On the strangeness of quantum probabilities (original) (raw)

The probabilistic roots of the quantum mechanical paradoxes

Indice 2 O. The goal of any mathematical investigation of the foundations of a physical theory is to clarify to what extent the mathematical formalism of that theory is uniquely determined by some clearly and explicitely stated physical assumptions. The achievement of that goal is particularly relevant in the case of the quantum theory where the novelty of the formalism, its being far away from any immediate intuition, the substantial failure met, for many years, by any attempt to deduce the quantum formalism from plausible physical assumptions, intersected with the never solved problems concerning the interpretation of the theory. That with quantum theory a new kind of probability theory was involved, was clear since the very beginnings of quantum mechanics (cf. [28]), even if it was not so clear which of the axioms of classifical probability had to be substituted, which physically meaningful statement had to replace it, how and if a physically meaningful statement could justify the apparently strange quantum mechanical formalism. The lack of clear answers to these questions had a tremendous impact on the process of interpretation and misinterpretation of quantum theory. The attempts to answer these questions motivated the development of a new branch of probability theory -quantum probability-and led to definite mathematical answers to these questions. In the present paper we want to discuss how these mathematical results allow to solve in a rather natural way some old problems concerning the interpretation of quantum theory and its mathematical foundations.

A detailed interpretation of probability, and its link with quantum mechanics

Eprint Arxiv 1011 6331, 2010

In the following we revisit the frequency interpretation of probability of Richard von Mises, in order to bring the essential implicit notions in focus. Following von Mises, we argue that probability can only be defined for events that can be repeated in similar conditions, and that exhibit 'frequency stabilization'. The central idea of the present article is that the mentioned 'conditions' should be well-defined and 'partitioned'. More precisely, we will divide probabilistic systems into object, environment, and probing subsystem, and show that such partitioning allows to solve a wide variety of classic paradoxes of probability theory. As a corollary, we arrive at the surprising conclusion that at least one central idea of the orthodox interpretation of quantum mechanics is a direct consequence of the meaning of probability. More precisely, the idea that the "observer influences the quantum system" is obvious if one realizes that quantum systems are probabilistic systems; it holds for all probabilistic systems, whether quantum or classical.

The Unreasonable Success of Quantum Probability II: Quantum Measurements as Universal Measurements

2014

In the first part of this two-part article, we have introduced and analyzed a multidimensional model, called the 'general tension-reduction' (GTR) model, able to describe general quantum-like measurements with an arbitrary number of outcomes, and we have used it as a general theoretical framework to study the most general possible condition of lack of knowledge in a measurement, so defining what we have called a 'universal measurement'. In this second part, we present the formal proof that universal measurements, which are averages over all possible forms of fluctuations, produce the same probabilities as measurements characterized by 'uniform' fluctuations on the measurement situation. Since quantum probabilities can be shown to arise from the presence of such uniform fluctuations, we have proven that they can be interpreted as the probabilities of a first-order non-classical theory, describing situations in which the experimenter lacks complete knowledge ab...

Quantum probability: New perspectives for the laws of chance

The main philosophical successes of quantum probability is the discovery that all the so-called quantum paradoxes have the same conceptual root and that such root is of probabilistic nature. This discovery marks the birth of quantum probability not as a purely mathematical (noncommutative) generalization of a classical theory, but as a conceptual turning point on the laws of chance, made necessary by experimental results.

Undecidability and the problem of outcomes in quantum measurements

2010

Abstract We argue that it is fundamentally impossible to recover information about quantum superpositions when a quantum system has interacted with a sufficiently large number of degrees of freedom of the environment. This is due to the fact that gravity imposes fundamental limitations on how accurate measurements can be. This leads to the notion of undecidability: there is no way to tell, due to fundamental limitations, if a quantum system evolved unitarily or suffered wavefunction collapse.

The unreasonable success of quantum probability I: Quantum measurements as uniform fluctuations

Journal of Mathematical Psychology, 2015

In the first part of this two-part article (Aerts & Sassoli de Bianchi, 2014), we have introduced and analyzed a multidimensional model, called the general tension-reduction (GTR) model, able to describe general quantum-like measurements with an arbitrary number of outcomes, and we have used it as a general theoretical framework to study the most general possible condition of lack of knowledge in a measurement, so defining what we have called a universal measurement. In this second part, we present the formal proof that universal measurements, which are averages over all possible forms of fluctuations, produce the same probabilities as measurements characterized by uniform fluctuations on the measurement situation. Since quantum probabilities can be shown to arise from the presence of such uniform fluctuations, we have proven that they can be interpreted as the probabilities of a first-order non-classical theory, describing situations in which the experimenter lacks complete knowledge about the nature of the interaction between the measuring apparatus and the entity under investigation. This same explanation can be applied-mutatis mutandis-to the case of cognitive measurements, made by human subjects on conceptual entities, or in decision processes, although it is not necessarily the case that the structure of the set of states would be in this case strictly Hilbertian. We also show that universal measurements correspond to maximally robust descriptions of indeterministic reproducible experiments, and since quantum measurements can also be shown to be maximally robust, this adds plausibility to their interpretation as universal measurements, and provides a further element of explanation for the great success of the quantum statistics in the description of a large class of phenomena.

Critique of “Elements of Quantum Probability”

Quantum Probability Communications, 2001

We analyse the thesis of that classical probability is unable to model the the stochastic nature of the Aspect experiment, in which violation of Bell's inequality was experimentally demonstrated. According to these authors the experiment shows the need to introduce the extension of classical probability known as Quantum Probability. We show that their argument depends on hidden assumptions and a highly restrictive view of the scope of classical probability. A careful probabilistic analysis shows, on the contrary, that it is classical deterministic physical thinking which cannot cope with the Aspect experiment and therefore needs revision. The ulterior aim of the paper is to help mathematical statisticians and probabilists to find their way into the fascinating world of quantum probability (thus: the same aim as that of Kümmerer and Maassen) by dismantling the bamboo curtain between ordinary and quantum probability which over the years has been built up as physicists and pure mathematicians have repeated to one another Feynman's famous dictum 'quantum probability is a different kind of probability'.