Quantum probability updating from zero priors (by-passing Cromwell’s rule)

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.

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

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.

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.

Quantum probability

Comparing large-scale graphs based on quantum probability theory

An operator-valued T1 theory for symmetric CZOs

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.

The Ellsberg paradox: A challenge to quantum decision theory?

Recent experimental findings clearly suggest that classical probability theory is still not successful to explain modalities in human cognition, especially in connection to decision making. The major problem seems to be the presence of epistemic uncertainty and its effects on cognition at any time point. Moreover, the stochasticity in the model arises due to the unknown path or trajectory (the definite state of mind at each point in time) a person follows. In fact, there exists more ambiguity than clarity in some aspects of cognition, for example, regarding cognitive affective decision-making behavior and subsequent choices (George et al. in Academy of Management Review 31: 347–365, 2006). The concept of ambiguity, its typical role and its function are not only long-standing debatable concerns in the humanities but also assume a similar level of importance, and are the subject of much debate, in contemporary neurological and cognitive approaches. For example, biologist Fredrick Grinnell made an argument clarifying that ambiguity “is inherent in carrying out and reporting research” (Grinnell in Science and Engineering Ethics 5:205, 1999). Not only that, but he also pointed out some parameters of ambiguity that support his view, for instance, the difficulty of distinguishing between data and noise in research “at the edge of discovery” (ibid. 207). Interestingly, as a positive side of ambiguity, he observed “while ambiguity in method and procedure—as an aspect of data and its interpretation—is clearly a potential problem, posing the risk of distortion, it is also associated here with creative insight.” The consideration of “black box model of human mental functions” produces much ambiguity. So, as we are aware of the intertwined mathematical, neurological and cognitive mysteries of the brain, ultimately, these very characteristics are challenges faced by the human cognition and give rise to a very complicated and complex integrative system of cognitive computation and affective perception. A generalized version of probability theory, borrowing the idea from the quantum paradigm, may be a plausible approach. Quantum theory enables a person to be in an indefinite state (superposition state) in the context of neurobiology, especially in relation to central nervous system (CNS), and this allows all these states to be potentially (of course, with proper probability amplitudes) expressible at each moment. Thus, a superposition state seems to provide a better representation of the conflict, ambiguity or uncertainty that a person experiences at each moment. This situation can be described in somewhat poetic language, i.e., the dynamical laws, as demonstrated by nature, allow equations but are only sometimes posed by the Mind of the experimentalist. This way, consciousness (as expressed in his introduction to consciousness or cognition can be expressed as only a critical description of human activity and not as a theory of mind. Some critical remarks about handling the ambiguity and contradictions due to emotions within quantum probability framework are made in this chapter. Mental states and related decision making have been widely discussed in various ancient and medieval Indian traditions. The existence of an indefinite state of mind was discussed by many Indian philosophical schools, including Buddhism. The concept of the neutral mind, as well as equanimity, is discussed in this chapter, especially because it has a potential to be related to modern neuroscience, as well as to quantum theory.

There has been a growing trend to develop cognitive models based on the mathematics of quantum theory. A common theme in the motivation of such models has been findings which apparently challenge the applicability of classical formalisms, specifically ones based on classical probability theory. Classical probability theory has had a singularly important place in cognitive theory, because of its (in general) descriptive success but, more importantly, because in decision situations with low, equivalent stakes it offers a multiply justified normative standard. Quantum cognitive models have had a degree of descriptive success and proponents of such models have argued that they reveal new intuitions or insights regarding decisions in uncertain situations. However, can quantum cognitive models further benefit from normative justifications analogous to those for classical probability models? If the answer is yes, how can we determine the rational status of a decision, which may be consistent with quantum theory, but inconsistent with classical probability theory? In this paper, we review the proposal from Pothos, Busemeyer, Shiffrin, and Yearsley (2017), that quantum decision models benefit from normative justification based on the Dutch Book Theorem, in exactly the same way as models based on classical probability theory.

In this chapter we present a brief introductions to Kolmogorov’s measure-theoretic and von Mises’ frequency probability theories. These are classical probability theories. A rather common opinion is that classical probability can not describe the probabilistic structure of quantum mechanics. Such an opinion is based on belief that classical probability can not reproduce the main distinguishing probabilistic features of quantum formalism, e.g., interference of probabilities and Born’s rule. It is also believed that the violation of Bell’s inequality can not be explained in the framework of classical probability theory. The aim of this book is to show that such a viewpoint to the relation between classical and quantum probabilities is not justified. We shall see that the mathematical formalisms of Kolmogorov’s measure-theoretic [Kolmogoroff, Grundbegriffe der Wahrscheinlichkeitsrechnung, 1933; Shiryaev in Ann. Probab. 17:866–944, 1989; Shiryaev, Probability, 1984] and Mises’ frequency [Von Mises in Math. Z. 5:52–99, 1919; Von Mises, Probability, Statistics and Truth, 1957; Von Mises, The Mathematical Theory of Probability and Statistics, 1964] models can be represented in the QL way. However, these models should be interpreted in the contextual framework.

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.

The origin of black hole entropy and the black hole information problem pro- vide important clues for trying to piece together a quantum theory of gravity. Thus far, discussions on this topic have mostly assumed that in a consistent theory of gravity and quantum mechanics, quantum theory will be unmodied. Here, we examine the black hole information problem in the context of generalisations of quantum theory. In particular, we examine black holes in the setting of generalised probabilistic theories, in which quantum theory and classical probability theory are special cases. We compute the time it takes information to escape a black hole, assuming that information is preserved. We nd that under some very general assumptions, the arguments of Page (that information should escape the black hole after half the Hawking photons have been emitted), and the black- hole mirror result of Hayden and Preskill (that information can escape quickly) need to be modied. The modication is determined entirely by what we call the Wootters-Hardy pa- rameter associated with a theory. We nd that although the information leaves the black hole after enough photons have been emitted, it is fairly generic that it fails to appear outside the black hole at this point | something impossible in quantum theory due to the no-hiding theorem. The information is neither inside the black hole, nor outside it, but is delocalised. Our central technical result is an information decoupling theorem which holds in the generalised probabilistic framework.

First, we discuss basic probability notions from the viewpoint of category theory. Our approach is based on the following four “sine quibus non” conditions: 1. (elementary) category theory is efficient (and suffices); 2. random variables, observables, probability measures, and states are morphisms; 3. classical probability theory and fuzzy probability theory in the sense of S. Gudder and S. Bugajski are special cases of a more general model; 4. a good model allows natural modifications. Second, we show that the category ID of D-posets of fuzzy sets and sequentially continuous D-homomorphisms allows to characterize the passage from classical to fuzzy events as the minimal generalization having nontrivial quantum character: a degenerated state can be transported to a nondegenerated one. Third, we describe a general model of probability theory based on the category ID so that the classical and fuzzy probability theories become special cases and the model allows natural modifications. Finally, we present a modification in which the closed unit interval [0,1] as the domain of traditional states is replaced by a suitable simplex.