53 research outputs found
Smearing of Observables and Spectral Measures on Quantum Structures
An observable on a quantum structure is any -homomorphism of quantum
structures from the Borel -algebra of the real line into the quantum
structure which is in our case a monotone -complete effect algebras
with the Riesz Decomposition Property. We show that every observable is a
smearing of a sharp observable which takes values from a Boolean
-subalgebra of the effect algebra, and we prove that for every element
of the effect algebra there is its spectral measure
A representation theorem for MV-algebras
An {\em MV-pair} is a pair where is a Boolean algebra and is
a subgroup of the automorphism group of satisfying certain conditions. Let
be the equivalence relation on naturally associated with . We
prove that for every MV-pair , the effect algebra is an MV-
effect algebra. Moreover, for every MV-effect algebra there is an MV-pair
such that is isomorphic to
Type-Decomposition of a Pseudo-Effect Algebra
The theory of direct decomposition of a centrally orthocomplete effect
algebra into direct summands of various types utilizes the notion of a
type-determining (TD) set. A pseudo-effect algebra (PEA) is a (possibly)
noncommutative version of an effect algebra. In this article we develop the
basic theory of centrally orthocomplete PEAs, generalize the notion of a TD set
to PEAs, and show that TD sets induce decompositions of centrally orthocomplete
PEAs into direct summands.Comment: 18 page
An Intrisic Topology for Orthomodular Lattices
We present a general way to define a topology on orthomodular lattices. We
show that in the case of a Hilbert lattice, this topology is equivalent to that
induced by the metrics of the corresponding Hilbert space. Moreover, we show
that in the case of a boolean algebra, the obtained topology is the discrete
one. Thus, our construction provides a general tool for studying orthomodular
lattices but also a way to distinguish classical and quantum logics.Comment: Under submission to the International Journal of Theoretical Physic
Orthocomplementation and compound systems
In their 1936 founding paper on quantum logic, Birkhoff and von Neumann
postulated that the lattice describing the experimental propositions concerning
a quantum system is orthocomplemented. We prove that this postulate fails for
the lattice L_sep describing a compound system consisting of so called
separated quantum systems. By separated we mean two systems prepared in
different ``rooms'' of the lab, and before any interaction takes place. In that
case the state of the compound system is necessarily a product state. As a
consequence, Dirac's superposition principle fails, and therefore L_sep cannot
satisfy all Piron's axioms. In previous works, assuming that L_sep is
orthocomplemented, it was argued that L_sep is not orthomodular and fails to
have the covering property. Here we prove that L_sep cannot admit and
orthocomplementation. Moreover, we propose a natural model for L_sep which has
the covering property.Comment: Submitted for the proceedings of the 2004 IQSA's conference in
Denver. Revised versio
Sharp and fuzzy observables on effect algebras
Observables on effect algebras and their fuzzy versions obtained by means of
confidence measures (Markov kernels) are studied. It is shown that, on effect
algebras with the (E)-property, given an observable and a confidence measure,
there exists a fuzzy version of the observable. Ordering of observables
according to their fuzzy properties is introduced, and some minimality
conditions with respect to this ordering are found. Applications of some
results of classical theory of experiments are considered.Comment: 23 page
The Expectation Monad in Quantum Foundations
The expectation monad is introduced abstractly via two composable
adjunctions, but concretely captures measures. It turns out to sit in between
known monads: on the one hand the distribution and ultrafilter monad, and on
the other hand the continuation monad. This expectation monad is used in two
probabilistic analogues of fundamental results of Manes and Gelfand for the
ultrafilter monad: algebras of the expectation monad are convex compact
Hausdorff spaces, and are dually equivalent to so-called Banach effect
algebras. These structures capture states and effects in quantum foundations,
and also the duality between them. Moreover, the approach leads to a new
re-formulation of Gleason's theorem, expressing that effects on a Hilbert space
are free effect modules on projections, obtained via tensoring with the unit
interval.Comment: In Proceedings QPL 2011, arXiv:1210.029
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