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A quantum theoretical explanation for probability judgment errors
A quantum probability model is introduced and used to explain human probability judgment errors including the conjunction, disjunction, inverse, and conditional fallacies, as well as unpacking effects and partitioning effects. Quantum probability theory is a general and coherent theory based on a set of (von Neumann) axioms which relax some of the constraints underlying classic (Kolmogorov) probability theory. The quantum model is compared and contrasted with other competing explanations for these judgment errors including the representativeness heuristic, the averaging model, and a memory retrieval model for probability judgments. The quantum model also provides ways to extend Bayesian, fuzzy set, and fuzzy trace theories. We conclude that quantum information processing principles provide a viable and promising new way to understand human judgment and reasoning
Hybridized solid-state qubit in the charge-flux regime
Most superconducting qubits operate in a regime dominated by either the
electrical charge or the magnetic flux. Here we study an intermediate case: a
hybridized charge-flux qubit with a third Josephson junction (JJ) added into
the SQUID loop of the Cooper-pair box. This additional JJ allows the optimal
design of a low-decoherence qubit. Both charge and flux noises are
considered. Moreover, we show that an efficient quantum measurement of either
the current or the charge can be achieved by using different area sizes for the
third JJ.Comment: 7 pages, 5 figures. Phys. Rev. B, in pres
A spectral-based numerical method for Kolmogorov equations in Hilbert spaces
We propose a numerical solution for the solution of the
Fokker-Planck-Kolmogorov (FPK) equations associated with stochastic partial
differential equations in Hilbert spaces.
The method is based on the spectral decomposition of the Ornstein-Uhlenbeck
semigroup associated to the Kolmogorov equation. This allows us to write the
solution of the Kolmogorov equation as a deterministic version of the
Wiener-Chaos Expansion. By using this expansion we reformulate the Kolmogorov
equation as a infinite system of ordinary differential equations, and by
truncation it we set a linear finite system of differential equations. The
solution of such system allow us to build an approximation to the solution of
the Kolmogorov equations. We test the numerical method with the Kolmogorov
equations associated with a stochastic diffusion equation, a Fisher-KPP
stochastic equation and a stochastic Burgers Eq. in dimension 1.Comment: 28 pages, 10 figure
Stability of real parametric polynomial discrete dynamical systems
We extend and improve the existing characterization of the dynamics of
general quadratic real polynomial maps with coefficients that depend on a
single parameter , and generalize this characterization to cubic real
polynomial maps, in a consistent theory that is further generalized to real
-th degree real polynomial maps. In essence, we give conditions for the
stability of the fixed points of any real polynomial map with real fixed
points. In order to do this, we have introduced the concept of Canonical
Polynomial Maps which are topologically conjugate to any polynomial map of the
same degree with real fixed points. The stability of the fixed points of
canonical polynomial maps has been found to depend solely on a special function
termed Product Position Function for a given fixed point. The values of this
product position determine the stability of the fixed point in question, when
it bifurcates, and even when chaos arises, as it passes through what we have
termed stability bands. The exact boundary values of these stability bands are
yet to be calculated for regions of type greater than one for polynomials of
degree higher than three.Comment: 23 pages, 4 figures, now published in Discrete Dynamics in Nature and
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Optomechanical-like coupling between superconducting resonators
We propose and analyze a circuit that implements a nonlinear coupling between
two superconducting microwave resonators. The resonators are coupled through a
superconducting quantum interference device (SQUID) that terminates one of the
resonators. This produces a nonlinear interaction on the standard
optomechanical form, where the quadrature of one resonator couples to the
photon number of the other resonator. The circuit therefore allows for
all-electrical realizations of analogs to optomechanical systems, with coupling
that can be both strong and tunable. We estimate the coupling strengths that
should be attainable with the proposed device, and we find that the device is a
promising candidate for realizing the single-photon strong-coupling regime. As
a potential application, we discuss implementations of networks of
nonlinearly-coupled microwave resonators, which could be used in
microwave-photon based quantum simulation.Comment: 10 pages, 7 figure
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