83,660 research outputs found
Conditional fiducial models
The fiducial is not unique in general, but we prove that in a restricted
class of models it is uniquely determined by the sampling distribution of the
data. It depends in particular not on the choice of a data generating model.
The arguments lead to a generalization of the classical formula found by Fisher
(1930). The restricted class includes cases with discrete distributions, the
case of the shape parameter in the Gamma distribution, and also the case of the
correlation coefficient in a bivariate Gaussian model. One of the examples can
also be used in a pedagogical context to demonstrate possible difficulties with
likelihood-, Bayesian-, and bootstrap-inference. Examples that demonstrate
non-uniqueness are also presented. It is explained that they can be seen as
cases with restrictions on the parameter space. Motivated by this the concept
of a conditional fiducial model is introduced. This class of models includes
the common case of iid samples from a one-parameter model investigated by
Hannig (2013), the structural group models investigated by Fraser (1968), and
also certain models discussed by Fisher (1973) in his final writing on the
subject
Coherent states for continuous spectrum operators with non-normalizable fiducial states
The problem of building coherent states from non-normalizable fiducial states
is considered. We propose a way of constructing such coherent states by
regularizing the divergence of the fiducial state norm. Then, we successfully
apply the formalism to particular cases involving systems with a continuous
spectrum: coherent states for the free particle and for the inverted oscillator
are explicitly provided. Similar ideas can be used for other
systems having non-normalizable fiducial states.Comment: 17 pages, typos corrected, references adde
Structure of Two-qubit Symmetric Informationally Complete POVMs
In the four-dimensional Hilbert space, there exist 16 Heisenberg--Weyl (HW)
covariant symmetric informationally complete positive operator valued measures
(SIC~POVMs) consisting of 256 fiducial states on a single orbit of the Clifford
group. We explore the structure of these SIC~POVMs by studying the symmetry
transformations within a given SIC~POVM and among different SIC~POVMs.
Furthermore, we find 16 additional SIC~POVMs by a regrouping of the 256
fiducial states, and show that they are unitarily equivalent to the original 16
SIC~POVMs by establishing an explicit unitary transformation. We then reveal
the additional structure of these SIC~POVMs when the four-dimensional Hilbert
space is taken as the tensor product of two qubit Hilbert spaces. In
particular, when either the standard product basis or the Bell basis are chosen
as the defining basis of the HW group, in eight of the 16 HW covariant
SIC~POVMs, all fiducial states have the same concurrence of . These
SIC~POVMs are particularly appealing for an experimental implementation, since
all fiducial states can be connected to each other with just local unitary
transformations. In addition, we introduce a concise representation of the
fiducial states with the aid of a suitable tabular arrangement of their
parameters.Comment: 10 pages, 1 figure, 5 table
A note on the time evolution of generalized coherent states
I consider the time evolution of generalized coherent states based on
non-standard fiducial vectors, and show that only for a restricted class of
fiducial vectors does the associated classical motion determine the quantum
evolution of the states. I discuss some consequences of this for path integral
representations.Comment: 9 pages. RevTe
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