4,942 research outputs found
On the meaning and interpretation of Tomography in abstract Hilbert spaces
The mechanism of describing quantum states by standard probability
(tomographic one) instead of wave function or density matrix is elucidated.
Quantum tomography is formulated in an abstract Hilbert space framework, by
means of the identity decompositions in the Hilbert space of hermitian linear
operators, with trace formula as scalar product of operators. Decompositions of
identity are considered with respect to over-complete families of projectors
labeled by extra parameters and containing a measure, depending on these
parameters. It plays the role of a Gram-Schmidt orthonormalization kernel. When
the measure is equal to one, the decomposition of identity coincides with a
positive operator valued measure (POVM) decomposition. Examples of spin
tomography, photon number tomography and symplectic tomography are reconsidered
in this new framework.Comment: Submitted to Phys. Lett.
Quantum Tomography twenty years later
A sample of some relevant developments that have taken place during the last
twenty years in classical and quantum tomography are displayed. We will present
a general conceptual framework that provides a simple unifying mathematical
picture for all of them and, as an effective use of it, three subjects have
been chosen that offer a wide panorama of the scope of classical and quantum
tomography: tomography along lines and submanifolds, coherent state tomography
and tomography in the abstract algebraic setting of quantum systems
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
Groupoids and the tomographic picture of quantum mechanics
The existing relation between the tomographic description of quantum states
and the convolution algebra of certain discrete groupoids represented on
Hilbert spaces will be discussed. The realizations of groupoid algebras based
on qudit, photon-number (Fock) states and symplectic tomography quantizers and
dequantizers will be constructed. Conditions for identifying the convolution
product of groupoid functions and the star--product arising from a
quantization--dequantization scheme will be given. A tomographic approach to
construct quasi--distributions out of suitable immersions of groupoids into
Hilbert spaces will be formulated and, finally, intertwining kernels for such
generalized symplectic tomograms will be evaluated explicitly
Local Analysis of Inverse Problems: H\"{o}lder Stability and Iterative Reconstruction
We consider a class of inverse problems defined by a nonlinear map from
parameter or model functions to the data. We assume that solutions exist. The
space of model functions is a Banach space which is smooth and uniformly
convex; however, the data space can be an arbitrary Banach space. We study
sequences of parameter functions generated by a nonlinear Landweber iteration
and conditions under which these strongly converge, locally, to the solutions
within an appropriate distance. We express the conditions for convergence in
terms of H\"{o}lder stability of the inverse maps, which ties naturally to the
analysis of inverse problems
Alternative commutation relations, star products and tomography
Invertible maps from operators of quantum obvservables onto functions of
c-number arguments and their associative products are first assessed. Different
types of maps like Weyl-Wigner-Stratonovich map and s-ordered quasidistribution
are discussed. The recently introduced symplectic tomography map of observables
(tomograms) related to the Heisenberg-Weyl group is shown to belong to the
standard framework of the maps from quantum observables onto the c-number
functions. The star-product for symbols of the quantum-observable for each one
of the maps (including the tomographic map) and explicit relations among
different star-products are obtained. Deformations of the Moyal star-product
and alternative commutation relations are also considered.Comment: LATEX plus two style files, to appear in J. Phys.
The SIC Question: History and State of Play
Recent years have seen significant advances in the study of symmetric
informationally complete (SIC) quantum measurements, also known as maximal sets
of complex equiangular lines. Previously, the published record contained
solutions up to dimension 67, and was with high confidence complete up through
dimension 50. Computer calculations have now furnished solutions in all
dimensions up to 151, and in several cases beyond that, as large as dimension
844. These new solutions exhibit an additional type of symmetry beyond the
basic definition of a SIC, and so verify a conjecture of Zauner in many new
cases. The solutions in dimensions 68 through 121 were obtained by Andrew
Scott, and his catalogue of distinct solutions is, with high confidence,
complete up to dimension 90. Additional results in dimensions 122 through 151
were calculated by the authors using Scott's code. We recap the history of the
problem, outline how the numerical searches were done, and pose some
conjectures on how the search technique could be improved. In order to
facilitate communication across disciplinary boundaries, we also present a
comprehensive bibliography of SIC research.Comment: 16 pages, 1 figure, many references; v3: updating bibliography,
dimension eight hundred forty fou
Information-theoretic postulates for quantum theory
Why are the laws of physics formulated in terms of complex Hilbert spaces?
Are there natural and consistent modifications of quantum theory that could be
tested experimentally? This book chapter gives a self-contained and accessible
summary of our paper [New J. Phys. 13, 063001, 2011] addressing these
questions, presenting the main ideas, but dropping many technical details. We
show that the formalism of quantum theory can be reconstructed from four
natural postulates, which do not refer to the mathematical formalism, but only
to the information-theoretic content of the physical theory. Our starting point
is to assume that there exist physical events (such as measurement outcomes)
that happen probabilistically, yielding the mathematical framework of "convex
state spaces". Then, quantum theory can be reconstructed by assuming that (i)
global states are determined by correlations between local measurements, (ii)
systems that carry the same amount of information have equivalent state spaces,
(iii) reversible time evolution can map every pure state to every other, and
(iv) positivity of probabilities is the only restriction on the possible
measurements.Comment: 17 pages, 3 figures. v3: some typos corrected and references updated.
Summarizes the argumentation and results of arXiv:1004.1483. Contribution to
the book "Quantum Theory: Informational Foundations and Foils", Springer
Verlag (http://www.springer.com/us/book/9789401773027), 201
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