1,545 research outputs found
Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements
We introduce the problem of constructing weighted complex projective
2-designs from the union of a family of orthonormal bases. If the weight
remains constant across elements of the same basis, then such designs can be
interpreted as generalizations of complete sets of mutually unbiased bases,
being equivalent whenever the design is composed of d+1 bases in dimension d.
We show that, for the purpose of quantum state determination, these designs
specify an optimal collection of orthogonal measurements. Using highly
nonlinear functions on abelian groups, we construct explicit examples from d+2
orthonormal bases whenever d+1 is a prime power, covering dimensions d=6, 10,
and 12, for example, where no complete sets of mutually unbiased bases have
thus far been found.Comment: 28 pages, to appear in J. Math. Phy
Quantum Fourier transform, Heisenberg groups and quasiprobability distributions
This paper aims to explore the inherent connection among Heisenberg groups,
quantum Fourier transform and (quasiprobability) distribution functions.
Distribution functions for continuous and finite quantum systems are examined
first as a semiclassical approach to quantum probability distribution. This
leads to studying certain functionals of a pair of "conjugate" observables,
connected via the quantum Fourier transform. The Heisenberg groups emerge
naturally from this study and we take a rapid look at their representations.
The quantum Fourier transform appears as the intertwining operator of two
equivalent representation arising out of an automorphism of the group.
Distribution functions correspond to certain distinguished sets in the group
algebra. The marginal properties of a particular class of distribution
functions (Wigner distributions) arise from a class of automorphisms of the
group algebra of the Heisenberg group. We then study the reconstruction of
Wigner function from the marginal distributions via inverse Radon transform
giving explicit formulas. We consider applications of our approach to quantum
information processing and quantum process tomography.Comment: 39 page
On the reconstruction of planar lattice-convex sets from the covariogram
A finite subset of is said to be lattice-convex if is
the intersection of with a convex set. The covariogram of
is the function associating to each u \in
\integer^d the cardinality of . Daurat, G\'erard, and Nivat and
independently Gardner, Gronchi, and Zong raised the problem on the
reconstruction of lattice-convex sets from . We provide a partial
positive answer to this problem by showing that for and under mild extra
assumptions, determines up to translations and reflections. As a
complement to the theorem on reconstruction we also extend the known
counterexamples (i.e., planar lattice-convex sets which are not
reconstructible, up to translations and reflections) to an infinite family of
counterexamples.Comment: accepted in Discrete and Computational Geometr
Combinatorics and Geometry of Transportation Polytopes: An Update
A transportation polytope consists of all multidimensional arrays or tables
of non-negative real numbers that satisfy certain sum conditions on subsets of
the entries. They arise naturally in optimization and statistics, and also have
interest for discrete mathematics because permutation matrices, latin squares,
and magic squares appear naturally as lattice points of these polytopes.
In this paper we survey advances on the understanding of the combinatorics
and geometry of these polyhedra and include some recent unpublished results on
the diameter of graphs of these polytopes. In particular, this is a thirty-year
update on the status of a list of open questions last visited in the 1984 book
by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure
A tomographic setting for quasi-distribution functions
The method of constructing the tomographic probability distributions
describing quantum states in parallel with density operators is presented.
Known examples of Husimi-Kano quasi-distribution and photon number tomography
are reconsidered in the new setting. New tomographic schemes based on coherent
states and nonlinear coherent states of deformed oscillators, including
q-oscillators, are suggested. The associated identity decompositions providing
Gram-Schmidt operators are explicitly given, and contact with the Agarwal-Wolf
-operator ordering theory is made.Comment: A slightly enlarged version in which contact with the Agarwal-Wolf
-operator ordering theory is mad
Lipschitz stability for the electrostatic inverse boundary value problem with piecewise linear conductivities
We consider the electrostatic inverse boundary value problem also known as
electrical impedance tomography (EIT) for the case where the conductivity is a
piecewise linear function on a domain and we show
that a Lipschitz stability estimate for the conductivity in terms of the local
Dirichlet-to-Neumann map holds true.Comment: 28 pages. arXiv admin note: text overlap with arXiv:1405.047
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