27 research outputs found
A note on infinite extreme correlation matrices
We give a characterization for the extreme points of the convex set of
correlation matrices with a countable index set. A Hermitian matrix is called a
correlation matrix if it is positive semidefinite with unit diagonal entries.
Using the characterization we show that there exist extreme points of any rank.Comment: 7 page
Completely positive maps on modules, instruments, extremality problems, and applications to physics
Convex sets of completely positive maps and positive semidefinite kernels are
considered in the most general context of modules over -algebras and a
complete charaterization of their extreme points is obtained. As a byproduct,
we determine extreme quantum instruments, preparations, channels, and extreme
autocorrelation functions. Various applications to quantum information and
measurement theories are given. The structure of quantum instruments is
analyzed thoroughly.Comment: 32 page
On the structure of covariant phase observables
We study the mathematical structure of covariant phase observables. Such an
observable can alternatively be expressed as a phase matrix, as a sequence of
unit vectors, as a sequence of phase states, or as an equivalent class of
covariant trace-preserving operations. Covariant generalized operator measures
are defined by structure matrices which form a W*-algebra with phase matrices
as its subset. The properties of the Radon-Nikodym derivatives of phase
probability measures are studied.Comment: 11 page
Complete measurements of quantum observables
We define a complete measurement of a quantum observable (POVM) as a
measurement of the maximally refined version of the POVM. Complete measurements
give information from the multiplicities of the measurement outcomes and can be
viewed as state preparation procedures. We show that any POVM can be measured
completely by using sequential measurements or maximally refinable instruments.
Moreover, the ancillary space of a complete measurement can be chosen to be
minimal.Comment: Based on talk given in CEQIP 2012 conferenc
Extreme commutative quantum observables are sharp
It is well known that, in the description of quantum observables, positive
operator valued measures (POVMs) generalize projection valued measures (PVMs)
and they also turn out be more optimal in many tasks. We show that a
commutative POVM is an extreme point in the convex set of all POVMs if and only
if it is a PVM. This results implies that non-commutativity is a necessary
ingredient to overcome the limitations of PVMs.Comment: 5 pages, minor corrections in v
Extreme phase and rotated quadrature measurements
We determine the extreme points of the convex set of covariant phase
observables. Such extremals describe the best phase parameter measurements of
laser light - the best in the sense that they are free from classical
randomness due to fluctuations in the measuring procedure. We also characterize
extreme fuzzy rotated quadratures
The Pegg-Barnett Formalism and Covariant Phase Observables
We compare the Pegg-Barnett (PB) formalism with the covariant phase
observable approach to the problem of quantum phase and show that PB-formalism
gives essentially the same results as the canonical (covariant) phase
observable. We also show that PB-formalism can be extended to cover all
covariant phase observables including the covariant phase observable arising
from the angle margin of the Husimi Q-function.Comment: 10 page
Quantum tomography, phase space observables, and generalized Markov kernels
We construct a generalized Markov kernel which transforms the observable
associated with the homodyne tomography into a covariant phase space observable
with a regular kernel state. Illustrative examples are given in the cases of a
'Schrodinger cat' kernel state and the Cahill-Glauber s-parametrized
distributions. Also we consider an example of a kernel state when the
generalized Markov kernel cannot be constructed.Comment: 20 pages, 3 figure
Density matrix reconstruction from displaced photon number distributions
We consider state reconstruction from the measurement statistics of phase
space observables generated by photon number states. The results are obtained
by inverting certain infinite matrices. In particular, we obtain reconstruction
formulas, each of which involves only a single phase space observable.Comment: 19 page
Continuous variable tomographic measurements
Using a recent result of Albini et al. to represent quantum homodyne
tomography in terms of a single observable (as a normalized positive operator
measure) we construct a generalized Markov kernel which transforms (the
measurement outcome statistics of) this observable into (the measurement
outcome statistics of) a covariant phase space observable. We also consider the
inverse question. Finally, we add some remarks on the quantum theoretical
justification of the experimental implementations of these observables in terms
of balanced homodyne and 8-port detection techniques, respectively.Comment: 9 page