On the relation between quantum mechanical probabilities and event frequencies (original) (raw)

Quantum Probabilities vs Event Frequencies

Brazilian Journal of Physics, 2005

Quantum probability is very different from classical probability. Part of this difference is manifested in the generic inability of stochastic processes to describe the results of multi-time measurements of quantum mechanical systems and the fact that the complex-valued temporal correlation functions of quantum theory have no interpretation in terms of multi-time measurements. By analysing experiments involving measurements at more than one moments of time, we conclude that this inequivalence must be manifested either as a failure of the quantum logic or as the inability to define probabilities in multi-time measurements because the relative frequencies do not converge. These alternatives can be empirically distinguished as they correspond to different behaviours of the statistical data in multi-time measurements.

Classical Vs Quantum Probability in Sequential Measurements

2005

We demonstrate in this paper that the probabilities for sequential measurements have features very different from those of single-time measurements. First, they cannot be modeled by a classical stochastic process. Second, they are contextual, namely they depend strongly on the specific measurement scheme through which they are determined. We construct Positive-Operator-Valued measures (POVM) that provide such probabilities. For observables with continuous spectrum, the constructed POVMs depend strongly on the resolution of the measurement device, a conclusion that persists even if we consider a quantum mechanical measurement device or the presence of an environment. We then examine the same issues in alternative interpretations of quantum theory. We first show that multi-time probabilities cannot be naturally defined in terms of a frequency operator. We next prove that local hidden variable theories cannot reproduce the predictions of quantum theory for sequential measurements, even when the degrees of freedom of the measuring apparatus are taken into account. Bohmian mechanics, however, does not fall in this category. We finally examine an alternative proposal that sequential measurements can be modeled by a process that does not satisfy the Kolmogorov axioms of probability. This removes contextuality without introducing non-locality, but implies that the empirical probabilities cannot be always defined (the event frequencies do not converge). We argue that the predictions of this hypothesis are not ruled out by existing experimental results (examining in particular the "which way" experiments); they are, however, distinguishable in principle. *

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...

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.

Conditional Probabilities and Collapse in Quantum Measurements

International Journal of Theoretical Physics, 2008

We show that including both the system and the apparatus in the quantum description of the measurement process, and using the concept of conditional probabilities, it is possible to deduce the statistical operator of the system after a measurement with a given result, which gives the probability distribution for all possible consecutive measurements on the system. This statistical operator, representing the state of the system after the first measurement, is in general not the same that would be obtained using the postulate of collapse.

On the strangeness of quantum probabilities

Quantum Studies: Mathematics and Foundations

Here we continue with the ideas expressed in "On the strangeness of quantum mechanics" [1] aiming to demonstrate more concretely how this philosophical outlook might be used as a key for resolving the measurement problem. We will address in detail the problem of determining how the concept of undecidability leads to substantial changes to classical theory of probability by showing how such changes produce a theory that coincides with the principles underlying quantum mechanics.

From quantum probabilities to classical facts

Arxiv preprint hep-th/9311090, 1993

Abstract: Model interactions between classical and quantum systems are briefly reviewed. These include: general measurement-like couplings, Stern-Gerlach experiment, model of a counter, quantum Zeno effect, piecewise deterministic Markov processes and meaning of ...

How Do the Probabilities Arise in Quantum Measurement?

Quanta, 2021

A satisfactory resolution of the persistent quantum measurement problem remains stubbornly unresolved in spite of an overabundance of efforts of many prominent scientists over the decades. Among others, one key element is considered yet to be resolved. It comprises of where the probabilities of the measurement outcome stem from. This article attempts to provide a plausible answer to this enigma, thus eventually making progress toward a cogent solution of the longstanding measurement problem.Quanta 2021; 10: 65–74.