This paper addresses the central question of what a coherent concept of probability might look like that would do justice to both classical probability theory, axiomatized by Kolmogorov, and quantum theory. At a time when quanta are receiving increased and expanded attention -- think, for example, of the advances in quantum computers or the promises associated with this new technology (National Academies of Sciences: Engineering, and Medicine, 2019) -- an adequate interpretation of probability, which is no less important, should be given due attention, particularly with regard to quantum theory.
This paper addresses the central question of what a coherent concept of probability might look like that would do justice to both classical probability theory, axiomatized by Kolmogorov, and quantum theory. At a time when quanta are receiving increased and expanded attention -- think, for example, of the advances in quantum computers or the promises associated with this new technology (National Academies of Sciences: Engineering, and Medicine, 2019) -- an adequate interpretation of probability, which is no less important, should be given due attention, particularly with regard to quantum theory.
Abstract By formulating the axioms of quantum mechanics, von Neumann also laid the foundations of a “quantum probability theory”. As such, it is regarded a generalization of the “classical probability theory” due to Kolmogorov. Outside of quantum physics, however, Kolmogorov’s axioms enjoy universal applicability. This raises the question of whether quantum physics indeed requires such a generalization of our conception of probability or if von Neumann’s axiomatization of quantum mechanics was contingent on the absence of a general theory of probability in the 1920s. This work argues in favor of the latter position. In particular, it shows how to construct a mathematically rigorous theory for non-relativistic N -body quantum systems subject to a time-independent scalar potential, which is based on Kolmogorov’s axioms and physically natural random variables. Though this theory is provably distinct from its quantum-mechanical analog, it nonetheless reproduces central predictions of the latter. Further work may make an empirical comparison possible. Moreover, the approach can in principle be adapted to other classes of quantum-mechanical models. Part II of this series discusses the empirical violation of Bell inequalities in the context of this approach. Part III addresses the projection postulate and the question of measurement.
Classical probability theory, as axiomatized in 1933 by Andrey Kolmogorov, has provided a useful and almost universally accepted theory for describing and quantifying uncertainty in scientific applications outside quantum mechanics. Recently, cognitive psychologists and mathematical economists have provided examples where classical probability theory appears inadequate but the probability theory underlying quantum mechanics appears effective. Formally, quantum probability theory is a generalization of classical probability. This article explores relationships between generalized probability theories, in particular quantum-like probability theories and those that do not have full complementation operators (e.g. event spaces based on intuitionistic logic), and discusses how these generalizations bear on important issues in the foundations of probability and the development of non-classical probability theories for the behavioural sciences.
In this chapter, we consider the way quantum theory represents deterministic, statistical, and probabilistic aspects of Nature. After a discussion in Section 3.1 of uncertainty in general, we discuss in Sections 3.2-3.4 several formal notions of classical probability, their relation to the probability concept used in applied statistics, and their dependence on the description used. Section 3.5 briefly discusses various examples, where probabilistic features emerge from fully deterministic situations-a theme to be taken up in much more detail in Chapter 11. Section 3.6 then shows how the statistical aspects of the quantum formalism naturally follow from the weak law of large numbers. The notions of c-probability in the classical case and of q-probability in the quantum case are formally defined in Section 3.7. Historically, the concept of classical probability (including its use in stochastic processes) was given an undisputed formal mathematical foundation in 1933 in terms of the measure-theoretic setting of Kolmogorov [164]. Apart from this traditional axiomatic foundation of classical probability theory, there is a less well-known equivalent axiomatic treatment by Whittle [302] in terms of expectations. Here probabilities appear as the expectations of statements, {0, 1}-valued random variables. von Plato [240] discusses the history of the concept of probability. Krüger et al. [169] discuss the history of probability in the various fields of application. Sklar [275] discusses the philosophical problems of the probability concept, with an emphasis on statistical mechanics.
In this work, we focus on the philosophical aspects and technical challenges that underlie the axiomatization of the non-Kolmogorovian probability framework, in connection with the problem of quantum contextuality. This fundamental feature of quantum theory has received a lot of attention recently, given that it might be connected to the speed-up of quantum computers-a phenomenon that is not fully understood. Although this problem has been extensively studied in the physics community, there are still many philosophical questions that should be properly formulated. We analyzed different problems from a conceptual standpoint using the non-Kolmogorovian probability approach as a technical tool.
The topic of probabilty in quantum mechanics is rather vast, and in this article, we shall choose to discuss it from the perspective of whether and in what sense quantum mechanics requires a generalisation of the usual (Kolmogorovian) concept of probability. We shall focus on the case of finite-dimensional quantum mechanics (which is analogous to that of discrete probability spaces), partly for simplicity and partly for ease of generalisation. While we shall largely focus on formal aspects of quantum probability (in particular the non-existence of joint distributions for incompatible observables), our discussion will relate also to notorious issues in the interpretation of quantum mechanics. Indeed, whether quantum probability can or cannot be ultimately reduced to classical probability connects rather nicely to the question of 'hidden variables' in quantum mechanics. An abridged version of this essay will appear in A. Hajek and C. Hitchcock (eds.), 'The Oxford Handbook of Probability and Philosophy', Oxford University Press. The present version includes in particular many more footnotes and references and the Appendix with the proofs of the Lemma and Proposition of Section 5.
Computational Thinking (CT) frameworks have expanded over the past two decades to address technological advances and sociotechnical concerns, yet they remain grounded in classical, deterministic computational logic.Quantum information science introduces fundamentally different principles for representing and processing information, requiring learners to engage with non-intuitive, probabilistic, and relational forms of reasoning that depart from classical computational models.Drawing on frameworks by Brennan and Resnick and Kafai et al., we explore what Quantum Computational Thinking (QCT) might entail and open a dialogue about how the learning sciences community might prepare learners for a quantummediated world, with attention to access, democratic participation, and the challenges of developing new intuitions. Computational thinkingNearly two decades ago, Wing (2006) introduced Computational Thinking (CT) as "a universally applicable attitude and skill set everyone, not just computer scientists, would be eager to learn and use" (p.33).Early work by Papert (1980/2020) framed CT primarily in terms of procedural thinking and programming.Subsequent scholars broadened its scope: Grover and Pea (2013) identified transferable competencies such as abstraction, decomposition, and debugging; Brennan and Resnick (2012) reframed CT as the interplay of concepts, practices, and perspectives, shifting attention toward epistemic and identity dimensions; and Kafai (2016) proposed computational participation to foreground the social, cultural, and ethical aspects of computing.Yet across these perspectives, CT remains grounded in classical, deterministic, binary logic.When these frameworks were established, quantum technologies were largely theoretical (Nielsen & Chuang, 2010).Today, quantum information science (QIS) is rapidly transitioning from theory to application, promising transformative impacts.The question is no longer whether quantum technologies will redefine the computational landscape, but how soon and for whom.This shift calls for re-examining what it means to think computationally.Quantum Computational Thinking (QCT) does not replace classical CT, but extends it to address a distinct class of problems that quantum computation is uniquely suited to tackle.In the next section, we explore what QCT entails and how quantum concepts, practices, and perspectives can deepen and extend CT for the emerging quantum era.