49 research outputs found
Weakly compact operators and the strong* topology for a Banach space
Peer reviewedPublisher PD
Complementary Observables in Quantum Mechanics
We review the notion of complementarity of observables in quantum mechanics,
as formulated and studied by Paul Busch and his colleagues over the years. In
addition, we provide further clarification on the operational meaning of the
concept, and present several characterisations of complementarity - some of
which new - in a unified manner, as a consequence of a basic factorisation
lemma for quantum effects. We work out several applications, including the
canonical cases of position-momentum, position-energy, number-phase, as well as
periodic observables relevant to spatial interferometry. We close the paper
with some considerations of complementarity in a noisy setting, focusing
especially on the case of convolutions of position and momentum, which was a
recurring topic in Paul's work on operational formulation of quantum
measurements and central to his philosophy of unsharp reality
Two questions on quantum statistics
The determination of a quantum observable from the first and second moments
of its measurement outcome statistics is investigated. Operational conditions
for the moments of a probability measure are given which suffice to determine
the probability measure. Differential operators are shown to lead to physically
relevant cases where the expectation values of large classes of noncommuting
observables do not distinguish superpositions of states and, in particular,
where the full moment information does not determine the probability measure.Comment: 8 page
Moment operators of the Cartesian margins of the phase space observables
The theory of operator integrals is used to determine the moment operators of
the Cartesian margins of the phase space observables generated by the mixtures
of the number states. The moments of the -margin are polynomials of the
position operator and those of the -margin are polynomials of the momentum
operator.Comment: 14 page
Diagonalization and representation results for nonpositive sesquilinear form measures
We study decompositions of operator measures and more general sesquilinear
form measures into linear combinations of positive parts, and their
diagonal vector expansions. The underlying philosophy is to represent as a
trace class valued measure of bounded variation on a new Hilbert space related
to . The choice of the auxiliary Hilbert space fixes a unique decomposition
with certain properties, but this choice itself is not canonical. We present
relations to Naimark type dilations and direct integrals.Comment: J. Math. Anal. Appl., in pres
Notes on coarse grainings and functions of observables
Using the Naimark dilation theory we investigate the question under what
conditions an observable which is a coarse graining of another observable is a
function of it. To this end, conditions for the separability and for the
Boolean structure of an observable are given
Semispectral measures as convolutions and their moment operators
The moment operators of a semispectral measure having the structure of the
convolution of a positive measure and a semispectral measure are studied, with
paying attention to the natural domains of these unbounded operators. The
results are then applied to conveniently determine the moment operators of the
Cartesian margins of the phase space observables.Comment: 7 page