In quantum physics the tests of most properties do not have predetermined outcomes. The latter have nevertheless well-defined probabilities of being realized during a test. Following Popper we interpret these probabilities as physical propensities. A first purpose of the present article is to formalize the propensity interpretation in the framework of state-property structures. Next, Gleason’s theorem asserts that in the Hilbert space there exists a unique propensity function (i.e., one probability measure for each state vector); the propensities are thus uniquely determined by the state vector. Conversely, we prove that if the state-property structure admits one and only one propensity function, then the set ℒ of all properties is a complete atomic orthomodular lattice. We point out that according to our assumption the probabilistic aspect of the system is entirely determined by its deterministic aspect. Assuming furthermore that each property can be ideally tested, it follows that ℒ is isomorphic to the direct union of Hilbertian space lattices. We recover thus the purely classical and purely quantum frameworks as the two extreme cases. The intermediate cases correspond to quantum mechanics with—possibly continuous—superselection variables. Finally, we prove that a system is classical, i.e., all properties are mutually compatible, if and only if the propensity function is dispersion free. In our approach the quantum probabilities appear thus as a generalization of classical determinism rather than a generalization of classical probabilities.
In quantum physics the tests of most properties do not have predetermined outcomes. The latter have nevertheless well-defined probabilities of being realized during a test. Following Popper we interpret these probabilities as physical propensities. A first purpose of the present article is to formalize the propensity interpretation in the framework of state-property structures. Next, Gleason’s theorem asserts that in the Hilbert space there exists a unique propensity function (i.e., one probability measure for each state vector); the propensities are thus uniquely determined by the state vector. Conversely, we prove that if the state-property structure admits one and only one propensity function, then the set ℒ of all properties is a complete atomic orthomodular lattice. We point out that according to our assumption the probabilistic aspect of the system is entirely determined by its deterministic aspect. Assuming furthermore that each property can be ideally tested, it follows that ℒ is isomorphic to the direct union of Hilbertian space lattices. We recover thus the purely classical and purely quantum frameworks as the two extreme cases. The intermediate cases correspond to quantum mechanics with—possibly continuous—superselection variables. Finally, we prove that a system is classical, i.e., all properties are mutually compatible, if and only if the propensity function is dispersion free. In our approach the quantum probabilities appear thus as a generalization of classical determinism rather than a generalization of classical probabilities.
Abstract Quantum theory is indeterministic, but not completely so. When a system is in a pure state there are properties it possesses with certainty, known as actual properties. The actual properties of a quantum system (in a pure state) fully determine the probability of finding the system to have any other property. We will call this principle, wherein the deterministic elements of a theory completely characterise the probabilistic elements, intermediate determinism. In dimensions of at least three, intermediate determinism in quantum theory is guaranteed by the structure of its lattice of properties. This observation follows from Gleason’s theorem, which is why it fails to hold in dimension two. In this work we extend the idea of intermediate determinism from properties to measurements. Under this extension intermediate determinism follows from the structure of quantum effects for separable Hilbert spaces of any dimension, including dimension two. Then, we find necessary and sufficient conditions for a general probabilistic theory to obey intermediate determinism. We show that, although related, both the no-restriction hypothesis and a Gleason-type theorem are neither necessary nor sufficient for intermediate determinism.
The postulates of quantum theory are rather abstract in comparison with those of other physical theories such as special relativity. This thesis considers two tools for investigating this discrepancy and makes a connection between them. The first of these tools, Gleason-type theorems, illustrates the interplay between postulates concerning observables, states and probabilities of measurement outcomes, demonstrating that they need not be entirely independent. Gleason’s original and remarkable result applied to observables described by projection-valued measures; however, the theorem does not hold in dimension two. Busch generalised the idea to observables described by positive operator measures, proving a result which holds for all separable Hilbert spaces. We show that Busch’s assumptions may be weakened without affecting the result. The manner in which we weaken the assumptions brings them closer to Gleason’s original treatment of projection-valued measures. We will then demonstrate the connection between Gleason-type theorems and Cauchy’s functional equation, a connection which yields an alternative proof of Busch’s result. The second tool we consider is the family of general probabilistic theories which offers a means of comparing quantum theory with reasonable alternatives. We identify a general probabilistic theory which reproduces the set of non-local correlations achievable in quantum theory, a property often thought to be particular to quantum theory. Finally, we connect these two tools by determining the class of general probabilistic theories which admit a Gleason-type theorem.
This introductory chapter sets out a general structure of physical theories which is applicable to both classical and quantum mechanics. We start with measurable properties of a given physical system, be it classical or quantum. These properties are called observables . We then introduce a definition of the state of a physical system in terms of measured values of a sufficiently large set of observables. A theory to describe the system should consists of four basic components: Basic mathematical framework This comprises a set of elements endowed with some specific mathematical structure and properties. In mathematics such a set is generally known as a space . Description of states States are described by elements of the space in the chosen mathematical framework. For this reason the space is called the state space of the system. Description of observables Observables are to be described by quantities defined on the state space. The description should yield all possible values of observables. The relationship between observables and states should be explicitly stated. The following two cases are of particular interest: 2 For a deterministic theory like classical mechanics a state should determine the values of all observables. For a probabilistic theory like quantum mechanics a state should determine the probability distribution of the values of all observables. Description of time evolution (dynamics).
The purpose of this paper is to revisit a neglected understanding of quantum mechanics, the so-called propensity theory and to propose a amended version of it. According to this version, the causal-structural approach adopted here, a quantum system has a physical structure, consisting of relata (components, say electrons) and relations (determining the interrelations between the components). This structure is mathematically encoded in wave functions and forms the base, on which propensities supervene. This paper proposes that propensities or intrinsic dispositions supervene on the structural properties of the system. A propensity is a disposition of a quantum system to display selective aspects in response to an encounter with a measurement situation. The selective propensities manifest themselves as probabilities, in accordance with the application of the Born rule to respective wave functions.
It is a scientific truism nowadays that quantum mechanical predictions are of statistical or rather of probabilistic nature: a quantum theory assigns a probability distribution to each measurable quantity (observable) of a physical system prepared according to a given preparation prescription (state). Such sets of probability distributions represent the most general quantitative predictions of physical theories. The aim of this study is to determine all theoretical frameworks which yield such a kind of output taking the probabilistic nature of quantitative physical predictions as initial hypothesis. Astonishingly, this project succeeds with a minimal number of additional assumptions, which arise quite naturally in the course of the analysis, and which essentially amount to regularity conditions. The result is that there exist only two classes of physical theories providing quantitative probabilistic predictions: classical theories where observables are represented by real functions over a phase space and states are represented by probability measures upon the phase space, and quantum theories where observables are represented by selfadjoint operators acting on a complex Hilbert space and states are represented by state operators. On empirical grounds, classical theories cannot provide the all-encompassing, fundamental frame for the mathematical description of physical systems. Quantum theories remain, and there is no more complete formal frame for quantitative physical theories. As an additional result, it is shown that there exist physical systems which possess a quantum mechanical but no classical description; a prominent example is the spin system of two spin-1/2 particles. The classical hidden variables model, which J. Bell relates to this system in order to prove his theorem, cannot be a true description of it because no such classical model exists, and Bell's theorem loses its foundation.
The Question of the Completeness of Quantum Mechanics When a classical system is in a pure state, that is, when its preparation represents maximal information, the value of every physical quantity is uniquely predicted. In other words, classical pure states are dispersion-free, and classical mechanics is deterministic. Only in case the state is nonpure, thus representing nonmaximal information about the system, does probability enter into the theory. Things are different with quantum systems. Even in case the state of the system is pure, it is impossible to predict a definite value of every physical quantity: only the probability distributions of the physical quantities are given. Pure quantum states have dispersion (though, of course, they can be dispersion-free for particular physical quantities), and probability enters into quantum theory at a much more fundamental level. The basic idea of so-called hidden-variable theories is to reject the fundamental role of probability in quantum theory and to argue that probability arises because quantum states, even the pure ones, do not represent the ultimate information about the system. Accordingly, it is conjectured that there are certain hidden variables, not yet subject to experimental detection, which would complete the information carried by the quantum states, thus giving rise to so precise a knowledge of the system that probabilities would disappear and precise values of all physical quantities would be uniquely determined.