8 research outputs found
Contextuality in Three Types of Quantum-Mechanical Systems
We present a formal theory of contextuality for a set of random variables
grouped into different subsets (contexts) corresponding to different, mutually
incompatible conditions. Within each context the random variables are jointly
distributed, but across different contexts they are stochastically unrelated.
The theory of contextuality is based on the analysis of the extent to which
some of these random variables can be viewed as preserving their identity
across different contexts when one considers all possible joint distributions
imposed on the entire set of the random variables. We illustrate the theory on
three systems of traditional interest in quantum physics (and also in
non-physical, e.g., behavioral studies). These are systems of the
Klyachko-Can-Binicioglu-Shumovsky-type, Einstein-Podolsky-Rosen-Bell-type, and
Suppes-Zanotti-Leggett-Garg-type. Listed in this order, each of them is
formally a special case of the previous one. For each of them we derive
necessary and sufficient conditions for contextuality while allowing for
experimental errors and contextual biases or signaling. Based on the same
principles that underly these derivations we also propose a measure for the
degree of contextuality and compute it for the three systems in question.Comment: Foundations of Physics 7, 762-78
Noncontextuality with Marginal Selectivity in Reconstructing Mental Architectures
We present a general theory of series-parallel mental architectures with
selectively influenced stochastically non-independent components. A mental
architecture is a hypothetical network of processes aimed at performing a task,
of which we only observe the overall time it takes under variable parameters of
the task. It is usually assumed that the network contains several processes
selectively influenced by different experimental factors, and then the question
is asked as to how these processes are arranged within the network, e.g.,
whether they are concurrent or sequential. One way of doing this is to consider
the distribution functions for the overall processing time and compute certain
linear combinations thereof (interaction contrasts). The theory of selective
influences in psychology can be viewed as a special application of the
interdisciplinary theory of (non)contextuality having its origins and main
applications in quantum theory. In particular, lack of contextuality is
equivalent to the existence of a "hidden" random entity of which all the random
variables in play are functions. Consequently, for any given value of this
common random entity, the processing times and their compositions (minima,
maxima, or sums) become deterministic quantities. These quantities, in turn,
can be treated as random variables with (shifted) Heaviside distribution
functions, for which one can easily compute various linear combinations across
different treatments, including interaction contrasts. This mathematical fact
leads to a simple method, more general than the previously used ones, to
investigate and characterize the interaction contrast for different types of
series-parallel architectures.Comment: published in Frontiers in Psychology: Cognition 1:12 doi:
10.3389/fpsyg.2015.00735 (special issue "Quantum Structures in Cognitive and
Social Science"
A Qualified Kolmogorovian Account of Probabilistic Contextuality
We describe a mathematical language for determining all possible patterns of
contextuality in the dependence of stochastic outputs of a system on its
deterministic inputs. The central notion is that of all possible couplings for
stochastically unrelated outputs indexed by mutually incompatible values of
inputs. A system is characterized by a pattern of which outputs can be
"directly influenced" by which inputs (a primitive relation, hypothetical or
normative), and by certain constraints imposed on the outputs (such as
Bell-type inequalities or their quantum analogues). The set of couplings
compatible with these constraints represents a form of contextuality in the
dependence of outputs on inputs with respect to the declared pattern of direct
influences.Comment: Lecture Notes in Computer Science 8369, 201-212 (2014