Classical Probability Research Articles

Уточнение методов расчета грузопотоков конвейерных транспортных систем угольных шахт

Productivity of a mining enterprise is limited by the possibility of its transportation subsystem, which efficiency improvement is an urgent scientific and practical task. Studying the processes of formation and transformation of material flows allows us to clarify the design methodology of continuous haulage systems. The normative methodology of belt conveyors selection is based on treating the material flows as normally distributed random variables, with the point value, i.e. the irregularity coefficient, being used as the main indicator. There are doubts in methodological justification of such an approach. The normal law of distribution in the classical variant adequately describes a random variable changing within infinite limits, and real mine material flows via the conveyor lines are random variables with unilateral constraints. The majority of tasks on finding the design ranges of material flows can be solved using experimentally established distribution functions (or probability density functions) of random quantities of material flows. Based on the publications in recent years, as well as on our own experimental studies, it is proposed to describe material flows by piecewise linear probability density functions, in particular the triangular ones, and to summarize material flows on the basis of general theoretical provisions of classical probability theory. The paper solves a particular problem of finding an analytical solution of the sum of two random material flows defined by the triangular distribution laws and compares the results of numerical integration of the probability densities of the material flows.

How Quantum Mechanics Requires Non-Additive Measures.

Measure theory is used in physics, not just to capture classical probability, but also to quantify the number of states. In previous works, we found that state quantification plays a foundational role in classical mechanics, and, therefore, we set ourselves to construct the quantum equivalent of the Liouville measure. Unlike the classical counterpart, this quantized measure is non-additive and has a unitary lower bound (i.e., no set of states can have less than one state). Conversely, requiring that state quantification is finite for finite continuous regions and that each state counts as one already implies non-additivity, which in turn implies the failure of classical theory. In this article we show these preliminary results and outline a new line of inquiry that may provide a different insight into the foundations of quantum theory. Additionally, this new approach may prove to be useful to those interested in a quantized theory of space-time, as we believe this requires a quantized measure for the quantification of the independent degrees of freedom.

Some convergence properties for arrays of rowwise asymptotically almost negatively associated random variables under sub-linear expectations

. In this article, under some suitable conditions, we study the L q convergence and complete q-th moment convergence for arrays of rowwise asymptotically almost negatively associated (AANA, for short) random variables under sub-linear expectations. Some general results on L q convergence and complete q-th moment convergence for arrays of rowwise AANA random variables under sub-linear expectations are established, which extend the corresponding ones from classic probability space to the case of sub-linear expectation space. As applications of our main results, we not only present a result on the complete consistency of the weighted estimator in a nonparametric regression model but also obtain the complete consistency of the estimators in errors-in-variables (EV, for short) regression models based on AANA errors under sub-linear expectations. We perform some numerical simulations to verify the validity of the theoretical results.

Localization of quantum walks with classical randomness: Comparison between manual methods and supervised machine learning.

A transition of quantum walk induced by classical randomness changes the probability distribution of the walker from a two-peak structure to a single-peak one when the random parameter exceeds a critical value. We first establish the generality of the localization by showing its emergence in the presence of random rotation or translation. The transition point can be located manually by examining the probability distribution, momentum of inertia, and inverse participation ratio. As a comparison, we implement three supervised machine learning methods, the support vector machine (SVM), multilayer perceptron neural network, and convolutional neural network with the same data and show they are able to identify the transition. While the SVM sometimes underestimates the exponents compared to the manual methods, the two neural-network methods show more deviations for the case with random translation due to the fluctuating probability distributions. Our work illustrates potentials and challenges facing machine learning of physical systems with mixed quantum and classical probabilities.

Further research on complete integral convergence for moving average process of ND random variables under sub-linear expectations

In this article, we establish some results on complete convergence and complete integral convergence of moving average process { X n = ∑ i = − ∞ ∞ a i Y i + n , n ≥ 1 } based on negatively dependent random variables under sub-linear expectations. We generalize corresponding conclusions obtained by Zhang and Ding, and extend convergence theorems for general function from classical probability space to the sub-linear expectation space.

An axiomatic approach to default risk and model uncertainty in rating systems

In this paper, we deal with an axiomatic approach to default risk. We introduce the notion of a default risk measure, which generalizes the classical probability of default (PD), and allows to incorporate model risk in various forms. We discuss different properties and representations of default risk measures via monetary risk measures, families of related tail risk measures, and Choquet capacities. In a second step, we turn our focus on default risk measures, which are given as worst-case PDs and distorted PDs. The latter are frequently used in order to take into account model risk for the computation of capital requirements through risk-weighted assets (RWAs), as demanded by the Capital Requirement Regulation (CRR). In this context, we discuss the impact of different default risk measures and margins of conservatism on the amount of risk-weighted assets.

Entanglement of Observables: Quantum Conditional Probability Approach

This paper is devoted to clarification of the notion of entanglement through decoupling it from the tensor product structure and treating as a constraint posed by probabilistic dependence of quantum observable A and B. In our framework, it is meaningless to speak about entanglement without pointing to the fixed observables A and B, so this is AB-entanglement. Dependence of quantum observables is formalized as non-coincidence of conditional probabilities. Starting with this probabilistic definition, we achieve the Hilbert space characterization of the AB-entangled states as amplitude non-factorisable states. In the tensor product case, AB-entanglement implies standard entanglement, but not vise verse. AB-entanglement for dichotomous observables is equivalent to their correlation, i.e., langle ABrangle _{psi} not = langle Arangle _{psi} langle Brangle _{psi} . We describe the class of quantum states that are A_{u} B_{u}-entangled for a family of unitary operators (u). Finally, observables entanglement is compared with dependence of random variables in classical probability theory.

On the Structure of Quantum Markov Chains on Cayley Trees Associated with Open Quantum Random Walks

Quantum Markov chains (QMCs) and open quantum random walks (OQRWs) represent different quantum extensions of the classical Markov chain framework. QMCs stand as a more profound layer within the realm of Markovian dynamics. The exploration of both QMCs and OQRWs has been a predominant focus over the past decade. Recently, a significant connection between QMCs and OQRWs has been forged, yielding valuable applications. This bridge is particularly impactful when studying QMCs on tree structures, where it intersects with the realm of phase transitions in models naturally arising from quantum statistical mechanics. Furthermore, it aids in elucidating statistical properties, such as recurrence and clustering. The objective of this paper centers around delving into the intricate structure of QMCs on Cayley trees in relation to OQRWs. The foundational elements of this class of QMCs are built upon using classical probability measures that encompass the hierarchical nature of Cayley trees. This exploration unveils the pivotal role that the dynamics of OQRWs play in shaping the behavior of the Markov chains under consideration. Moreover, the analysis extends to their classical counterparts. The findings are further underscored by the examination of notable examples, contributing to a comprehensive understanding of the outcomes.

Classicality and Bell’s theorem

A widespread view among physicists is that Bell’s theorem rests on an implicit assumption of “classicality,” in addition to locality. According to this understanding, the violation of Bell’s inequalities poses no challenge to locality, but simply reinforces the fact that quantum mechanics is not classical. The paper provides a critical analysis of this view. First we characterize the notion of classicality in probabilistic terms. We argue that classicality thus construed has nothing to do with the validity of classical physics, nor of classical probability theory, contrary to what many believe. At the same time, we show that the probabilistic notion of classicality is not an additional premise of Bell’s theorem, but a mathematical corollary of locality in conjunction with the standard auxiliary assumptions of Bell. Accordingly, any theory that claims to get around the derivation of Bell’s inequalities by giving up classicality, in fact has to give up one of those standard assumptions. As an illustration of this, we look at two recent interpretations of quantum mechanics, Reinhard Werner’s operational quantum mechanics and Robert Griffiths’ consistent histories approach, that are claimed to be local and non-classical, and identify which of the standard assumptions of Bell’s theorem each of them is forced to give up. We claim that while in operational quantum mechanics the Common Cause Principle is violated, the consistent histories approach is conspiratorial. Finally, we relate these two options to the idea of realism, a notion that is also often identified as an implicit assumption of Bell’s theorem.

Phase space perspective on a model for isomerization in an optical cavity.

Explanation for the modification of rates and mechanism of reactions carried out in optical cavities still eludes us. Several studies indicate that the cavity-mediated changes in the nature of vibrational energy flow within a molecule may play a significant role. Here, we study a model polaritonic system, proposed and analyzed earlier by Fischer et al., J. Chem. Phys. 156, 154305 (2022), comprising a one-dimensional isomerization mode coupled to a single photon mode in a lossless cavity. We show that the isomerization probability in the presence of virtual photons, for specific cavity-system coupling strengths and cavity frequencies, can exhibit suppression or enhancement for different choices of the initial reactant vibropolariton wavepacket. We observe a qualitative agreement between the classical and quantum average isomerization probabilities in the virtual photon case. A significant part of the effects due to coupling to the cavity can be rationalized in terms of a "chaos-order-chaos" transition of the classical phase space and the phase space localization nature of the polariton states that dominantly participate in the quantum isomerization dynamics. On the other hand, for initial states with zero photons (i.e., a "dark cavity"), the isomerization probability is suppressed when the cavity frequency is tuned near to the fundamental frequency of the reactive mode. The classical-quantum correspondence in the zero photon case is unsatisfactory. In this simple model, we find that the suppression or enhancement of isomerization arises due to the interplay between cavity-system energy flow dynamics and quantum tunneling.

Stochastic Quantum Probability and Subatomic Particle Experiment Design

This article is the third of a trilogy of articles on the nature of probability in quantum mechanics. The first article [1] began by noting that superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into the position probability density of an atomic particle, which happens nowhere else in probability theory. It went on to also note that there is an unexplained coincidence in quantum mechanics in that the interference term in the squared amplitude of superposed wavefunctions gives the squared amplitude the form of a variance of a sum of correlated random variables and went on to examine whether there could be an archetypical variable, in the Platonic sense of true form, behind quantum probability that would reconcile quantum probability with classic probability. This examination found such a variable, which can encompass both local and nonlocal quantum events. The second article [2] provided evidence that quantum probability has such a stochastic nature. This evidence was based on the number of electrons that need to be sent through a two-slit interferometer to gain a clear pattern of self-interference, which when compared with the number that would be expected to be sufficient in order for the position probability distribution of the self-interference wavefunction to take clear shape suggests that there is more variability present than that described by the formulation of quantum mechanics, which implies the presence of an underlying and as yet unrecognized physical process. This final article completes the trilogy by considering how a key aspect of experimental design would be affected by the increased variability that would be present if quantum probability is itself stochastic in the manner suggested in the previous two articles.

A tomographic approach to the sum uncertainty relation and quantum entanglement in continuous variable systems

Entropic uncertainty relations (EURs) have been examined in various contexts, primarily in qubit systems, including their links with entanglement, as they subsume the Heisenberg uncertainty principle. With their genesis in the Shannon entropy, EURs find applications in quantum information and quantum optics. EURs are state-dependent, and the state has to be reconstructed from tomograms (which are histograms readily available from experiments). This is a challenge when the Hilbert space is large, as in continuous variable (CV) systems and certain hybrid quantum (HQ) systems. A viable alternative approach therefore is to extract as much information as possible about the unknown quantum state directly from appropriate tomograms. Many variants of EURs can be extracted from tomograms, even for CV systems. In earlier work we have defined many tomographic entanglement indicators (TEIs) that can be readily calculated from tomograms without knowledge of the density matrix, and have reported on their efficacy as entanglement indicators in various contexts in both CV and HQ systems. The specific objectives of the present work are as follows: (i) To use the tomographic approach to investigate the links between EURs and TEIs in CV and HQ systems as they evolve in time. (ii) To identify the TEI that most closely tracks the temporal evolution of EURs. We consider two generic systems. The first is a multilevel atom modeled as a nonlinear oscillator interacting with a quantized radiation field. The second is the Λ-atom interacting with two radiation fields. The former model accomodates investigations on the role of the initial state of the field and the ratio of the strengths of interaction and nonlinearity in the connection between TEIs and EURs. The second model opens up the possibility of examining the connection between mixed state bipartite entanglement and EURs, when the number of atomic levels is finite. Since the tomogram respects the requirements of classical probability theory, this effort also sheds light on the extent to which TEIs reflect the temporal behaviour of those EURs which are rooted in the Shannon entropy.

Complete convergence theorems for moving average process generated by independent random variables under sub-linear expectations

The research of convergence properties of moving average process is a challenging field of limit theorems. The aim of this article is to provide a method to prove the complete convergence and complete integral convergence of moving average process for independent random variables in sub-linear expectation space. The results obtained in the article are the extensions of some complete convergence theorems under classical probability space.

MAD (about median) vs. quantile-based alternatives for classical standard deviation, skewness, and kurtosis

In classical probability and statistics, one computes many measures of interest from mean and standard deviation. However, mean, and especially standard deviation, are overly sensitive to outliers. One way to address this sensitivity is by considering alternative metrics for deviation, skewness, and kurtosis using mean absolute deviations from the median (MAD). We show that the proposed measures can be computed in terms of the sub-means of the appropriate left and right sub-ranges. They can be interpreted in terms of average distances of values of these sub-ranges from their respective medians. We emphasize that these measures utilize only the first-order moment within each sub-range and, in addition, are invariant to translation or scaling. The obtained formulas are similar to the quantile measures of deviation, skewness, and kurtosis but involve computing sub-means as opposed to quantiles. While the classical skewness can be unbounded, both the MAD-based and quantile skewness always lies in the range [−1, 1]. In addition, while both the classical kurtosis and quantile-based kurtosis can be unbounded, the proposed MAD-based alternative for kurtosis lies in the range [0, 1]. We present a detailed comparison of MAD-based, quantile-based, and classical metrics for the six well-known theoretical distributions considered. We illustrate the practical utility of MAD-based metrics by considering the theoretical properties of the Pareto distribution with high concentrations of density in the upper tail, as might apply to the analysis of wealth and income. In summary, the proposed MAD-based alternatives provide a universal scale to compare deviation, skewness, and kurtosis across different distributions.

Modeling mental peculiarities of a decision maker by a Fourier-holography technique

A task of modeling individual mental features of a decision-maker using a Fourier holography setup is considered. The problem is considered for a situation when current conditions of decision-making contradict to the previously learned rule of decision-making logic modeled by the non-cooperative game "Prisoner's Dilemma". The approach to the problem is based on a hypothesis of the correlation between mental features and the properties of the neural network as a material carrier of intelligence. The 6f Fourier holography scheme of the resonant architecture is considered as a three-layer neural network implementing a neuro-physiologically motivated concept of the "excitation ring" proposed by A.M. Ivanitsky. We analytically assess the dependence of the validity limits of the classical total probability formula for a disjunction of incompatible events on the characteristics of low-frequency filters in holograms and the correlation radii of the training image of the basic decision rule. Analytical results are confirmed by results of the numerical simulation.

A Variation of the Algorithm to Achieve the Maximum Entropy for Belief Functions.

Evidence theory (TE), based on imprecise probabilities, is often more appropriate than the classical theory of probability (PT) to apply in situations with inaccurate or incomplete information. The quantification of the information that a piece of evidence involves is a key issue in TE. Shannon's entropy is an excellent measure in the PT for such purposes, being easy to calculate and fulfilling a wide set of properties that make it axiomatically the best one in PT. In TE, a similar role is played by the maximum of entropy (ME), verifying a similar set of properties. The ME is the unique measure in TE that has such axiomatic behavior. The problem of the ME in TE is its complex computational calculus, which makes its use problematic in some situations. There exists only one algorithm for the calculus of the ME in TE with a high computational cost, and this problem has been the principal drawback found with this measure. In this work, a variation of the original algorithm is presented. It is shown that with this modification, a reduction in the necessary steps to attain the ME can be obtained because, in each step, the power set of possibilities is reduced with respect to the original algorithm, which is the key point of the complexity found. This solution can provide greater applicability of this measure.

Charge transport, information scrambling and quantum operator-coherence in a many-body system with U(1) symmetry

In this work, we derive an exact hydrodynamical description for the coupled, charge and operator dynamics, in a quantum many-body system with U(1) symmetry. Using an emergent symmetry in the complex Brownian SYK model with charge conservation, we map the operator dynamics in the model to the imaginary-time dynamics of an SU(4) spin-chain. We utilize the emergent SU(4) description to demonstrate that the U(1) symmetry causes quantum-coherence to persist even after disorder-averaging, in sharp contrast to models without symmetries. In line with this property, we write down a ‘restricted’ Fokker-Planck equation for the out-of-time ordered correlator (OTOC) in the large-N limit, which permits a classical probability description strictly in the incoherent sector of the global operator-space. We then exploit this feature to describe the OTOC in terms of a Fisher-Kolmogorov-Petrovsky-Piskun (FKPP)-equation which couples the operator with the charge and is valid at all time-scales and for arbitrary charge-density profiles. The coupled equations obtained belong to a class of models also used to describe the population dynamics of bacteria embedded in a diffusive media. We simulate them to explore operator-dynamics in a background of non-uniform charge configuration, which reveals that the charge transport can strongly affect dynamics of operators, including those that have no overlap with the charge.

The joy of mathematical discovery: To the 120th anniversary of Academician A.N. Kolmogorov

This article is dedicated to the memory of the outstanding mathematician of the 20th century, Andrei Nikolaevich Kolmogorov, whose 120th birthday is celebrated this year. The author describes in detail the formation of Kolmogorov as a scientist, the significance of Moscow State University in the scientific life and pedagogical activity of the famous mathematician. Kolmogorov’s contribution to such branches of mathematics as classical analysis, topology, geometry, approximation theory, functional analysis, and probability theory is analyzed, and the contribution of his scientific school to Russian science, including not only mathematics and physics but also the humanities, is shown.

Capacity inequalities and strong laws for m-widely acceptable random variables under sub-linear expectations

In this work, a generalised concept of dependent random variables, i.e., m-widely acceptable random variables, is put forward and some inequalities such as exponential inequality and Rosenthal-type inequality are established under sub-linear expectations. By virtue of these inequalities, we obtain a general result on complete convergence and the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of m-widely acceptable random variables, which extend and improve some existing ones in classical probability space as well as in sub-linear expectation space. As applications of the main results, the complete consistency and strong consistency of weighted estimators in nonparametric regression model under sub-linear expectations are also investigated.

Quantum Circuit Components for Cognitive Decision-Making

This paper demonstrates that some non-classical models of human decision-making can be run successfully as circuits on quantum computers. Since the 1960s, many observed cognitive behaviors have been shown to violate rules based on classical probability and set theory. For example, the order in which questions are posed in a survey affects whether participants answer 'yes' or 'no', so the population that answers 'yes' to both questions cannot be modeled as the intersection of two fixed sets. It can, however, be modeled as a sequence of projections carried out in different orders. This and other examples have been described successfully using quantum probability, which relies on comparing angles between subspaces rather than volumes between subsets. Now in the early 2020s, quantum computers have reached the point where some of these quantum cognitive models can be implemented and investigated on quantum hardware, by representing the mental states in qubit registers, and the cognitive operations and decisions using different gates and measurements. This paper develops such quantum circuit representations for quantum cognitive models, focusing particularly on modeling order effects and decision-making under uncertainty. The claim is not that the human brain uses qubits and quantum circuits explicitly (just like the use of Boolean set theory does not require the brain to be using classical bits), but that the mathematics shared between quantum cognition and quantum computing motivates the exploration of quantum computers for cognition modeling. Key quantum properties include superposition, entanglement, and collapse, as these mathematical elements provide a common language between cognitive models, quantum hardware, and circuit implementations.