96 research outputs found
Probability representation and quantumness tests for qudits and two-mode light states
Using tomographic-probability representation of spin states, quantum behavior
of qudits is examined. For a general j-qudit state we propose an explicit
formula of quantumness witnetness whose negative average value is incompatible
with classical statistical model. Probability representations of quantum and
classical (2j+1)-level systems are compared within the framework of quantumness
tests. Trough employing Jordan-Schwinger map the method is extended to check
quantumness of two-mode light states.Comment: 5 pages, 2 figures, PDFLaTeX, Contribution to the 11th International
Conference on Squeezed States and Uncertainty Relations (ICSSUR'09), June
22-26, 2009, Olomouc, Czech Republi
Scaling Separability Criterion: Application To Gaussian States
We introduce examples of three- and four-mode entangled Gaussian mixed states
that are not detected by the scaling and Peres-Horodecki separability criteria.
The presented modification of the scaling criterion resolves this problem. Also
it is shown that the new criterion reproduces the main features of the scaling
pictures for different cases of entangled states, while the previous versions
lead to completely different outcomes. This property of the presented scheme is
evidence of its higher generality.Comment: 7 pages, 4 figure
Inverse spin-s portrait and representation of qudit states by single probability vectors
Using the tomographic probability representation of qudit states and the
inverse spin-portrait method, we suggest a bijective map of the qudit density
operator onto a single probability distribution. Within the framework of the
approach proposed, any quantum spin-j state is associated with the
(2j+1)(4j+1)-dimensional probability vector whose components are labeled by
spin projections and points on the sphere. Such a vector has a clear physical
meaning and can be relatively easily measured. Quantum states form a convex
subset of the 2j(4j+3) simplex, with the boundary being illustrated for qubits
(j=1/2) and qutrits (j=1). A relation to the (2j+1)^2- and
(2j+1)(2j+2)-dimensional probability vectors is established in terms of spin-s
portraits. We also address an auxiliary problem of the optimum reconstruction
of qudit states, where the optimality implies a minimum relative error of the
density matrix due to the errors in measured probabilities.Comment: 23 pages, 4 figures, PDF LaTeX, submitted to the Journal of Russian
Laser Researc
Separability and entanglement of four-mode Gaussian states
The known Peres-Horodecki criterion and scaling criterion of separability are
considered on examples of three-mode and four-mode Gaussian states of
electromagnetic field. It is shown that the principal minors of the photon
quadrature dispersion matrix are sensitive to the change of scaling parameters.
An empirical observation has shown that the bigger the modulus of negative
principal minors, the more entangled the state.Comment: 14 pages, 11 figure
Symmetric informationally complete positive operator valued measure and probability representation of quantum mechanics
Symmetric informationally complete positive operator valued measures
(SIC-POVMs) are studied within the framework of the probability representation
of quantum mechanics. A SIC-POVM is shown to be a special case of the
probability representation. The problem of SIC-POVM existence is formulated in
terms of symbols of operators associated with a star-product quantization
scheme. We show that SIC-POVMs (if they do exist) must obey general rules of
the star product, and, starting from this fact, we derive new relations on
SIC-projectors. The case of qubits is considered in detail, in particular, the
relation between the SIC probability representation and other probability
representations is established, the connection with mutually unbiased bases is
discussed, and comments to the Lie algebraic structure of SIC-POVMs are
presented.Comment: 22 pages, 1 figure, LaTeX, partially presented at the Workshop
"Nonlinearity and Coherence in Classical and Quantum Systems" held at the
University "Federico II" in Naples, Italy on December 4, 2009 in honor of
Prof. Margarita A. Man'ko in connection with her 70th birthday, minor
misprints are corrected in the second versio
On calculating the mean values of quantum observables in the optical tomography representation
Given a density operator the optical tomography map defines a
one-parameter set of probability distributions on the real line allowing to reconstruct . We
introduce a dual map from the special class of quantum observables
to a special class of generalized functions such that the
mean value is given by the formula
. The class
includes all the symmetrized polynomials of canonical variables
and .Comment: 8 page
Time-Dependent Invariants and Green's Functions in the Probability Representation of Quantum Mechanics
In the probability representation of quantum mechanics, quantum states are
represented by a classical probability distribution, the marginal distribution
function (MDF), whose time dependence is governed by a classical evolution
equation. We find and explicitly solve, for a wide class of Hamiltonians, new
equations for the Green's function of such an equation, the so-called classical
propagator. We elucidate the connection of the classical propagator to the
quantum propagator for the density matrix and to the Green's function of the
Schr\"odinger equation. Within the new description of quantum mechanics we give
a definition of coherence solely in terms of properties of the MDF and we test
the new definition recovering well known results. As an application, the forced
parametric oscillator is considered . Its classical and quantum propagator are
found, together with the MDF for coherent and Fock states.Comment: 29 pages, RevTex, 6 eps-figures, to appear on Phys. Rev.
Unitary and Non-Unitary Matrices as a Source of Different Bases of Operators Acting on Hilbert Spaces
Columns of d^2 x N matrices are shown to create different sets of N operators
acting on -dimensional Hilbert space. This construction corresponds to a
formalism of the star-product of operator symbols. The known bases are shown to
be partial cases of generic formulas derived by using d^2 x N matrices as a
source for constructing arbitrary bases. The known examples of the SIC-POVM,
MUBs, and the phase-space description of qubit states are considered from the
viewpoint of the developed unified approach. Star-product schemes are
classified with respect to associated d^2 x N matrices. In particular, unitary
matrices correspond to self-dual schemes. Such self-dual star-product schemes
are shown to be determined by dequantizers which do not form POVM.Comment: 12 pages, 1 figure, 1 table, to appear in Journal of Russian Laser
Researc
Frank-Condon principle and adjustment of optical waveguides with nonhomogeneous refractive indices
The adjustment of two different selfocs is considered using both exact
formulas for the mode-connection coefficients expressed in terms of Hermite
polynomials of several variables and a qualitative approach based on the
Frank-Condon principle. Several examples of the refractive-index dependence are
studied and illustrative plots for these examples are presented. The connection
with the tomographic approach to quantum states of a two-dimensional oscillator
and the Frank-Condon factors is established.Comment: 8 pages, 4 figures, published version (layout of figures changed,
typos corrected, references added
- …