6,165 research outputs found
Comment on "Typicality for Generalized Microcanonical Ensemble"
The validity of the so-called "typicality" argument for a generalised
microcanonical ensemble proposed recently is examined.Comment: Version to appear in PR
Information geometry of density matrices and state estimation
Given a pure state vector |x> and a density matrix rho, the function
p(x|rho)= defines a probability density on the space of pure states
parameterised by density matrices. The associated Fisher-Rao information
measure is used to define a unitary invariant Riemannian metric on the space of
density matrices. An alternative derivation of the metric, based on square-root
density matrices and trace norms, is provided. This is applied to the problem
of quantum-state estimation. In the simplest case of unitary parameter
estimation, new higher-order corrections to the uncertainty relations,
applicable to general mixed states, are derived.Comment: published versio
Statistical Geometry in Quantum Mechanics
A statistical model M is a family of probability distributions, characterised
by a set of continuous parameters known as the parameter space. This possesses
natural geometrical properties induced by the embedding of the family of
probability distributions into the Hilbert space H. By consideration of the
square-root density function we can regard M as a submanifold of the unit
sphere in H. Therefore, H embodies the `state space' of the probability
distributions, and the geometry of M can be described in terms of the embedding
of in H. The geometry in question is characterised by a natural Riemannian
metric (the Fisher-Rao metric), thus allowing us to formulate the principles of
classical statistical inference in a natural geometric setting. In particular,
we focus attention on the variance lower bounds for statistical estimation, and
establish generalisations of the classical Cramer-Rao and Bhattacharyya
inequalities. The statistical model M is then specialised to the case of a
submanifold of the state space of a quantum mechanical system. This is pursued
by introducing a compatible complex structure on the underlying real Hilbert
space, which allows the operations of ordinary quantum mechanics to be
reinterpreted in the language of real Hilbert space geometry. The application
of generalised variance bounds in the case of quantum statistical estimation
leads to a set of higher order corrections to the Heisenberg uncertainty
relations for canonically conjugate observables.Comment: 32 pages, LaTex file, Extended version to include quantum measurement
theor
Metric approach to quantum constraints
A new framework for deriving equations of motion for constrained quantum
systems is introduced, and a procedure for its implementation is outlined. In
special cases the framework reduces to a quantum analogue of the Dirac theory
of constrains in classical mechanics. Explicit examples involving spin-1/2
particles are worked out in detail: in one example our approach coincides with
a quantum version of the Dirac formalism, while the other example illustrates
how a situation that cannot be treated by Dirac's approach can nevertheless be
dealt with in the present scheme.Comment: 13 pages, 1 figur
Constrained quantum dynamics
We give an overview of the two different methods that have been introduced in
order to describe the dynamics of constrained quantum systems; the symplectic
formulation and the metric formulation. The symplectic method extends the work
of Dirac on constrained classical systems to quantum systems, whereas the
metric approach is purely a quantum mechanical method having no immediate
classical counterpart. Two examples are provided that illustrate the nonlinear
motion induced by the constraint.Comment: 8 pages, 4 figures. Based on the talk presented at the DICE2008
conference in Castiglioncello, September 200
Geometric Phase and Modulo Relations for Probability Amplitudes as Functions on Complex Parameter Spaces
We investigate general differential relations connecting the respective
behavior s of the phase and modulo of probability amplitudes of the form
\amp{\psi_f}{\psi}, where is a fixed state in Hilbert space
and is a section of a holomorphic line bundle over some complex
parameter space. Amplitude functions on such bundles, while not strictly
holomorphic, nevertheless satisfy generalized Cauchy-Riemann conditions
involving the U(1) Berry-Simon connection on the parameter space. These
conditions entail invertible relations between the gradients of the phase and
modulo, therefore allowing for the reconstruction of the phase from the modulo
(or vice-versa) and other conditions on the behavior of either polar component
of the amplitude. As a special case, we consider amplitude functions valued on
the space of pure states, the ray space , where
transition probabilities have a geometric interpretation in terms of geodesic
distances as measured with the Fubini-Study metric. In conjunction with the
generalized Cauchy-Riemann conditions, this geodesic interpretation leads to
additional relations, in particular a novel connection between the modulus of
the amplitude and the phase gradient, somewhat reminiscent of the WKB formula.
Finally, a connection with geometric phases is established.Comment: 11 pages, 1 figure, revtex
The Information Geometry of the Ising Model on Planar Random Graphs
It has been suggested that an information geometric view of statistical
mechanics in which a metric is introduced onto the space of parameters provides
an interesting alternative characterisation of the phase structure,
particularly in the case where there are two such parameters -- such as the
Ising model with inverse temperature and external field .
In various two parameter calculable models the scalar curvature of
the information metric has been found to diverge at the phase transition point
and a plausible scaling relation postulated: . For spin models the necessity of calculating in
non-zero field has limited analytic consideration to 1D, mean-field and Bethe
lattice Ising models. In this letter we use the solution in field of the Ising
model on an ensemble of planar random graphs (where ) to evaluate the scaling behaviour of the scalar curvature, and find
. The apparent discrepancy is traced
back to the effect of a negative .Comment: Version accepted for publication in PRE, revtex
Hidden variable interpretation of spontaneous localization theory
The spontaneous localization theory of Ghirardi, Rimini, and Weber (GRW) is a
theory in which wavepacket reduction is treated as a genuine physical process.
Here it is shown that the mathematical formalism of GRW can be given an
interpretation in terms of an evolving distribution of particles on
configuration space similar to Bohmian mechanics (BM). The GRW wavefunction
acts as a pilot wave for the set of particles. In addition, a continuous stream
of noisy information concerning the precise whereabouts of the particles must
be specified. Nonlinear filtering techniques are used to determine the dynamics
of the distribution of particles conditional on this noisy information and
consistency with the GRW wavefunction dynamics is demonstrated. Viewing this
development as a hybrid BM-GRW theory, it is argued that, besides helping to
clarify the relationship between the GRW theory and BM, its merits make it
worth considering in its own right.Comment: 13 page
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