458 research outputs found
Parallelism of quantum computations from prequantum classical statistical field theory (PCSFT)
This paper is devoted to such a fundamental problem of quantum computing as
quantum parallelism. It is well known that quantum parallelism is the basis of
the ability of quantum computer to perform in polynomial time computations
performed by classical computers for exponential time. Therefore better
understanding of quantum parallelism is important both for theoretical and
applied research, cf. e.g. David Deutsch \cite{DD}. We present a realistic
interpretation based on recently developed prequantum classical statistical
field theory (PCSFT). In the PCSFT-approach to QM quantum states (mixed as well
as pure) are labels of special ensembles of classical fields. Thus e.g. a
single (!) ``electron in the pure state'' can be identified with a
special `` electron random field,'' say Quantum computer
operates with such random fields. By one computational step for e.g. a Boolean
function the initial random field is
transformed into the final random field ``containing all
values'' of This is the objective of quantum computer's ability to operate
quickly with huge amounts of information -- in fact, with classical random
fields
Contextuality versus Incompatibility
Our aim is to compare the fundamental notions of quantum physics -
contextuality vs. incompatibility. One has to distinguish two different notions
of contextuality, {\it Bohr-contextuality} and {\it Bell-contextuality}. The
latter is defined operationally via violation of noncontextuality (Bell type)
inequalities. This sort of contextuality will be compared with incompatibility.
It is easy to show that, for quantum observables, there is {\it no
contextuality without incompatibility.} The natural question arises: What is
contextuality without incompatibility? (What is "dry-residue"?) Generally this
is the very complex question. We concentrated on contextuality for four quantum
observables. We shown that in the CHSH-scenarios (for "natural quantum
observables") {\it contextuality is reduced to incompatibility.} However,
generally contextuality without incompatibility may have some physical content.
We found a mathematical constraint extracting the contextuality component from
incompatibility. However, the physical meaning of this constraint is not clear.
In appendix 1, we briefly discuss another sort of contextuality based on the
Bohr's complementarity principle which is treated as the {\it
contextuality-incompatibility principle}. Bohr-contextuality plays the crucial
role in quantum foundations. Incompatibility is, in fact, a consequence of
Bohr-contextuality. Finally, we remark that outside of physics, e.g., in
cognitive psychology and decision making Bell-contextuality cleaned of
incompatibility can play the important role.Comment: discussion of 3-types of contextuality (introduction and appendix 1),
discussion on Suppes-Zanotti and Boole inequalities vs. original Bell
inequality for correlated observables (appendix 2
EPR-Bohm experiment, interference of probabilities, imprecision of time
We demonstrate that the EPR-Bohm probabilities can be easily obtained in the
classical (but contextual) probabilistic framework by using the formula of
interference of probabilities. From this point of view the EPR-Bell experiment
is just an experiment on interference of probabilities. We analyse the time
structure of contextuality in the EPR-Bohm experiment. The conclusion is that
quantum mechanics does not contradict to a local realistic model in which
probabilities are calculated as averages over conditionings/measurements for
pairs of instances of time If we restrict our consideration only to
simultaneous measurements at the fixed instance of time we would get
contradiction with Bell's theorem. One of implications of this fact might be
the impossibility to define instances of {\it time with absolute precision} on
the level of the contextual microscopic realistic model.Comment: We analyze time structure of conditioning in the EPR-Bohm experimen
Comment on Hess-Philipp anti-Bell and Gill-Weihs-Zeilinger-Zukowski anti-Hess-Philipp arguments
We present comparative analysis of Gill-Weihs-Zeilinger-Zukowski arguments
directed against Hess-Philipp anti-Bell arguments. In general we support
Hess-Philipp viewpoint to the sequence of measurements in the EPR-Bohm
experiments as stochastic time-like process. On the other hand, we support
Gill-Weihs-Zeilinger-Zukowski arguments against the use of time-like
correlations as the factor blocking the derivation of Bell-type inequalities.
We presented our own time-analysis of measurements in the EPR-Bohm experiments
based on the frequency approach to probability. Our analysis gives strong
arguments in favour of local realism. Moreover, our frequency analysis supports
the original EPR-idea that quantum mechnaics is not complete
The Quantum-Like Brain Operating on Subcognitive and Cognitive Time Scales
We propose a {\it quantum-like} (QL) model of the functioning of the brain.
It should be sharply distinguished from the reductionist {\it quantum} model.
By the latter cognition is created by {\it physical quantum processes} in the
brain. The crucial point of our modelling is that discovery of the mathematical
formalism of quantum mechanics (QM) was in fact discovery of a very general
formalism describing {\it consistent processing of incomplete information}
about contexts (physical, mental, economic, social). The brain is an advanced
device which developed the ability to create a QL representation of contexts.
Therefore its functioning can also be described by the mathematical formalism
of QM. The possibility of such a description has nothing to do with composing
of the brain of quantum systems (photons, electrons, protons,...). Moreover, we
shall propose a model in that the QL representation is based on conventional
neurophysiological model of the functioning of the brain. The brain uses the QL
rule (given by von Neumann trace formula) for calculation of {\it approximative
averages} for mental functions, but the physical basis of mental functions is
given by neural networks in the brain. The QL representation has a {\it
temporal basis.} Any cognitive process is based on (at least) two time scales:
subcognitive time scale (which is very fine) and cognitive time scale (which is
essentially coarser)
Probabilistic pathway representation of cognitive information
We present for mental processes the program of mathematical mapping which has
been successfully realized for physical processes. We emphasize that our
project is not about mathematical simulation of brain's functioning as a
complex physical system, i.e., mapping of physical and chemical processes in
the brain on mathematical spaces. The project is about mapping of purely mental
processes on mathematical spaces. We present various arguments -- philosophic,
mathematical, information, and neurophysiological -- in favor of the -adic
model of mental space. -adic spaces have structures of hierarchic trees and
in our model such a tree hierarchy is considered as an image of neuronal
hierarchy. Hierarchic neural pathways are considered as fundamental units of
information processing. As neural pathways can go through whole body, the
mental space is produced by the whole neural system. Finally, we develop
Probabilistic Neural Pathway Model in that Mental States are represented by
probability distributions on mental space
Randomness: quantum versus classical
Recent tremendous development of quantum information theory led to a number
of quantum technological projects, e.g., quantum random generators. This
development stimulates a new wave of interest in quantum foundations. One of
the most intriguing problems of quantum foundations is elaboration of a
consistent and commonly accepted interpretation of quantum state. Closely
related problem is clarification of the notion of quantum randomness and its
interrelation with classical randomness. In this short review we shall discuss
basics of classical theory of randomness (which by itself is very complex and
characterized by diversity of approaches) and compare it with irreducible
quantum randomness. The second part of this review is devoted to the
information interpretation of quantum mechanics (QM) in the spirit of Zeilinger
and Brukner (and QBism of Fuchs et al.) and physics in general (e.g., Wheeler's
"it from bit") as well as digital philosophy of Chaitin (with historical
coupling to ideas of Leibnitz). Finally, we continue discussion on
interrelation of quantum and classical randomness and information
interpretation of QM.Comment: arXiv admin note: text overlap with arXiv:1410.577
Classical probabilistic realization of "Random Numbers Certified by Bell's Theorem"
We question the commonly accepted statement that random numbers certified by
Bell's theorem carry some special sort of randomness, so to say, quantum
randomness or intrinsic randomness. We show that such numbers can be easily
generated by classical random generators
CHSH inequality: Quantum probabilities as classical conditional probabilities
The celebrating theorem of A. Fine implies that the CHSH inequality is
violated if and only if the joint probability distribution for the quadruples
of observables involved the EPR-Bohm-Bell experiment does not exist, i.e., it
is impossible to use the classical probabilistic model (Kolmogorov, 1933). In
this note we demonstrate that, in spite of Fine's theorem, the results of
observations in the EPR-Bohm-Bell experiment can be described in the classical
probabilistic framework. However, the "quantum probabilities" have to be
interpreted as conditional probabilities, where conditioning is with respect to
fixed experimental settings. Our approach is based on the complete account of
randomness involved in the experiment. The crucial point is that randomness of
selections of experimental settings has to be taken into account. This approach
can be applied to any complex experiment in which statistical data are
collected for various (in general incompatible) experimental settings. Finally,
we emphasize that our construction of the classical probability space for the
EPR-Bohm-Bell experiment cannot be used to support the hidden variable approach
to the quantum phenomena. The classical random parameter involved in
our considerations cannot be identified with the hidden variable
which is used the Bell-type considerations.Comment: presented at the conference Quantum Theory: from Problems to
Applications, Vaxjo, Sweden, June 2014, and during the course of lectures on
inter-relation between classical and quantum randomness given at Institute
for Quantum Optics and Quantum Information of Austrian Academy of Science,
May 201
Violation of the Bell's type inequalities as a local expression of incompatibility
By filtering out the philosophic component we can be said that the EPR-paper
was directed against the straightforward interpretation of the Heisenberg's
uncertainty principle or more generally the Bohr's complementarity principle.
The latter expresses contextuality of quantum measurements: dependence of
measurement's output on the complete experimental arrangement. However, Bell
restructured the EPR-argument against complementarity to justify nonlocal
theories with hidden variables of the Bohmian mechanics' type. Then this Bell's
kind of nonlocality - {\it subquantum nonlocality} - was lifted to the level of
quantum theory - up to the terminology {\it "quantum nonlocality"}. The aim of
this short note is to explain that Bell's test is simply a special {\it test of
local incompatibility of quantum observables}, similar to interference
experiments, e.g., the two-slit experiment
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