1,078 research outputs found
Suboptimal quantum-error-correcting procedure based on semidefinite programming
In this paper, we consider a simplified error-correcting problem: for a fixed
encoding process, to find a cascade connected quantum channel such that the
worst fidelity between the input and the output becomes maximum. With the use
of the one-to-one parametrization of quantum channels, a procedure finding a
suboptimal error-correcting channel based on a semidefinite programming is
proposed. The effectiveness of our method is verified by an example of the
bit-flip channel decoding.Comment: 6 pages, no figure, Some notations differ from those in the PRA
versio
Likelihood Analysis of Power Spectra and Generalized Moment Problems
We develop an approach to spectral estimation that has been advocated by
Ferrante, Masiero and Pavon and, in the context of the scalar-valued covariance
extension problem, by Enqvist and Karlsson. The aim is to determine the power
spectrum that is consistent with given moments and minimizes the relative
entropy between the probability law of the underlying Gaussian stochastic
process to that of a prior. The approach is analogous to the framework of
earlier work by Byrnes, Georgiou and Lindquist and can also be viewed as a
generalization of the classical work by Burg and Jaynes on the maximum entropy
method. In the present paper we present a new fast algorithm in the general
case (i.e., for general Gaussian priors) and show that for priors with a
specific structure the solution can be given in closed form.Comment: 17 pages, 4 figure
The Euclidean distance degree of an algebraic variety
The nearest point map of a real algebraic variety with respect to Euclidean
distance is an algebraic function. For instance, for varieties of low rank
matrices, the Eckart-Young Theorem states that this map is given by the
singular value decomposition. This article develops a theory of such nearest
point maps from the perspective of computational algebraic geometry. The
Euclidean distance degree of a variety is the number of critical points of the
squared distance to a generic point outside the variety. Focusing on varieties
seen in applications, we present numerous tools for exact computations.Comment: to appear in Foundations of Computational Mathematic
Neutrino mixing, interval matrices and singular values
We study the properties of singular values of mixing matrices embedded within
an experimentally determined interval matrix. We argue that any physically
admissible mixing matrix needs to have the property of being a contraction.
This condition constrains the interval matrix, by imposing correlations on its
elements and leaving behind only physical mixings that may unveil signs of new
physics in terms of extra neutrino species. We propose a description of the
admissible three-dimensional mixing space as a convex hull over experimentally
determined unitary mixing matrices parametrized by Euler angles which allows us
to select either unitary or nonunitary mixing matrices. The unitarity-breaking
cases are found through singular values and we construct unitary extensions
yielding a complete theory of minimal dimensionality larger than three through
the theory of unitary matrix dilations. We discuss further applications to the
quark sector.Comment: Misprints correcte
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