|
|
Dr Michael Hall
Research Interests
- Geometric approach to entropy: V=exp(S)
Shannon derived the form of the statistical entropy, uniquely up to
a multiplicative constant, from several axioms for the 'randomness'
of a probability distribution. A geometric alternative is derive the
form of the entropy, uniquely up to an additive constant, from several
axioms for the effective 'volume' occupied by a probability distribution.
This leads to geometric interpretations and derivations in a wide variety
of contexts in which entropy appears, including quantum and classical
information theory, statistical mechanics, uncertainty relations, and
dynamical processes (see, eg, http://xxx.lanl.gov/abs/physics/9903045).
- Optimal estimates and exact uncertainty relations
Consider an estimate of one quantum observable, made on the basis of
a measurement of a second observable on a given quantum state. There
is a fundamental lower bound for the noise or inaccuracy of any such
estimate, that vanishes when the observables are compatible, and which
is far stronger than the corresponding Heisenberg uncertainty relation.
Moreover, the lower bound is achievable for the case of a complete measurement
on a pure state, corresponding to an `exact uncertainty' relation, and
is formally related to properties of `weak values'. For conjugate observables,
this exact uncertainty relation reduces to an equality that links the
variance of the estimated observable and the Fisher information of the
measured observable. The optimal estimate of a quadrature observable,
from optical heterodyne detection on a single mode field, is related
to the derivative of the logarithm of the Husimi Q function (see, eg,
http://xxx.lanl.gov/abs/quant-ph/0107149
and http://xxx.lanl.gov/abs/quant-ph/0309091).
-
Joint measurement uncertainty relations
Any measurement provides information, and this information may be used
as the basis for simultaneously estimating the values of any two observables.
One can then ask for universal relations holding for the uncertainties
and inaccuracies of these joint estimates. Such relations have been
found, both for optimal and non-optimal estimates, and greatly generalise
previously known relations for so-called `unbiased' joint measurements.
Applications include uncertainty relations for joint measurements made
using EPR-correlated particles (see, eg, http://xxx.lanl.gov/abs/quant-ph/0309091).
- Quantum mechanics from an exact uncertainty principle
If one assumes that nonclassical fluctuations are added to the momentum
of an ensemble of classical particles, with the fluctuation strengh
determined solely by the position uncertainty of the ensemble, then
the modifed equations of motion for the ensemble are equivalent to the
Schroedinger equation. Thus statistical uncertainty can be taken as
the conceptual basis for moving from classical to quantum mechanics.
Results have been generalised to bosonic fields. For the case of quantum
gravity, the Wheeler-DeWitt equation is obtained with a unique operator
ordering (see, eg, http://xxx.lanl.gov/abs/quant-ph/0102069
and http://xxx.lanl.gov/abs/gr-qc/0408098).
- Algebra for generalised quantum observables (POVMs)
It is trivial to add and multiply Hermitian observables, but not so
for general quantum observables described by positive operator valued
measures. However, it has recently been shown that binary algebraic
operations can in fact be defined (such as the sum and the symmetric
product). The approach relies on the simultaneous promotion of the observables
to Hermitian observables on an extended Hilbert space, and has been
applied to obtain new uncertainty relations and metrics for POVM observables
(see, eg, http://xxx.lanl.gov/abs/quant-ph/0302007).
- Configuration space ensembles
Suppose that the configuration of a physical system (eg, a particle
or a field) is inherently imprecise, requiring that it be described
by a probability density P on the configuration space. Assuming that
the dynamics of this fundamental quanity, P, is determined by an action
principle, implies the existence of a conjugate quantity S and an `ensemble
Hamiltonian' H. Conservation of probability further requires the invariance
of H under addition of a constant to S. The resulting formalism is very
general, and incorporates both quantum and classical ensembles, for
the cases of both continuous and discrete configuration spaces. The
fundamental quantum-classical difference is the presence of an extra
term in the quantum ensemble Hamiltonian. Moreover, unlike the Dirac
approach, classical and quantum constraints can be treated on an equal
footing (see, eg, http://xxx.lanl.gov/abs/quant-ph/0404123
and http://xxx.lanl.gov/abs/gr-qc/0408098).
|
