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Andrei Yu. Khrennikov

Publications and source records attributed to Andrei Yu. Khrennikov.

17 recordsLinked to original sources
arXiv2019

Multidimensional nonlinear pseudo-differential evolution equation with p-adic spatial variables

We study the Cauchy problem for $p$-adic nonlinear evolutionary pseudo-differential equations for complex-valued functions of a real positive time variable and p-adic spatial variables. Among the equations under consideration there is the p-adic analog of the porous medium equation (or more generally, the nonlinear filtration equation) which arise in numerous application in mathematical physics and mathematical biology. Our approach is based on the construction of a linear Markov semigroup on a p-adic ball and the proof of m-accretivity of the appropriate nonlinear operator. The latter result is equivalent to the existence and uniqueness of a mild solution of the Cauchy problem of a nonlinear equation of the porous medium type.

math.AP↗
arXiv2018

On the p-Adic Navier-Stokes Equation

We prove the local solvability of the p-adic analog of the Navier-Stokes equation. This equation describes, within the p-adic model of porous medium, the flow of a fluid in capillaries.

math-ph↗
arXiv2017

p-Adic Analogue of the Porous Medium Equation

We consider a nonlinear evolution equation for complex-valued functions of a real positive time variable and a p-adic spatial variable. This equation is a non-Archimedean counterpart of the fractional porous medium equation. Developing, as a tool, an $L^1$-theory of Vladimirov's p-adic fractional differentiation operator, we prove m-accretivity of the appropriate nonlinear operator, thus obtaining the existence and uniqueness of a mild solution. We give also an example of an explicit solution of the p-adic porous medium equation.

math.AP↗
arXiv2017

Test of the no-signaling principle in the Hensen "loophole-free CHSH experiment"

We analyze the data from the loophole-free CHSH experiment performed by Hensen et al., and show that it is actually not exempt of an important loophole. By increasing the size of the sample of event-ready detections, one can exhibit in the experimental data a violation of the no-signaling principle with a statistical significance at least similar to that of the reported violation of the CHSH inequality, if not stronger.

quant-ph↗
arXiv2017

Ultrametrics in the genetic code and the genome

Ultrametric approach to the genetic code and the genome is considered and developed. $p$-Adic degeneracy of the genetic code is pointed out. Ultrametric tree of the codon space is presented. It is shown that codons and amino acids can be treated as $p$-adic ultrametric networks. Ultrametric modification of the Hamming distance is defined and noted how it can be useful. Ultrametric approach with $p$-adic distance is an attractive and promising trend towards investigation of bioinformation.

q-bio.OT↗
arXiv2015

Summation of p-Adic Functional Series in Integer Points

Summation of a large class of the functional series, which terms contain factorials, is considered. We first investigated finite partial sums for integer arguments. These sums have the same values in real and all p-adic cases. The corresponding infinite functional series are divergent in the real case, but they are convergent and have p-adic invariant sums in p-adic cases. We found polynomials which generate all significant ingredients of these series and make connection between their real and p-adic properties. In particular, we found connection of one of our integer sequences with the Bell numbers.

math.NT↗
arXiv2011

Double blinding-attack on entanglement-based quantum key distribution protocols

We propose a double blinding-attack on entangled-based quantum key distribution protocols. The principle of the attack is the same as in existing blinding attack except that instead of blinding the detectors on one side only, Eve is blinding the detectors of both Alice and Bob. In the BBM92 protocol, the attack allows Eve to get a full knowledge of the key and remain undetected even if Alice and Bob are using 100% efficient detectors. The attack can be easily extended to Ekert protocol, with an efficiency as high as 85.3%.

quant-ph↗
arXiv2011

Multiple-Photon Absorption Attack on Entanglement-Based Quantum Key Distribution Protocols

In elaborating on the multiple-photon absorption attack on Ekert protocol proposed in arXiv:1011.4740, we show that it can be used in other entanglement-based protocols, in particular the BBM92 protocol. In this attack, the eavesdropper (Eve) is assumed to be in control of the source, and she sends pulses correlated in polarization (but not entangled) containing several photons at frequencies for which only multiple-photon absorptions are possible in Alice's and Bob's detectors. Whenever the photons stemming from one 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 is enough to reproduce the type of statistics on the detected pulses that are considered as indicating a secure quantum key distribution, even though the source is actually a mixture of separable states. The violation of Bell inequalities measured by Alice and Bob increases with the order of the multiple-photon absorption that Eve can drive into their detectors, while the measured quantum bit error rate decreases as a function of the same variable. We show that the attack can be successful even in the simplest case of a two-photon absorption or three-photon absorption attack, and we discuss possible countermeasures, in particular the use of a fair sampling test.

quant-ph↗
arXiv2010

A Fair Sampling Test for Ekert Protocol

We propose a local scheme to enhance the security of quantum key distribution in Ekert protocol (E91). Our proposal is a fair sampling test meant to detect an eavesdropping attempt that would use a biased sample to mimic an apparent violation of Bell inequalities. The test is local and non disruptive: it can be unilaterally performed at any time by either Alice or Bob during the production of the key, and together with the Bell inequality test.

quant-ph↗
arXiv2007

New possible properties of atomic nuclei investigated by non linear methods: Fractal and recurrence quantification analysis

For the first time we apply the methodologies of nonlinear analysis to investigate atomic matter. We use these methods in the analysis of Atomic Weights and of Mass Number of atomic nuclei. Using the AutoCorrelation Function and Mutual Information we establish the presence of nonlinear effects in the mechanism of increasing mass of atomic nuclei considered as a function of the atomic number. We find that increasing mass is divergent, possibly chaotic. We also investigate the possible existence of a Power Law for atomic nuclei and, using also the technique of the variogram, we conclude that a fractal regime could superintend to the mechanism of increasing mass for nuclei. Finally, using the Hurst exponent, evidence is obtained that the mechanism of increasing mass in atomic nuclei is in the fractional Brownian regime. The most interesting results are obtained by using Recurrence Quantification Analysis (RQA). New recurrences, psudoperiodicities, self-resemblance and class of self-similarities are identified with values of determinism showing oscillating values indicating the presence of more or less stability during the process of increasing mass of atomic nuclei. In brief, new regimes of regularities are identified for atomic nuclei that deserve to be studied by future researches. In particular an accurate analysis of binding energy values by nonlinear methods is further required.

physics.gen-ph↗
arXiv2007

Brain as Quantum-like Machine for Transferring Time into Mind

We propose a model of processing of information in the brain which has the following distinguishing features: a). It is quantum-like (QL). The brain uses the quantum rule (given by von Neumann trace formula) for calculation of averages for psychological functions. b). Those functions are considered as self-observations of the brain. c). The QL-representation has the temporal basis. The brain is a machine transferring time into cognition. d). Any cognitive process is based on (at least) two time scales: precognitive time scale (which is very fine) and cognitive time scale (which is essentially coarser). To couple our model to physiology, behavioral science, and psychology, we consider a number of known fundamental time scales in the brain. Although the elaboration of those scales was based on advanced experimental research, there are still many controversial approaches and results. The temporal structure of the brain functioning is very complex.

q-bio.NC↗
arXiv2006

Is the Fair Sampling Assumption supported by EPR Experiments?

We analyze optical EPR experimental data performed by Weihs et al in Innsbruck 1997-1998. We show that for some linear combinations of the raw coincidence rates, the experimental results display some anomalous behavior that a more general source state (like non-maximally entangled state) cannot straightforwardly account for. We attempt to explain these anomalies by taking account of the relative efficiencies of the four channels. For this purpose, we use the fair sampling assumption, and assume explicitly that the detection efficiencies for the pairs of entangled photons can be written as a product of the two corresponding detection efficiencies for the single photons. We show that this explicit use of fair sampling cannot be maintained to be a reasonable assumption as it leads to an apparent violation of the no-signalling principle.

quant-ph↗
arXiv2006

On some detailed examples of quantum like structures containing quantum potential states in the sphere of biological dynamics

In the first part of the present paper we give an analysis of the ontic nature of quantum states to be intended as potentialities and of the central role of spin to be considered as the basic essence of quantum mechanical reality: using an algebraic quantum like structure we give mathematical proof on the transition from potentiality to actualization : we recall here what was recently given by us in arXiv quant-ph/0607196. However, as may be expected, it is not so easy to introduce examples containing an adequate description of ontic potentialities through detailed models of systems. The central aim of this paper is to attempt to reach this objective giving direct cases of systems in which ontic potentialities act jointly to actualization. Our aim is to provide evidence for the possible importance of potential states in the sphere of the biological dynamics giving detailed examples of interest for biological studies. We outline the possible implications of potentialities at the level of linear and non linear biological dynamics.

physics.gen-ph↗
arXiv2005

On the classical limit of the hyperbolic quantum mechanics

We demonstrated that classical mechanics have, besides the well known quantum deformation, another deformation -- so called hyperbolic quantum mechanics. The classical Poisson bracket can be obtained as the limit $h\to 0$ not only of the ordinary Moyal bracket, but also hyperbolic analogue of the Moyal bracket. Thus there are two different deformations of classical phase-space: complex Hilbert space and hyperbolic Hilbert space (module over a so called hyperbolic algebra -- the two dimensional Clifford algebra). To prove the correspondence principle we use the calculus over the hyperbolic algebra similar to functional superanalysis of Vladimirov-Volovich. Ordinary (complex) and hyperbolic quantum mechanics are characterized by two types of interference perturbation of the classical formula of total probability: ordinary $\cos$-interference and hyperbolic $\cosh$-interference.

quant-ph↗
arXiv2004

On the representation of contextual probabilistic dynamics in the complex Hilbert space: linear and nonlinear evolutions, Schroedinger dynamics

We constructed the representation of contextual probabilistic dynamics in the complex Hilbert space. Thus dynamics of the wave function can be considered as Hilbert space projections of realistic dynamics in a ``prespace''. The basic condition for representing of the prespace-dynamics is the law of statistical conservation of energy -- conservation of probabilities. Construction of the dynamical representation is an important step in the development of contextual statistical viewpoint to quantum processes. But the contextual statistical model is essentially more general than the quantum one. Therefore in general the Hilbert space projection of the ``prespace'' dynamics can be nonlinear and even irreversible (but it is always unitary). There were found conditions of linearity and reversibility of the Hilbert space dynamical projection. We also found conditions for the conventional Schroedinger dynamics (including time-dependent Hamiltonians). We remark that in general even the Schroedinger dynamics is based just on the statistical conservation of energy; for individual systems the law of conservation of energy can be violated (at least in our theoretical model).

quant-ph↗
arXiv2003

Marginality in non-compatible random events

We present a way of introducing joint distibution function and its marginal distribution functions for non-compatible observables. Each such marginal distribution function has the property of commutativity. Models based on this approach can be used to better explain some classical phenomena in stochastic processes.

quant-ph↗