80 research outputs found
Wigner function and the probability representation of quantum states
The relation of the Wigner function with the fair probability distribution
called tomographic distribution or quantum tomogram associated with the quantum
state is reviewed. The connection of the tomographic picture of quantum
mechanics with the integral Radon transform of the Wigner quasidistribution is
discussed. The Wigner--Moyal equation for the Wigner function is presented in
the form of kinetic equation for the tomographic probability distribution both
in quantum mechanics and in the classical limit of the Liouville equation. The
calculation of moments of physical observables in terms of integrals with the
state tomographic probability distributions is constructed having a standard
form of averaging in the probability theory. New uncertainty relations for the
position and momentum are written in terms of optical tomograms suitable for
direct experimental check. Some recent experiments on checking the uncertainty
relations including the entropic uncertainty relations are discussed.Comment: 4 p, contribution to the Proceedings of the conference "Wigner 111 -
Colourful and Deep (November 11-13 Budapest, Hungary
Tomographic entropic inequalities in the probability representation of quantum mechanics
A review of the tomographic-probability representation of classical and
quantum states is presented. The tomographic entropies and entropic uncertainty
relations are discussed in connection with ambiguities in the interpretation of
the state tomograms which are considered either as a set of the probability
distributions of random variables depending on extra parameters or as a single
joint probability distribution of these random variables and random parameters
with specific properties of the marginals. Examples of optical tomograms of
photon states, symplectic tomograms, and unitary spin tomograms of qudits are
given. A new universal integral inequality for generic wave function is
obtained on the base of tomographic entropic uncertainty relations.Comment: 9 pages; to be published in AIP Conference Proceedings as a
contribution to the conference "Beauty in Physics: Theory and Experiment"
(the Hacienda Cocoyoc, Morelos, Mexico, May 14--18, 2012
Correlations in a system of classical--like coins simulating spin-1/2 states in the probability representation of quantum mechanics
An analog of classical "hidden variables" for qubit states is presented. The
states of qubit (two-level atom, spin-1/2 particle) are mapped onto the states
of three classical--like coins. The bijective map of the states corresponds to
the presence of correlations of random classical--like variables associated
with the coin positions "up" or "down" and the observables are mapped onto
quantum observables described by Hermitian matrices. The connection of the
classical--coin statistics with the statistical properties of qubits is found
Conventional Quantum Mechanics Without Wave Function and Density Matrix
The tomographic invertable map of the Wigner function onto the positive
probability distribution function is studied. Alternatives to the Schr\"odinger
evolution equation and to the energy level equation written for the positive
probability distribution are discussed. Instead of the transition probability
amplitude (Feynman path integral) a transition probability is introduced.
A new formulation of the conventional quantum mechanics (without wave
function and density matrix) based on the ``probability representation'' of
quantum states is given. An equation for the propagator in the new formulation
of quantum mechanics is derived. Some paradoxes of quantum mechanics are
reconsidered.Comment: Latex, 37 pages, to be published as lectures given at the Latin
American School of Physics, (XXXI ELAF, July-August, 1998, Mexico) in
Casta\~nos, O., Hacyan, S., Lopez-Pe\~na, R., (Eds.), AIP Conference
Proceedings (American Institute of Physics, New York, 1999
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