scholarly journals Probabilistic theories and reconstructions of quantum theory

2021 ◽  
Author(s):  
Markus Müller
Keyword(s):  
Probability Theory ◽  
Quantum Theory ◽  
Three Dimensional ◽  
Complex Numbers ◽  
Classical Probability ◽  
Quantum Bit ◽  
Classical Probability Theory ◽  
Generalized Probabilistic Theories ◽  
Usual Representation ◽  
Probabilistic Theories

These lecture notes provide a basic introduction to the framework of generalized probabilistic theories (GPTs) and a sketch of a reconstruction of quantum theory (QT) from simple operational principles. To build some intuition for how physics could be even more general than quantum, I present two conceivable phenomena beyond QT: superstrong nonlocality and higher-order interference. Then I introduce the framework of GPTs, generalizing both quantum and classical probability theory. Finally, I summarize a reconstruction of QT from the principles of Tomographic Locality, Continuous Reversibility, and the Subspace Axiom. In particular, I show why a quantum bit is described by a Bloch ball, why it is three-dimensional, and how one obtains the complex numbers and operators of the usual representation of QT.

Reasoning About Measurements

2021 ◽  
pp. 31-92
Author(s):  
Jochen Rau
Keyword(s):  
Probability Theory ◽  
Quantum Theory ◽  
Physical Theory ◽  
Logical Structure ◽  
Statistical Independence ◽  
Classical Probability ◽  
Multiple Systems ◽  
Classical Probability Theory ◽  
Operational Requirement

This chapter explains the approach of ‘operationalism’, which in a physical theory admits only concepts associated with concrete experimental procedures, and lays out its consequences for propositions about measurements, their logical structure, and states. It illustrates these with toy examples where the ability to perform measurements is limited by design. For systems composed of several constituents this chapter introduces the notions of composite and reduced states, statistical independence, and correlations. It examines what it means for multiple systems to be prepared identically, and how this is represented mathematically. The operational requirement that there must be procedures to measure and prepare a state is examined, and the ensuing constraints derived. It is argued that these constraint leave only one alternative to classical probability theory that is consistent, universal, and fully operational, namely, quantum theory.

QUANTUM DISCORD DERIVED FROM TSALLIS ENTROPY

2013 ◽  
Vol 11 (01) ◽  
pp. 1350013 ◽  
Author(s):  
JACEK JURKOWSKI
Keyword(s):  
Probability Theory ◽  
Quantum Theory ◽  
Quantum Discord ◽  
Tsallis Entropy ◽  
Classical Probability ◽  
Analytical Expression ◽  
Classical Probability Theory ◽  
Definition Of ◽  
Isotropic States

Due to some ambiguity in the definition of mutual Tsallis entropy in classical probability theory, its generalization to quantum theory is discussed and, as a consequence, two types of generalized quantum discords, called q-discords, are defined in terms of quantum Tsallis entropy. Both q-discords for two-qubit Werner and isotropic states are determined and compared and it is shown that one of them is non-negative, at least for states under investigation, for all q > 0. Finally, an analytical expression for q-discord of certain family of two-qubit X-states is presented. Using this example, we show that both types of q-discords can take negative values for some q > 1, hence their use as correlations measures is rather limited.

scholarly journals Compatibility for probabilistic theories

2016 ◽  
Vol 66 (2) ◽  
Author(s):  
Stan Gudder
Keyword(s):  
Probability Theory ◽  
Quantum Mechanics ◽  
Quantum Logic ◽  
Probabilistic Theory ◽  
Classical Probability ◽  
Classical Probability Theory ◽  
Positive Vector ◽  
Vector Valued ◽  
Pt Quantum Mechanics ◽  
Probabilistic Theories

AbstractWe define an index of compatibility for a probabilistic theory (PT). Quantum mechanics with index 0 and classical probability theory with index 1 are at the two extremes. In this way, quantum mechanics is at least as incompatible as any PT. We consider a PT called a concrete quantum logic that may have compatibility index strictly between 0 and 1, but we have not been able to show this yet. Finally, we show that observables in a PT can be represented by positive, vector-valued measures.

scholarly journals What Is Rational and Irrational in Human Decision Making

2021 ◽  
Vol 3 (1) ◽  
pp. 242-252
Author(s):  
Emmanuel M. Pothos ◽  
Oliver J. Waddup ◽  
Prince Kouassi ◽  
James M. Yearsley
Keyword(s):  
Probability Theory ◽  
Quantum Theory ◽  
Cognitive Theory ◽  
Cognitive Models ◽  
Probability Models ◽  
Classical Probability ◽  
Important Place ◽  
Classical Probability Theory ◽  
Human Decision ◽  
Normative Standard

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.

scholarly journals Upgrading Probability via Fractions of Events

2016 ◽  
Vol 24 (1) ◽  
pp. 29-41 ◽  
Author(s):  
Roman Frič ◽  
Martin Papčo
Keyword(s):  
Probability Theory ◽  
Random Variables ◽  
Random Variable ◽  
Classical Probability ◽  
Classical Probability Theory ◽  
Black And White ◽  
Random Events ◽  
Indicator Functions ◽  
Quantum Phenomena ◽  
Special Case

Abstract The influence of “Grundbegriffe” by A. N. Kolmogorov (published in 1933) on education in the area of probability and its impact on research in stochastics cannot be overestimated. We would like to point out three aspects of the classical probability theory “calling for” an upgrade: (i) classical random events are black-and-white (Boolean); (ii) classical random variables do not model quantum phenomena; (iii) basic maps (probability measures and observables { dual maps to random variables) have very different “mathematical nature”. Accordingly, we propose an upgraded probability theory based on Łukasiewicz operations (multivalued logic) on events, elementary category theory, and covering the classical probability theory as a special case. The upgrade can be compared to replacing calculations with integers by calculations with rational (and real) numbers. Namely, to avoid the three objections, we embed the classical (Boolean) random events (represented by the f0; 1g-valued indicator functions of sets) into upgraded random events (represented by measurable {0; 1}-valued functions), the minimal domain of probability containing “fractions” of classical random events, and we upgrade the notions of probability measure and random variable.

scholarly journals Algebras with two multiplications and their cumulants

2019 ◽  
Vol 52 (2) ◽  
pp. 157-186
Author(s):  
Adam Burchardt
Keyword(s):  
Probability Theory ◽  
Linear Space ◽  
Structure Constant ◽  
Classical Probability ◽  
Classical Probability Theory ◽  
Sum Of Products ◽  
Probabilistic Concept ◽  
Algebraic Formula

Abstract Cumulants are a notion that comes from the classical probability theory; they are an alternative to a notion of moments. We adapt the probabilistic concept of cumulants to the setup of a linear space equipped with two multiplication structures. We present an algebraic formula which involves those two multiplications as a sum of products of cumulants. In our approach, beside cumulants, we make use of standard combinatorial tools as forests and their colourings. We also show that the resulting statement can be understood as an analogue of Leonov–Shiryaev’s formula. This purely combinatorial presentation leads to some conclusions about structure constant of Jack characters.

Uncertainty about the value of quantum probability for cognitive modeling

2013 ◽  
Vol 36 (3) ◽  
pp. 279-280 ◽  
Author(s):  
Christina Behme
Keyword(s):  
Probability Theory ◽  
Cognitive Modeling ◽  
Quantum Probability ◽  
Convincing Evidence ◽  
Logical Possibility ◽  
Classical Probability ◽  
Classical Probability Theory ◽  
Human Rationality

AbstractI argue that the overly simplistic scenarios discussed by Pothos & Busemeyer (P&B) establish at best that quantum probability theory (QPT) is a logical possibility allowing distinct predictions from classical probability theory (CPT). The article fails, however, to provide convincing evidence for the proposal that QPT offers unique insights regarding cognition and the nature of human rationality.

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