arXiv subjects

Andrei Khrennikov

Publications and source records attributed to Andrei Khrennikov.

At least 127 records · Page 7Linked to original sources

arXiv2011

Classical signal viewpoint to bunching and anti-bunching

The similarity between classical wave mechanics and quantum mechanics (QM) played an important role in the development of QM (starting with works of De Broglie, Schrödinger, "late Einstein", Lamb, Lande, Mandel, Marshall, Santos, Boyer, and many others). We present a new wave-type approach, so called prequantum classical statistical field theory (PCSFT). PCSFT explores an analogy between some quantum phenomena and classical theory of random fields. Quantum systems are interpreted as symbolic representations of such fields (not only for photons, cf. Lande and Lamb, but even for massive particles). All quantum averages and correlations (including composite systems in entangled states) can be represented as averages and correlations for classical random fields. In this paper PCSFT is used to provide a classical signal representation of bunching and anti-bunching. At least the latter is typically considered as essentially quantum (nonclassical) phenomenon.

quant-ph↗

arXiv2011

Born's rule from measurements of classical signals by threshold detectors which are properly calibrated

The very old problem of the statistical content of quantum mechanics (QM) is studied in a novel framework. The Born's rule (one of the basic postulates of QM) is derived from theory of classical random signals. We present a measurement scheme which transforms continuous signals into discrete clicks and reproduces the Born's rule. This is the sheme of threshold type detection. Calibration of detectors plays a crucial role.

quant-ph↗

arXiv2011

Bounds for the Distance Dependence of Correlation Functions of Entangled Photons in Waveguides

The distance dependence of the probability of observing two photons in a waveguide is investigated. The Glauber correlation functions of the entangled photons in waveguides are considered and the spatial and temporal dependence of the correlation functions is evaluated. We derive upper bounds to the distance dependence of the probability of observing two photons. These inequalities should be possible to observe in experiments.

quant-ph↗

arXiv2011

A general quantum information model for the contextual dependent systems breaking the classical probability law

There exist several phenomena (systems) breaking the classical probability laws. Such systems are contextual dependent adaptive systems. In this paper, we present a new mathematical formula to compute the probability in those systems by using the concepts of the adaptive dynamics and quantum information theory -- quantum channels and the lifting. The basic examples of the contextual dependent phenomena can be found in quantum physics. And recently similar examples were found in biological and psychological sciences. Our novel approach is motivated by traditional quantum probability, but it is general enough to describe aforementioned phenomena outside of quantum physics.

quant-ph↗

arXiv2011

Totality of Subquantum Nonlocal Correlations

In a series of previous papers we developed a purely field model of microphenomena, so called prequantum classical statistical field theory (PCSFT). This model not only reproduces important probabilistic predictions of QM including correlations for entangled systems, but it also gives a possibility to go beyond quantum mechanics (QM), i.e., to make predictions of phenomena which could be observed at the subquantum level. In this paper we discuss one of such predictions - existence of nonlocal correlations between prequantum random fields corresponding to {\it all} quantum systems. (And by PCSFT quantum systems are represented by classical Gaussian random fields and quantum observables by quadratic forms of these fields.) The source of these correlations is the common background field. Thus all prequantum random fields are "entangled", but in the sense of classical signal theory. On one hand, PCSFT demystifies quantum nonlocality by reducing it to nonlocal classical correlations based on the common random background. On the other hand, it demonstrates total generality of such correlations. They exist even for distinguishable quantum systems in factorizable states (by PCSFT terminology - for Gaussian random fields with covariance operators corresponding to factorizable quantum states).

quant-ph↗

arXiv2011

Violation of Bell's inequality and postulate on simultaneous measurement of compatible observables

We discuss coupling of violation of Bell's inequality and non-Kolmogorovness of statistical data in the EPR-Bohm experiment. We emphasize that nonlocalty and "death of realism" are only sufficient, but not necessary conditions of non-Kolmogorovness. There can be found other sufficient conditions of non-Kolmogorovness and, hence, violation of Bell's inequality. We find one important source of non-Kolmogorovness by analyzing axiomatics of quantum mechanics. We pay attention to the postulate (due to von Neumann and Dirac) on simultaneous measurement of quantum observables given by commuting operators. This postulate is criticized as nonphysical. We propose a new interpretation of the Born-von Neumann-Dirac rule for calculation of the joint probability distribution for such observables. It gets a natural physical interpretation by considering conditional measurement scheme. We use this argument (i.e., rejection of the postulate on simultaneous measurement to motivate non-Kolmogorovness of the probabilistic structure of the EPR-Bohm experiment.

quant-ph↗

arXiv2010

Quantum-like model of processing of information in the brain based on classical electromagnetic field

We propose a model of quantum-like (QL) processing of mental information. This model is based on quantum information theory. However, in contrast to models of "quantum physical brain" reducing mental activity (at least at the highest level) to quantum physical phenomena in the brain, our model matches well with the basic neuronal paradigm of the cognitive science. QL information processing is based (surprisingly) on classical electromagnetic signals induced by joint activity of neurons. This novel approach to quantum information is based on representation of quantum mechanics as a version of classical signal theory which was recently elaborated by the author. The brain uses the QL representation (QLR) for working with abstract concepts; concrete images are described by classical information theory. Two processes, classical and QL, are performed parallely. Moreover, information is actively transmitted from one representation to another. A QL concept given in our model by a density operator can generate a variety of concrete images given by temporal realizations of the corresponding (Gaussian) random signal. This signal has the covariance operator coinciding with the density operator encoding the abstract concept under consideration. The presence of various temporal scales in the brain plays the crucial role in creation of QLR in the brain.

physics.gen-ph↗

arXiv2010

Violation of Bell Inequalities with a Mixture of Separable States in a Multiple-Photon Absorption Attack on Ekert Protocol

We propose a new type of attack on Ekert protocol in which, rather surprisingly, Eve drives a violation of Bell inequalities in Alice's and Bob's detectors with a mixture of separable states. She does so by sending correlated pulses containing several photons at frequencies where only multiple-photon absorptions are possible in their detectors. Whenever the photons stemming from a same pulse are dispatched in such a way that the number of photons is insufficient to trigger a multiple-photon absorption in either channel, the pulse remains undetected. We show that this simple feature leads to violations of Bell inequalities that can match closely those predicted for entangled states, even in the simplest cases of two-photon and three-photon absorptions.

quant-ph↗

arXiv2010

Classical Signal Model for Quantum Channels

Recently it was shown that the main distinguishing features of quantum mechanics (QM) can be reproduced by a model based on classical random fields, so called prequantum classical statistical field theory (PCSFT). This model provides a possibility to represent averages of quantum observables, including correlations of observables on subsystems of a composite system (e.g., entangled systems), as averages with respect to fluctuations of classical (Gaussian) random fields. In this note we consider some consequences of PCSFT for quantum information theory. They are based on the observation \cite{W} of two authors of this paper that classical Gaussian channels (important in classical signal theory) can be represented as quantum channels. Now we show that quantum channels can be represented as classical linear transformations of classical Gaussian signal

quant-ph↗

arXiv2010

Towards violation of Born's rule: description of a simple experiment

Recently a new model with hidden variables of the wave type was elaborated, so called prequantum classical statistical field theory (PCSFT). Roughly speaking PCSFT is a classical signal theory applied to a special class of signals -- "quantum systems". PCSFT reproduces successfully all probabilistic predictions of QM, including correlations for entangled systems. This model peacefully coexists with all known no-go theorems, including Bell's theorem. In our approach QM is an approximate model. All probabilistic predictions of QM are only (quite good) approximations of "real physical averages". The latter are averages with respect to fluctuations of prequantum fields. In particular, Born's rule is only an approximate rule. More precise experiments should demonstrate its violation. We present a simple experiment which has to produce statistical data violating Born's rule. Since the PCSFT-presentation of this experiment may be difficult for experimenters, we reformulate consequences of PCSFT in terms of the conventional wave function. In general, deviation from Born's rule is rather small. We found an experiment amplifying this deviation. We start with a toy example in section 2. Then we present a more realistic example based on Gaussian states with very small dispersion, see section 3.

quant-ph↗

arXiv2010

Demystification of quantum entanglement

One of the crucial differences between mathematical models of classical and quantum mechanics is the use of the tensor product of the state spaces of subsystems as the state space of the corresponding composite system. (To describe an ensemble of classical composite systems one uses random variables taking values in the Cartesian product of the state spaces of subsystems.) We show that, nevertheless, it is possible to establish a natural correspondence between the classical and quantum probabilistic descriptions of composite systems. Quantum averages for composite systems (including entangled) can be represented as averages with respect to classical random fields. It is essentially what Albert Einstein was dreamed of. Quantum mechanics is represented as classical statistical mechanics with infinite-dimensional phase space. While the mathematical construction is completely rigorous, its physical interpretation is a complicated problem (which will not be discussed in this paper).

physics.gen-ph↗

arXiv2009

An analog of Heisenberg uncertainty relation in prequantum classical field theory

Prequantum classical statistical field theory (PCSFT) is a model which provides a possibility to represent averages of quantum observables, including correlations of observables on subsystems of a composite system, as averages with respect to fluctuations of classical random fields. PCSFT is a classical model of the wave type. For example, "electron" is described by electronic field. In contrast to QM, this field is a real physical field and not a field of probabilities. An important point is that the prequantum field of e.g. electron contains the irreducible contribution of the background field, vacuum fluctuations. In principle, the traditional QM-formalism can be considered as a special regularization procedure: subtraction of averages with respect to vacuum fluctuations. In this paper we derive a classical analog of the Heisenberg-Robertson inequality for dispersions of functionals of classical (prequantum) fields. PCSFT Robertson-like inequality provides a restriction on the product of classical dispersions. However, this restriction is not so rigid as in QM. The quantum dispersion corresponds to the difference between e.g. the electron field dispersion and the dispersion of vacuum fluctuations. Classical Robertson-like inequality contains these differences. Hence, it does not imply such a rigid estimate from below for dispersions as it was done in QM.

quant-ph↗

arXiv2009

Financial heat machine

We consider dynamics of financial markets as dynamics of expectations and discuss such a dynamics from the point of view of phenomenological thermodynamics. We describe a financial Carnot cycle and the financial analogue of a heat machine. We see, that while in physics a perpetuum mobile is absolutely impossible, in economics such mobile may exist under some conditions. Our thermodynamical model for the financial market induces a rather unusual interpretation of the role of financial crises. In contrast to the common point of view, in our model financial crises play a crucial role in functioning of the modern financial market. This is an important (concluding) stage of any financial cycle that is analogous to the stage of cooling in the ordinary Carnot cycle. A financial cycle could not be completed without such a stage as well as the ordinary Carnot cycle. Thus, in spite its destructive (at the first sight) consequences the stage or financial crises is as well important as the stage of "boiling of the financial market" ("heating of expectations")

cond-mat.other↗

arXiv2009

p-adic probability prediction of correlations between particles in the two-slit and neutron interferometry experiments

We start from Feynman`s idea to use negative probabilities to describe the two slit experiment and other quantum interfernce experiments. Formally by using negative probability distributions we can explain the results of the two slit experiment on the basis of the pure corpuscular picture of quantum mechnanics. However, negative probabilities are absurd objects in the framework of the standard Kolmogorov theory of probability. We present a large class of non-Kolmogorovean probability models where negative probabilities are well defined on the frequency basis. These are models with probabilities which belong to the so-called field of $p$-adic numbers. However, these models are characterized by correlations between trails. Therefore, we predict correlations between particles in interference experiments. In fact, our predictions are similar to the predictions of the so-called nonergodic interpretation of quantum mechanics, which was proposed by V. Buonomano. We propose the concrete experiments (in particular, in the framework of the neutron interferometry) to verify our predictions on the correlations.

math-ph↗

arXiv2009

A hidden variables model for interference phenomena based on $p$-adic random dynamical systems

We propose a model based on random dynamical systems (RDS) in information spaces (realized as rings of $p$-adic integers) which supports Buonomano's non-ergodic interpretation of quantum mechanics. In this model the memory system of an equipment works as a dynamical system perturbed by noise. Interference patterns correspond to attractors of RDS. There exists a large class of $p$-adic RDS for which interference patterns cannot be disturbed by noise. Therefore, if the equipment is described by such a RDS then the result of statistical experiment does not depend on noise in the equipment. On the one hand, we support the corpuscular model, because a quantum particle can be described as a corpuscular object. On the other hand, our model does not differ strongly from the wave model, because a quantum particle interacts with the whole equipment. Hence the interaction has nonlocal character. For example, in the two slit experiment a quantum particle interacts with both slits (but it passes only one of them).

math-ph↗

arXiv2009

Von Neumann and Luders postulates and quantum information theory

This note is devoted to some foundational aspects of quantum mechanics (QM) related to quantum information (QI) theory, especially quantum teleportation and ``one way quantum computing.'' We emphasize the role of the projection postulate (determining post-measurement states) in QI and the difference between its Lüders and von Neumann versions. These projection postulates differ crucially in the case of observables with degenerate spectra. Such observables play the fundamental role in operations with entangled states: any measurement on one subsystem is represented by an observable with degenerate spectrum in the Hilbert space of a composite system. If von Neumann was right and Lüders was wrong the canonical schemes of quantum teleportation and ``one way quantum computing'' would not work. Surprisingly, we found that, in fact, von Neumann's description of measurements via refinement implies (under natural assumptions) Lüders projection postulate. It seems that this important observation was missed during last 70 years. This result closed the problem of the proper use of the projection postulate in quantum information theory. One can proceed with Lüders postulate (as people in quantum information really do).

quant-ph↗

arXiv2009

Detection model based on representation of quantum particles by classical random fields: Born's rule and beyond

Recently a new attempt to go beyond quantum mechanics (QM) was presented in the form of so called prequantum classical statistical field theory (PCSFT). Its main experimental prediction is violation of Born's rule which provides only an approximative description of real probabilities. We expect that it will be possible to design numerous experiments demonstrating violation of Born's rule. Moreover, recently the first experimental evidence of violation was found in the triple slits interference experiment, see \cite{WWW}. Although this experimental test was motivated by another prequantum model, it can be definitely considered as at least preliminary confirmation of the main prediction of PCSFT. In our approach quantum particles are just symbolic representations of "prequantum random fields," e.g., "electron-field" or "neutron-field"; photon is associated with classical random electromagnetic field. Such prequantum fields fluctuate on time and space scales which are essentially finer than scales of QM, cf. `t Hooft's attempt to go beyond QM \cite{H1}--\cite{TH2}. In this paper we elaborate a detection model in the PCSFT-framework. In this model classical random fields (corresponding to "quantum particles") interact with detectors inducing probabilities which match with Born's rule only approximately. Thus QM arises from PCSFT as an approximative theory. New tests of violation of Born's rule are proposed.

quant-ph↗

arXiv2009

Design of an Experiment to Test Quantum Probabilistic Behavior of the Financial market

The recent crash demonstrated (once again) that the description of the financial market by present financial mathematics cannot be considered as totally satisfactory. We remind that nowadays financial mathematics is heavily based on the use of random variables and stochastic processes which are described by Kolmogorov's measure-theoretic model for probability ("classical probabilistic model"). I speculate that the present financial crises is a sign (a kind of experiment to test validity of classical probability theory at the financial market) that the use of this model in finances should be either totally rejected or at least completed. One of the best candidates for a new probabilistic financial model is quantum probability or its generalizations, so to say quantum-like (QL) models. Speculations that the financial market may be nonclassical have been present in scientific literature for many years. The aim of this note is to move from the domain of speculation to rigorous statistical arguments in favor of probabilistic nonclassicality of the financial market. I design a corresponding statistical test which is based on violation of the formula of total probability (FTP). The latter is the basic in classical probability and its violation would be a strong sign in favor of QL behavior at the market.

physics.gen-ph↗