Comparing large-scale graphs based on quantum probability theory

In the light of Quantum Physics, probabilistic technique is said to support a backbone in the domain of Quantum Information Retrieval (QIR). This well known domain is based on the Quantum Probability Theory, which is used to examine the probability of relevance of a document given a user's query. The Quantum Probability Ranking Principle (QPRP), captures dependencies between documents in absolute way via “quantum interference” by considering a single vector i.e., one-dimension subspace. The HAL semantic space can be used to capture meaning of the document of information needs by representing the document into a high dimensional semantic space. This paper presents an investigation of the Quantum Probability Ranking Principle with the concept of HAL semantic space such that which can act as a bridge the gap between Quantum Probability Theory and HAL semantic space, that can results to agreeable performance in an ad-hoc retrieval task of document ranking.

Non-Commutativity, Incompatibility, Emergent Behavior and Decision Support Systems

The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933. Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well. To precisely this end, von Neumann initiated the study of what are now called von Neumann algebras and, with Murray, made a first classification of such algebras into three types. The nonrelativistic quantum theory of systems with finitely many degrees of freedom deals exclusively with type I algebras. However, for the description of further quantum systems, the other types of von Neumann algebras are indispensable. The paper reviews quantum probability theory in terms of general von Neumann algebras, stressing the similarity of the conceptual structure of classical and noncommutative probability theories and emphasizing the correspondence between the classical and quantum concepts, though also indicating the nonclassical nature of quantum probabilistic predictions. In addition, differences between the probability theories in the type I, II and III settings are explained. A brief description is given of quantum systems for which probability theory based on type I algebras is known to be insufficient. These illustrate the physical significance of the previously mentioned differences.

The main thesis of the present paper is that: Quantum Probability is not a generalization of classical probability, but it is a deeper level of it. Classical random variables have an intrinsic (microscopic) non–commutative structure that generalize usual quantum theory. The study of this generalization is the core of the non–linear quantization program.

As was recently shown by the authors, quantum probability theory can be used for the modelling of the process of decision-making (e.g. probabilistic risk analysis) for macroscopic geophysical structures such as hydrocarbon reservoirs. This approach can be considered as a geophysical realization of Hilbert's programme on axiomatization of statistical models in physics (the famous sixth Hilbert problem). In this conceptual paper, we continue development of this approach to decision-making under uncertainty which is generated by complexity, variability, heterogeneity, anisotropy, as well as the restrictions to accessibility of subsurface structures. The belief state of a geological expert about the potential of exploring a hydrocarbon reservoir is continuously updated by outputs of measurements, and selection of mathematical models and scales of numerical simulation. These outputs can be treated as signals from the information environment E The dynamics of the belief state can be modelled with the aid of the theory of open quantum systems: a quantum state (representing uncertainty in beliefs) is dynamically modified through coupling with E; stabilization to a steady state determines a decision strategy. In this paper, the process of decision-making about hydrocarbon reservoirs (e.g. 'explore or not?'; 'open new well or not?'; 'contaminated by water or not?'; 'double or triple porosity medium?') is modelled by using the Gorini-Kossakowski-Sudarshan-Lindblad equation. In our model, this equation describes the evolution of experts' predictions about a geophysical structure. We proceed with the information approach to quantum theory and the subjective interpretation of quantum probabilities (due to quantum Bayesianism).This article is part of the theme issue 'Hilbert's sixth problem'.

The Ellsberg paradox: A challenge to quantum decision theory?

How quickly can we compute the nonnegative rank (r) of an m x n matrix? This problem ---- and the companion problem of finding a nonnegative matrix factorization with minimum inner-dimension ---- has a rich history, with applications in quantum mechanics, probability theory, data analysis, communication complexity and polyhedral combinatorics. Here we will survey the recent progress on this question, that has essentially resolved its worst-case complexity.

Cromwell’s rule (also known as the zero priors paradox) refers to the constraint of classical probability theory that if one assigns a prior probability of 0 or 1 to a hypothesis, then the posterior has to be 0 or 1 as well (this is a straightforward implication of how Bayes’ rule works). Relatedly, hypotheses with a very low prior cannot be updated to have a very high posterior without a tremendous amount of new evidence to support them (or to make other possibilities highly improbable). Cromwell’s rule appears at odds with our intuition of how humans update probabilities. In this work, we report two simple decision making experiments, which seem to be inconsistent with Cromwell’s rule. Quantum probability theory, the rules for how to assign probabilities from the mathematical formalism of quantum mechanics, provides an alternative framework for probabilistic inference. An advantage of quantum probability theory is that it is not subject to Cromwell’s rule and it can accommodate changes from zero or very small priors to significant posteriors. We outline a model of decision making, based on quantum theory, which can accommodate the changes from priors to posteriors, observed in our experiments.

The classical entropies were formulated by means of the classical probability theory. The essence of the classical probability theory corresponds to the use of c‐number with commutative property. On the base of the classical probability theory, the classical information theory can be discussed by using the classical entropies. On the other hand, the principle of the quantum phenomenon seems to be having non‐commutative behavior. It leads to the use of the q‐operator with non‐commutative property to introduce the quantum entropies based on the quantum probability theory. In order to discuss the efficiency of the information communication processes, a measure of complexity of initial state itself and a measure of transmitted complexity through communication channels are necessary. Shannon's entropy and the mutual entropy represent these measures of complexities in classical communication system, respectively. Von Neumann entropy and Ohya mutual entropy relate to these measures of complexities in quantum communication system, respectively. Recently, several mutual entropy type measures (Lindblad—Nielsen entropy and Coherent entropy) were proposed by using an entropy exchange with respect to a channel and an initial state. In this paper, we will show the most suitable one for discuss the efficiency of information transmission for quantum processes. In classical information theory, one of the most important tools to analyze communication processes are the classical coding theorems, which was investigated through the classical dynamical entropy. The classical dynamical entropy was discussed in [1] with respect to a measure preserving transformation defined on a message space for finite partitions of the measure space. In quantum information theory, Emch [2] and Conne, Stormer [3] first studied a complexity of quantum dynamical processes. After that, Ohya introduced a S‐ mixing entropy in general quantum systems, which was natural extension of the von Neumann entropy. Using the S—mixing entropy, Ohya defined a mean entropy and a mean mutual entropy for the quantum dynamical systems. Conne, Narnhoffer and Thirring introduced a dynamical entropy (CNT entropy) [4] and several researchers discussed the CNT entropy [5,6,7,8]. Alicki and Fannes defined a dynamical entropy (AF entropy) [9] using a finite operational partitions of unity. In 1995, Voiculescu proposed a dynamical entropy (dynamical approximation entropy) [10] in C* and W* algebras. Through the quantum Markov process, Accardi, Ohya and Watanabe defined a dynamical entropy (AOW entropy) [11,12] in 1997. Base on transition expectations and liftings in the sense of Accardi and Ohya [13], Kossakowski, Ohya and Watanabe introduced a dynamical entropy (KOW entropy) [14] with respect to completely positive maps in 1999. In this paper, we will discuss the complexity of the quantum dynamical process to calculate the generalized AOW entropy given by KOW entropy.

The method of the quantum probability theory only requires simple structural data of graph and allows us to avoid a heavy combinational argument often necessary to obtain full description of spectrum of the adjacency matrix. In the present paper, by using the idea of calculation of the probability amplitudes for continuous-time quantum walk in terms of the quantum probability theory, we investigate quantum central limit theorem for continuous-time quantum walks on odd graphs.

Contextual decisions and beliefs and their impact upon market outcomes are at the core of research in behavioural finance. We describe some of the notable probabilistic fallacies that underpin investor behaviour and the consequent deviation of asset prices from the rational expectations equilibrium. In real financial markets, the complexity of financial products and the surrounding ambiguity calls for a more general formalization of agents belief formation than offered by the standard probability theory and dynamic models based on classical stochastic processes. The main advantage of quantum probability (QP) is that it can capture contextuality of beliefs through the notion of non-commuting prospect observables. QP has the potential to model myopia in asset return evaluation, as well as inter-asset valuation. Moreover, the interference term of the agents’ comparison state can provide a quantitative description of their vacillating ambiguity perception characterized by non-additive beliefs of agents. Some of the implications of non-classicality in beliefs for the composite market outcomes can also be modelled with the aid of QP. As a final step we also discuss the contributions of the growing body of psychological studies that reveal a true (quantum type) contextuality in human preference statistics showing that the classical probability theory is too restrictive to capture the very strong non-classical correlations between preference outcomes and beliefs.

Principal concepts of quantum probability theory are considered. Some properties of a priori and a posteriori quantum probabilities are studied. Concepts of regular and nondestructive measurement procedures are analyzed. Effects associated with compatibility and lack of correlation of quantum procedures are studied. Markov sequences of quantum procedures for quantum systems with finite or countable number of states is defined. The computation process on the n-qubit quantum computer is showed to represent operations that obey laws of quantum probability theory.

Equipped with quantum probability theory, quantum language models (QLMs) aimed at a principled approach to modeling term dependency have drawn increasing attention. However, even though they are theoretically more general and have effective performance, current QLMs do not take context information into account. The most important element, namely density matrix, is constructed as a summation of word projectors, whose representation is independent of context information. To address this problem, we propose a Context based Quantum Language Model (C-QLM). Between word embedding and sentence density matrix, a bidirectional long short term memory network is adopted to learn the hidden context information. Then a set of vectors is utilized to extract density matrices’s features for question and answer sentences which can be used to calculate the matching score. Experiment results on TREC-QA and WIKI-QA datasets demonstrate the effectiveness of our proposed model.

7 - Embarking on theoretical studies for off-shell science

Multi-task learning has demonstrated remarkable success across a broad spectrum of natural language processing tasks, particularly with neural network-based methods. Despite these advances, a fundamental gap remains in explaining the relationship between task-relatedness and model effectiveness. To address this issue, we propose a novel approach for implicitly modeling task-relatedness by leveraging a quantum physical mathematical framework. In this paper, we introduce a complex-valued neural network designed to encapsulate and analyze task-relatedness. Within this framework, sentences originating from diverse tasks are encoded as mixed quantum systems, represented on a meticulously defined Semantic Hilbert Space. This allows the network to interpret inter-task relationships through the explicit physical semantics of well-constrained components grounded in quantum probability theory. By adhering to these rigorous principles, our model not only establishes a robust method for quantifying task-relatedness but also fosters a deeper, self-explanatory understanding of the underlying processes. To validate the efficacy of our approach, we conducted extensive experiments across five benchmark text classification tasks. The results demonstrate both the superior performance and the interpretability of the proposed model, highlighting its potential as a self-explanatory system for multi-task learning in NLP.