32 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
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
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
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
When do pieces determine the whole? Extreme marginals of a completely positive map
We will consider completely positive maps defined on tensor products of von Neumann algebras and taking values in the algebra of bounded operators on a Hilbert space and particularly certain convex subsets of the set of such maps. We show that when one of the marginal maps of such a map is an extreme point, then the marginals uniquely determine the map. We will further prove that when both of the marginals are extreme, then the whole map is extreme. We show that this general result is the common source of several well-known results dealing with, e.g., jointly measurable observables. We also obtain new insight especially in the realm of quantum instruments and their marginal observables and channels. © 2014 World Scientific Publishing Company.</p
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