47 research outputs found
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
BALANCE EQUATIONS-BASED PROPERTIES OF THE RABI HAMILTONIAN
A stationary physical system satisfies peculiar balance conditions involving mean values of appropriate
observables. In this paper, we show how to deduce such quantitative links, named balance equations,
demonstrating as well their usefulness in bringing to light physical properties of the system without
solving the Schr¨odinger equation. The knowledge of such properties in the case of the Rabi Hamiltonian
is exploited to provide arguments to make easier the variational engineering of the ground state of this
model
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
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
A note on entropic uncertainty relations of position and momentum
We consider two entropic uncertainty relations of position and momentum
recently discussed in literature. By a suitable rescaling of one of them, we
obtain a smooth interpolation of both for high-resolution and low-resolution
measurements respectively. Because our interpolation has never been mentioned
in literature before, we propose it as a candidate for an improved entropic
uncertainty relation of position and momentum. Up to now, the author has
neither been able to falsify nor prove the new inequality. In our opinion it is
a challenge to do either one.Comment: 2 pages, 2 figures, 2 references adde
On supersymmetric quantum mechanics
This paper constitutes a review on N=2 fractional supersymmetric Quantum
Mechanics of order k. The presentation is based on the introduction of a
generalized Weyl-Heisenberg algebra W_k. It is shown how a general Hamiltonian
can be associated with the algebra W_k. This general Hamiltonian covers various
supersymmetrical versions of dynamical systems (Morse system, Poschl-Teller
system, fractional supersymmetric oscillator of order k, etc.). The case of
ordinary supersymmetric Quantum Mechanics corresponds to k=2. A connection
between fractional supersymmetric Quantum Mechanics and ordinary supersymmetric
Quantum Mechanics is briefly described. A realization of the algebra W_k, of
the N=2 supercharges and of the corresponding Hamiltonian is given in terms of
deformed-bosons and k-fermions as well as in terms of differential operators.Comment: Review paper (31 pages) to be published in: Fundamental World of
Quantum Chemistry, A Tribute to the Memory of Per-Olov Lowdin, Volume 3, E.
Brandas and E.S. Kryachko (Eds.), Springer-Verlag, Berlin, 200