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Andrei Khrennikov

Publications and source records attributed to Andrei Khrennikov.

At least 109 records · Page 6Linked to original sources
arXiv2013

Classical random field viewpoint to Gorini-Kossakowski-Sudarshan-Lindblad equation and its linear and nonlinear generalizations

We show that the basic equation of theory of open systems, Gorini-Kossakowski-Sudarshan-Lindblad equation, as well as its linear and nonlinear generalizations have a natural classical probabilistic interpretation - in the framework of prequantum classical statistical field theory. The latter gives an example of a classical probabilistic model (with random fields as subquantum variables) reproducing the basic probabilistic predictions of quantum mechanics.

quant-ph↗
arXiv2013

Criteria of ergodicity for $p$-adic dynamical systems in terms of coordinate functions

This paper is devoted to the problem of ergodicity of $p$-adic dynamical systems. Our aim is to present criteria of ergodicity in terms of coordinate functions corresponding to digits in the canonical expansion of $p$-adic numbers. The coordinate representation can be useful, e.g., for applications to cryptography. Moreover, by using this representation we can consider non-smooth $p$-adic transformations. The basic technical toolsare van der Put series and usage of algebraic structure (permutations) induced by coordinate functions with partially frozen variables. We illustrate the basic theorems by presenting concrete classes of ergodic functions.

math.DS↗
arXiv2012

Measurement problem: from de Broglie to theory of classical random fields interacting with threshold detectors

The quantum measurement problem as was formulated by von Neumann in 1933 can be solved by going beyond the operational quantum formalism. In our "prequantum model" quantum systems are symbolic representations of classical random fields. The Schrödinger's dynamics is a special form of the linear dynamics of classical fields. Measurements are described as interactions of classical fields with detectors. Discontinuity, the "collapse of the wave function", has the trivial origin: usage of threshold type detectors. The von Neumann projection postulate can be interpreted as the formal mathematical encoding of the absence of coincidence detection in measurement on a single quantum system, e.g., photon's polarization measurement. Our model, prequantum classical statistical field theory (PCSFT), in combination with measurements by threshold detectors satisfies the quantum restriction on coincidence detections: the second order coherence is less than one (opposite to all known semiclassical and classical feld models). The basic rule of quantum probability, the Born's rule, is derived from properties of prequantum random felds interacting with threshold type detectors. Comparison with De Broglie's views to quantum mechanics as theory of physical waves with singularities is presented.

quant-ph↗
arXiv2012

Born's rule from statistical mechanics of classical fields: from hitting times to quantum probabilities

We show that quantum probabilities can be derived from statistical mechanics of classical fields. We consider Brownian motion in the space of fields and show that such a random field interacting with threshold type detectors produces clicks at random moments of time. And the corresponding probability distribution can be approximately described by the formalism of quantum mechanics. Hence, probabilities in quantum mechanics and classical statistical mechanics differ not so much as it is typically claimed. The temporal structure of the "prequantum random field" (which is the $L_2$-valued Wiener process) plays the crucial role. Moments of detector's clicks are mathematically described as hitting times which are actively used in classical theory of stochastic processes. Born's rule appears as an approximate rule. In principle, the difference between the "precise detection probability rule" derived in this paper and the conventional Born's rule can be tested experimentally. In our model the presence of the random gain in detectors playes a crucial role. We also stress the role of the detection threshold. It is not merely a technicality, but the fundamental element of the model.

quant-ph↗
arXiv2012

T-functions revisited: New criteria for bijectivity/transitivity

The paper presents new criteria for bijectivity/transitivity of T-functions and fast knapsack-like algorithm of evaluation of a T-function. Our approach is based on non-Archimedean ergodic theory: Both the criteria and algorithm use van der Put series to represent 1-Lipschitz $p$-adic functions and to study measure-preservation/ergodicity of these.

cs.CR↗
arXiv2012

Criteria of measure-preserving for $p$-adic dynamical systems in terms of the van der Put basis

This paper is devoted to (discrete) $p$-adic dynamical systems, an important domain of algebraic and arithmetic dynamics. We consider the following open problem from theory of $p$-adic dynamical systems. Given continuous function $f:Z_p > Z_p.$ Let us represent it via special convergent series, namely van der Put series. How can one specify whether this function is measure-preserving or not for an arbitrary $p$? In this paper, for any prime $p$ we present a complete description of all compatible measure-preserving functions in the additive form representation. In addition we prove the criterion in terms of coefficients with respect to the van der Put basis determining whether a compatible function $f:Z_p > Z_p$ preserves the Haar measure.

math.DS↗
arXiv2012

Vaxjo Interpretation of Wave Function: 2012

We discuss the problem of interpretation of the wave function. The latest version of the Växjö interpretation is presented. It differs essentially from the original Växjö interpretation (2001). The main distinguishing feature of the present Växjö interpretation is the combination of realism on the subquantum level with nonobjectivity of quantum observables (i.e., impossibility to assign their values before measurements). Hence, realism is destroyed by detectors transforming continuous subquantum reality into discrete events, clicks of detectors. The Växjö interpretation-2012 is fundamentally contextual in the sense that the value of an observable depends on measurement context. This is contextuality in Bohr's sense. It is more general than Bell's contextuality based on joint measurements of compatible observables.

quant-ph↗
arXiv2012

Remote State Preparation of Mental Information

The aim of this paper is to define in theoretical terms and summarise the available experimental evidence that physical and mental "objects", if considered "information units", may present similar classical and quantum models of communication beyond their specific characteristics. Starting with the Remote State Preparation protocol, a variant of the Teleportation protocol, for which formal models and experimental evidence are already available in quantum mechanics, we outline a formal model applied to mental information we defined Remote State Preparation of Mental Information (RSPMI), and we summarise the experimental evidence supporting the feasibility of a RSPMI protocol. The available experimental evidence offers strong support to the possibility of real communication at distance of mental information promoting the integration between disciplines that have as their object of knowledge different aspects of reality, both physical and the mental, leading to a significant paradigm shift in cognitive and information science.

quant-ph↗
arXiv2012

The quantum-like description of the dynamics of party governance in the US political system

This paper is devoted to the application of the mathematical formalism of quantum mechanics to social (political) science. By using the quantum dynamical equations we model the process of decision making in US elections. The crucial point we attempt to make is that the voter's mental state can be represented as a superposition of two possible choices for either republicans or democrats. However, reality dictates a more complicated situation: typically a voter participates in two elections, i.e. the congress and the presidential elections. In both elections he/she has to decide between two choices. This very feature of the US election system requires that the mental state is represented by a 2-qubit state corresponding to the superposition of 4 different choices (e.g. for republicans in the congress; for the president as a democrat). The main issue of this paper is to describe the dynamics of the voters' mental states taking in account the mental and socio- political environment. What is truly novel in this paper is that instead of using Schrödinger's equation to describe the dynamics in an absence of interactions, we here apply the quantum master equation. This equation describes quantum decoherence, i.e., resolution from superposition to a definite choice.

physics.gen-ph↗
arXiv2012

Decompositions of Gelfand-Shilov kernels into kernels of similar class

We prove that any linear operator with kernel in a Gelfand-Shilov space is a composition of two operators with kernels in the same Gelfand-Shilov space. We also give links on numerical approximations for such compositions. We apply these composition rules to establish Schatten-von Neumann properties for such operators.

math.FA↗
arXiv2012

Ergodicity criteria for non-expanding transformations of 2-adic spheres

In the paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of discrete dynamical systems $ $ on 2-adic spheres $\mathbf S_{2^{-r}}(a)$ of radius $2^{-r}$, $r\ge 1$, centered at some point $a$ from the ultrametric space of 2-adic integers $\mathbb Z_2$. The map $f\colon\mathbb Z_2\to\mathbb Z_2$ is assumed to be non-expanding and measure-preserving; that is, $f$ satisfies a Lipschitz condition with a constant 1 with respect to the 2-adic metric, and $f$ preserves a natural probability measure on $\mathbb Z_2$, the Haar measure $μ_2$ on $\mathbb Z_2$ which is normalized so that $μ_2(\mathbb Z_2)=1$.

math.DS↗
arXiv2012

Fisher Information for Inverse Problems and Trace Class Operators

This paper provides a mathematical framework for Fisher information analysis for inverse problems based on Gaussian noise on infinite-dimensional Hilbert space. The covariance operator for the Gaussian noise is assumed to be trace class, and the Jacobian of the forward operator Hilbert-Schmidt. We show that the appropriate space for defining the Fisher information is given by the Cameron-Martin space. This is mainly because the range space of the covariance operator always is strictly smaller than the Hilbert space. For the Fisher information to be well-defined, it is furthermore required that the range space of the Jacobian is contained in the Cameron-Martin space. In order for this condition to hold and for the Fisher information to be trace class, a sufficient condition is formulated based on the singular values of the Jacobian as well as of the eigenvalues of the covariance operator, together with some regularity assumptions regarding their relative rate of convergence. An explicit example is given regarding an electromagnetic inverse source problem with "external" spherically isotropic noise, as well as "internal" additive uncorrelated noise.

math-ph↗
arXiv2012

Extended Tensor Products and Generalization of the Notion of Entanglement

Motivated by the novel applications of the mathematical formalism of quantum theory and its generalizations in cognitive science, psychology, social and political sciences, and economics, we extend the notion of the tensor product and entanglement. We also study the relation between conventional entanglement of complex qubits and our generalized entanglement. Our construction can also be used to describe entanglement in the framework of non-Archimedean physics. It is also possible to construct tensor products of non-Archimedean (e.g., $p$-adic) and complex Hilbert spaces.

physics.gen-ph↗
arXiv2011

Bell argument: Locality or Realism? Time to make the choice

This paper discusses a possible resolution of the nonobjectivity-nonlocality dilemma in quantum mechanics in 'the light of experimental tests of the Bell inequality for two entangled photons and a Bell-like inequality for a single neutron. My conclusion is that these experiments show that quantum mechanics is nonobjective: that is, the values of physical observables cannot be assigned to a system before measurement. Bell's assumption of nonlocality has to be rejected as having no direct experimental confirmation, at least thus far. I also consider the relationships between nonobjectivity and contextuality. Specifically, I analyze the impact of the Kochen-Specker theorem on the problem of contextuality of quantum observables. I argue that, just as von Neumann's "no-go" theorem, the Kochen-Specker theorem is based on assumptions that do not correspond to the real physical situation. Finally, I present a theory of measurement based on a classical, purely wave model (pre-quantum classical statistical field theory), a model that reproduces quantum probabilities. In this model continuous fields are transformed into discrete clicks of detectors. While this model is classical, it is nonobjective. In this case, nonobjectivity is the result of the dependence of experimental outcomes on the context of measurement, in accordance with Bohr's view.

quant-ph↗
arXiv2011

Classical signal model reproducing quantum probabilities for single and coincidence detections

We present a simple classical (random) signal model reproducing Born's rule. The crucial point of our approach is that the presence of detector's threshold and calibration procedure have to be treated not as simply experimental technicalities, but as the basic counterparts of the theoretical model. We call this approach threshold signal detection model (TSD). The experiment on coincidence detection which was done by Grangier in 1986 \cite{Grangier} played a crucial role in rejection of (semi-)classical field models in favor of quantum mechanics (QM): impossibility to resolve the wave-particle duality in favor of a purely wave model. QM predicts that the relative probability of coincidence detection, the coefficient $g^{(2)}(0),$ is zero (for one photon states), but in (semi-)classical models $g^{(2)}(0)\geq 1.$ In TSD the coefficient $g^{(2)}(0)$ decreases as $1/{\cal E}_d^2,$ where ${\cal E}_d>0$ is the detection threshold. Hence, by increasing this threshold an experimenter can make the coefficient $g^{(2)}(0)$ essentially less than 1. The TSD-prediction can be tested experimentally in new Grangier type experiments presenting a detailed monitoring of dependence of the coefficient $g^{(2)}(0)$ on the detection threshold.

quant-ph↗
arXiv2011

Quantum probabilities and violation of CHSH-inequality from classical random signals and threshold type properly calibrated detectors

We present a purely wave model (based on classical random field) which reproduces quantum probabilities (given by the fundamental law of quantum mechanics, Born's rule) including probabilities for joint detection of a pair of quantum observables (e.g., spin or polarization projections). The crucial point of our approach is that the presence of detector's threshold and calibration procedure have to be treated not as simply experimental technicalities, but as the basic counteparts of the theoretical model. The presence of the background field (vacuum fluctuations) is also the key-element of our prequantum model. It is of the classical signal type and the methods of classical signal theory (including statistical radiophysics) are used for its development. We stress that our prequantum model is not objective, i.e., the values of observables (clicks of detectors) cannot be assigned in advance, i.e., before measurement. Hence, the dilemma, nonobjectivity or nonlocality, is resolved in favor of nonobjectivity (our model is local of the classical field type). In particular, we reproduce the probabilities for the EPR-experiment for photon polarization and, hence, violate CHSH inequality for classical random signals (measured by the threshold type and properly calibrated detectors acting in the presence of the background field).

quant-ph↗