to a special case of nonnegativity of . We
also provide an example to show that this sufficient condition does not hold in
general.Comment: 11 page
Nonclassicality and the concept of local constraints on the photon number distribution
We exploit results from the classical Stieltjes moment problem to bring out
the totality of all the information regarding phase insensitive nonclassicality
of a state as captured by the photon number distribution p_n. Central to our
approach is the realization that n !p_n constitutes the sequence of moments of
a (quasi) probability distribution, notwithstanding the fact that p_n can by
itself be regarded as a probability distribution. This leads to classicality
restrictions on p_n that are local in n involving p_n's for only a small number
of consecutive n's, enabling a critical examination of the conjecture that
oscillation in p_n is a signature of nonclassicality.Comment: Five pages in revtex with one ps figure included using eps
A Complete Characterization of the Gap between Convexity and SOS-Convexity
Our first contribution in this paper is to prove that three natural sum of
squares (sos) based sufficient conditions for convexity of polynomials, via the
definition of convexity, its first order characterization, and its second order
characterization, are equivalent. These three equivalent algebraic conditions,
henceforth referred to as sos-convexity, can be checked by semidefinite
programming whereas deciding convexity is NP-hard. If we denote the set of
convex and sos-convex polynomials in variables of degree with
and respectively, then our main
contribution is to prove that if and
only if or or . We also present a complete
characterization for forms (homogeneous polynomials) except for the case
which is joint work with G. Blekherman and is to be published
elsewhere. Our result states that the set of convex forms in
variables of degree equals the set of sos-convex forms if
and only if or or . To prove these results, we present
in particular explicit examples of polynomials in
and
and forms in
and , and a
general procedure for constructing forms in from nonnegative but not sos forms in variables and degree .
Although for disparate reasons, the remarkable outcome is that convex
polynomials (resp. forms) are sos-convex exactly in cases where nonnegative
polynomials (resp. forms) are sums of squares, as characterized by Hilbert.Comment: 25 pages; minor editorial revisions made; formal certificates for
computer assisted proofs of the paper added to arXi
Nonclassical Moments and their Measurement
Practically applicable criteria for the nonclassicality of quantum states are
formulated in terms of different types of moments. For this purpose the moments
of the creation and annihilation operators, of two quadratures, and of a
quadrature and the photon number operator turn out to be useful. It is shown
that all the required moments can be determined by homodyne correlation
measurements. An example of a nonclassical effect that is easily characterized
by our methods is amplitude-squared squeezing.Comment: 12 pages, 6 figure
Peres-Horodecki separability criterion for continuous variable systems
The Peres-Horodecki criterion of positivity under partial transpose is
studied in the context of separability of bipartite continuous variable states.
The partial transpose operation admits, in the continuous case, a geometric
interpretation as mirror reflection in phase space. This recognition leads to
uncertainty principles, stronger than the traditional ones, to be obeyed by all
separable states. For all bipartite Gaussian states, the Peres-Horodecki
criterion turns out to be necessary and sufficient condition for separability.Comment: 6 pages, no figure
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