64,139 research outputs found
On the Foundations of the Brussels Operational-Realistic Approach to Cognition
The scientific community is becoming more and more interested in the research
that applies the mathematical formalism of quantum theory to model human
decision-making. In this paper, we provide the theoretical foundations of the
quantum approach to cognition that we developed in Brussels. These foundations
rest on the results of two decade studies on the axiomatic and
operational-realistic approaches to the foundations of quantum physics. The
deep analogies between the foundations of physics and cognition lead us to
investigate the validity of quantum theory as a general and unitary framework
for cognitive processes, and the empirical success of the Hilbert space models
derived by such investigation provides a strong theoretical confirmation of
this validity. However, two situations in the cognitive realm, 'question order
effects' and 'response replicability', indicate that even the Hilbert space
framework could be insufficient to reproduce the collected data. This does not
mean that the mentioned operational-realistic approach would be incorrect, but
simply that a larger class of measurements would be in force in human
cognition, so that an extended quantum formalism may be needed to deal with all
of them. As we will explain, the recently derived 'extended Bloch
representation' of quantum theory (and the associated 'general
tension-reduction' model) precisely provides such extended formalism, while
remaining within the same unitary interpretative framework.Comment: 21 page
Extended statistical modeling under symmetry; the link toward quantum mechanics
We derive essential elements of quantum mechanics from a parametric structure
extending that of traditional mathematical statistics. The basic setting is a
set of incompatible experiments, and a transformation group
on the cartesian product of the parameter spaces of these experiments.
The set of possible parameters is constrained to lie in a subspace of , an
orbit or a set of orbits of . Each possible model is then connected to a
parametric Hilbert space. The spaces of different experiments are linked
unitarily, thus defining a common Hilbert space . A state is
equivalent to a question together with an answer: the choice of an experiment
plus a value for the corresponding parameter. Finally,
probabilities are introduced through Born's formula, which is derived from a
recent version of Gleason's theorem. This then leads to the usual formalism of
elementary quantum mechanics in important special cases. The theory is
illustrated by the example of a quantum particle with spin.Comment: The paper has been withdrawn because it is outdate
A new class of entanglement measures
We introduce new entanglement measures on the set of density operators on
tensor product Hilbert spaces. These measures are based on the greatest cross
norm on the tensor product of the sets of trace class operators on Hilbert
space. We show that they satisfy the basic requirements on entanglement
measures discussed in the literature, including convexity, invariance under
local unitary operations and non-increase under local quantum operations and
classical communication.Comment: Revised version accepted by J Math Phys, 12 pages, LaTeX, contains
Sections 1-5 & 7 of the previous version. The previous Section 6 is now in
quant-ph/0105104 and the previous Section 8 is superseded by quant-ph/010501
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