197 research outputs found
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
Spontaneous Collapse of Unstable Quantum Superposition State: A Single-Particle Model of Modified Quantum Dynamics
We propose a modified dynamics of quantum mechanics, in which classical
mechanics of a point mass derives intrinsically in a massive limit of a
single-particle model. On the premise that a position basis plays a special
role in wavefunction collapse, we deduce to formalize spontaneous localization
of wavefunction on the analogy drawn from thermodynamics, in which a
characteristic energy scale and a time scale are introduced to separate quantum
and classical regimes.Comment: 2figs., contribution to Xth ICQO 200
Martingale Models for Quantum State Reduction
Stochastic models for quantum state reduction give rise to statistical laws
that are in most respects in agreement with those of quantum measurement
theory. Here we examine the correspondence of the two theories in detail,
making a systematic use of the methods of martingale theory. An analysis is
carried out to determine the magnitude of the fluctuations experienced by the
expectation of the observable during the course of the reduction process and an
upper bound is established for the ensemble average of the greatest
fluctuations incurred. We consider the general projection postulate of L\"uders
applicable in the case of a possibly degenerate eigenvalue spectrum, and derive
this result rigorously from the underlying stochastic dynamics for state
reduction in the case of both a pure and a mixed initial state. We also analyse
the associated Lindblad equation for the evolution of the density matrix, and
obtain an exact time-dependent solution for the state reduction that explicitly
exhibits the transition from a general initial density matrix to the L\"uders
density matrix. Finally, we apply Girsanov's theorem to derive a set of simple
formulae for the dynamics of the state in terms of a family of geometric
Brownian motions, thereby constructing an explicit unravelling of the Lindblad
equation.Comment: 30 pages LaTeX. Submitted to Journal of Physics
Quantum noise and stochastic reduction
In standard nonrelativistic quantum mechanics the expectation of the energy
is a conserved quantity. It is possible to extend the dynamical law associated
with the evolution of a quantum state consistently to include a nonlinear
stochastic component, while respecting the conservation law. According to the
dynamics thus obtained, referred to as the energy-based stochastic Schrodinger
equation, an arbitrary initial state collapses spontaneously to one of the
energy eigenstates, thus describing the phenomenon of quantum state reduction.
In this article, two such models are investigated: one that achieves state
reduction in infinite time, and the other in finite time. The properties of the
associated energy expectation process and the energy variance process are
worked out in detail. By use of a novel application of a nonlinear filtering
method, closed-form solutions--algebraic in character and involving no
integration--are obtained for both these models. In each case, the solution is
expressed in terms of a random variable representing the terminal energy of the
system, and an independent noise process. With these solutions at hand it is
possible to simulate explicitly the dynamics of the quantum states of
complicated physical systems.Comment: 50 page
Interest Rates and Information Geometry
The space of probability distributions on a given sample space possesses
natural geometric properties. For example, in the case of a smooth parametric
family of probability distributions on the real line, the parameter space has a
Riemannian structure induced by the embedding of the family into the Hilbert
space of square-integrable functions, and is characterised by the Fisher-Rao
metric. In the nonparametric case the relevant geometry is determined by the
spherical distance function of Bhattacharyya. In the context of term structure
modelling, we show that minus the derivative of the discount function with
respect to the maturity date gives rise to a probability density. This follows
as a consequence of the positivity of interest rates. Therefore, by mapping the
density functions associated with a given family of term structures to Hilbert
space, the resulting metrical geometry can be used to analyse the relationship
of yield curves to one another. We show that the general arbitrage-free yield
curve dynamics can be represented as a process taking values in the convex
space of smooth density functions on the positive real line. It follows that
the theory of interest rate dynamics can be represented by a class of processes
in Hilbert space. We also derive the dynamics for the central moments
associated with the distribution determined by the yield curve.Comment: 20 pages, 3 figure
Unconditionally secure quantum bit commitment is impossible
The claim of quantum cryptography has always been that it can provide
protocols that are unconditionally secure, that is, for which the security does
not depend on any restriction on the time, space or technology available to the
cheaters. We show that this claim does not hold for any quantum bit commitment
protocol. Since many cryptographic tasks use bit commitment as a basic
primitive, this result implies a severe setback for quantum cryptography. The
model used encompasses all reasonable implementations of quantum bit commitment
protocols in which the participants have not met before, including those that
make use of the theory of special relativity.Comment: 4 pages, revtex. Journal version replacing the version published in
the proceedings of PhysComp96. This is a significantly improved version which
emphasis the generality of the resul
Quantization of Nonstandard Hamiltonian Systems
The quantization of classical theories that admit more than one Hamiltonian
description is considered. This is done from a geometrical viewpoint, both at
the quantization level (geometric quantization) and at the level of the
dynamics of the quantum theory. A spin-1/2 system is taken as an example in
which all the steps can be completed. It is shown that the geometry of the
quantum theory imposes restrictions on the physically allowed nonstandard
quantum theories.Comment: Revtex file, 23 pages, no figure
A generalized quantum microcanonical ensemble
We discuss a generalized quantum microcanonical ensemble. It describes
isolated systems that are not necessarily in an eigenstate of the Hamilton
operator. Statistical averages are obtained by a combination of a time average
and a maximum entropy argument to resolve the lack of knowledge about initial
conditions. As a result, statistical averages of linear observables coincide
with values obtained in the canonical ensemble. Non-canonical averages can be
obtained by taking into account conserved quantities which are non-linear
functions of the microstate.Comment: improved version, new titl
Alternative new notation for quantum information theory
A new notation has been introduced for the quantum information theory. By
this notation,some calculations became simple in quantum information theory
such as quantum swapping, quantum teleportation.Comment: submitte
Teleportation and Secret Sharing with Pure Entangled States
We present two optimal methods of teleporting an unknown qubit using any pure
entangled state. We also discuss how such methods can also have succesful
application in quantum secret sharing with pure multipartite entangled states.Comment: Latex, 13 pages, submitted to PRA. One sub section already appeared
in the archive: quant-ph /990701
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