5 research outputs found
Unambiguous coherent state identification: Searching a quantum database
We consider an unambiguous identification of an unknown coherent state with
one of two unknown coherent reference states. Specifically, we consider two
modes of an electromagnetic field prepared in unknown coherent states alpha_1
and alpha_2, respectively. The third mode is prepared either in the state
alpha_1 or in the state alpha_2. The task is to identify (unambiguously) which
of the two modes are in the same state. We present a scheme consisting of three
beamsplitters capable to perform this task. Although we don't prove the
optimality, we show that the performance of the proposed setup is better than
the generalization of the optimal measurement known for a finite-dimensional
case. We show that a single beamsplitter is capable to perform an unambiguous
quantum state comparison for coherent states optimally. Finally we propose an
experimental setup consisting of 2N-1 beamsplitters for unambiguous
identification among N unknown coherent states. This setup can be considered as
a search in a quantum database. The elements of the database are unknown
coherent states encoded in different modes of an electromagnetic field. The
task is to specify the two modes that are excited in the same, though unknown,
coherent state.Comment: version accepted for publication, 12 pages, 3 figure
Quantum Entanglement and fixed point Hopf bifurcation
We present the qualitative differences in the phase transitions of the
mono-mode Dicke model in its integrable and chaotic versions. We show that a
first order phase transition occurs in the integrable case whereas a second
order in the chaotic one. This difference is also reflected in the classical
limit: for the integrable case the stable fixed point in phase space suffers a
bifurcation of Hopf type whereas for the second one a pitchfork type
bifurcation has been reported
Universal geometric approach to uncertainty, entropy and information
It is shown that for any ensemble, whether classical or quantum, continuous
or discrete, there is only one measure of the "volume" of the ensemble that is
compatible with several basic geometric postulates. This volume measure is thus
a preferred and universal choice for characterising the inherent spread,
dispersion, localisation, etc, of the ensemble. Remarkably, this unique
"ensemble volume" is a simple function of the ensemble entropy, and hence
provides a new geometric characterisation of the latter quantity. Applications
include unified, volume-based derivations of the Holevo and Shannon bounds in
quantum and classical information theory; a precise geometric interpretation of
thermodynamic entropy for equilibrium ensembles; a geometric derivation of
semi-classical uncertainty relations; a new means for defining classical and
quantum localization for arbitrary evolution processes; a geometric
interpretation of relative entropy; and a new proposed definition for the
spot-size of an optical beam. Advantages of the ensemble volume over other
measures of localization (root-mean-square deviation, Renyi entropies, and
inverse participation ratio) are discussed.Comment: Latex, 38 pages + 2 figures; p(\alpha)->1/|T| in Eq. (72) [Eq. (A10)
of published version
Spreading lengths of Hermite polynomials
The Renyi, Shannon and Fisher spreading lengths of the classical or
hypergeometric orthogonal polynomials, which are quantifiers of their
distribution all over the orthogonality interval, are defined and investigated.
These information-theoretic measures of the associated Rakhmanov probability
density, which are direct measures of the polynomial spreading in the sense of
having the same units as the variable, share interesting properties: invariance
under translations and reflections, linear scaling and vanishing in the limit
that the variable tends towards a given definite value. The expressions of the
Renyi and Fisher lengths for the Hermite polynomials are computed in terms of
the polynomial degree. The combinatorial multivariable Bell polynomials, which
are shown to characterize the finite power of an arbitrary polynomial, play a
relevant role for the computation of these information-theoretic lengths.
Indeed these polynomials allow us to design an error-free computing approach
for the entropic moments (weighted L^q-norms) of Hermite polynomials and
subsequently for the Renyi and Tsallis entropies, as well as for the Renyi
spreading lengths. Sharp bounds for the Shannon length of these polynomials are
also given by means of an information-theoretic-based optimization procedure.
Moreover, it is computationally proved the existence of a linear correlation
between the Shannon length (as well as the second-order Renyi length) and the
standard deviation. Finally, the application to the most popular
quantum-mechanical prototype system, the harmonic oscillator, is discussed and
some relevant asymptotical open issues related to the entropic moments
mentioned previously are posed.Comment: 16 pages, 4 figures. Journal of Computational and Applied Mathematics
(2009), doi:10.1016/j.cam.2009.09.04