A probabilistic propositional logic, endowed with an epistemic component for asserting (non-)compatibility of diagonizable and bounded observables, is presented and illustrated for reasoning about the random results of projective measurements made on a given quantum state. Simultaneous measurements are assumed to imply that the underlying observables are compatible. A sound and weakly complete axiomatization is provided relying on the decidable first-order theory of real closed ordered fields. The proposed logic is proved to be a conservative extension of classical propositional logic.
A probabilistic propositional logic, endowed with an epistemic component for asserting (non-)compatibility of diagonizable and bounded observables, is presented and illustrated for reasoning about the random results of projective measurements made on a given quantum state. Simultaneous measurements are assumed to imply that the underlying observables are compatible. A sound and weakly complete axiomatization is provided relying on the decidable first-order theory of real closed ordered fields. The proposed logic is proved to be a conservative extension of classical propositional logic.
In order to find a physical axiomatization of quantum theory, physical theories are often considered as a special case of a much more general framework of generalized probabilistic theories. We first present a detailed introduction to generalized probabilistic theories before we examine new approaches towards a physical axiomatization of quantum theory. The main result of the thesis is that one postulate concerning post-measurement states (which have gained almost no attention in this context so far) is sufficient to rule out all probabilistic theories with a polytopic state space, except for classical theory. The postulate states that a measurement with a definite outcome does not alter the state. This rules out almost all toy theories considered so far, for instance the PR-box, but is compatible with quantum theory. Thus, this result can be seen as an approach towards a classification of quantum theory in physical terms. It only deals with local state spaces, in contrast to other approaches that are mostly concerned with nonlocality in composite systems.
Statistical physical theories are frequently formulated in terms of probabilistic structures founded on a ’’logic of experimentally verifiable propositions.’’ It is argued that to each experimentally verifiable proposition there corresponds an experimental procedure which, in general, alters the state of the system, and is completely characterized by a ’’measurement transformation’’ or ’’operation.’’ An analysis of the relations among these experimental procedures leads us to a ’’logic of operations’’ which is quite different from the ’’lattice theoretic logics’’ that are often considered (albeit inadequate empirical justification), as models for the calculus of experimentally verifiable propositions of quantum theory. It is seen that the quantum probability theory based on the logic of operations provides the proper mathematical framework for discussing the statistics of successive observations in quantum theory. We also indicate how a theory of quantum stochastic processes can be formulated in a way similar to the Kolmogorov formulation of the classical theory.
We propose a new link between mathematical undecidability and quantum physics. We demonstrate that the states of elementary quantum systems are capable of encoding mathematical axioms and show that quantum measurements are capable of revealing whether a given proposition is decidable or not within the axiomatic system. Whenever a mathematical proposition is undecidable within the axioms encoded in the state, the measurement associated with the proposition gives random outcomes. Our results support the view that quantum randomness is irreducible and a manifestation of mathematical undecidability. Whenever a proposition and a given set of axioms together contain more information than the axioms themselves, the proposition can neither be proved nor disproved from the axioms { it is mathematically undecidable [1, 2]. Here we propose a novel link between mathematical undecidability and quantum physics. We demonstrate that the states of elementary quantum systems are capable of encoding mathematical axioms. Quantum mechanics imposes an upper limit on how much information can be encoded in a quantum state [3, 4], thus limiting the information content of the set of axioms. We show that quantum measurements are capable of revealing whether a given proposition is decidable or not. Whenever a mathematical proposition is undecidable within the system of axioms encoded in the state, the measurement associated with the proposition gives random outcomes. This allows for an experimental test of mathematical undecidability by realizing in the laboratory both the actual quantum states and the required quantum measurements. (To illustrate these ideas, we conducted experiments using the polarization of photons.) Our results support the view that quantum randomness is irreducible [5, 6] and a manifestation of mathematical undecidability. Any formal system is based on axioms, which are propositions that are dened to be true. A proposition is logically independent from a given set of axioms if it can neither be proved nor disproved from the axioms. If a proposition is independent from the axioms, neither the proposition itself nor its negation creates an inconsistency together with the axiomatic system. ^
Several attempts have been made to establish the logical structure of quantum mechanics on an axiomatic foundation. It has been shown that, whereas the logical structure of the propositional calculus of classical mechanics is that of a Boolean lattice, the logical structure of the propositional calculus of quantum mechanics is non-Boolean. Moreover, a probability model of a quantum system, namely, the quantum probability model, has a mathematical structure completely different from that of the Kolmogorovian probability model, which is the standard probability model of a classical system. These results, which are still controversial, are analysed and put in critical perspective.
The non-classical nature of quantum theory has led to a number of efforts to formalize the logic of statements whose universe of discourse is the subatomic (see, for example, [3], [6], [8], and [10]).It might be hoped that ideally the fruit of such efforts would be a formalism not totally divorced from the classical logic, but incorporating or incorporated in it in such a manner as to unify the logic of the micro-and the macro-worlds.Indeed, this has been the general direction taken by these efforts, including that which we principally examine and expand upon here.*Josef Jauch, [6], pp.67-89, developed a "propositional calculus of quantum mechanics".He objected to calling a system like his a "logic of quantum mechanics"; however, our use of 'logic' here is not, I think, the use to which Jauch objected.A logic, for our present purposes, is a mathematical formalism having a reasonable interpretation as a theory of deduction; its actual application as normative of argumentation in reality is a matter of empirical judgment.This seems close to the way Jauch wished to use "propositional calculus" but more general, allowing for extension to a quantification theory of quantum mechanics, for example.Although Jauch's presentation of a propositional calculus of quantum mechanics is mathematically interesting and suggestive, there are a number of problems connected with it.It is our purpose to engage some of these problems and to present a rigorously defined formalism intended to come close to doing the job intended by Jauch for his logic.The success or lack of success of this effort as related to actual application of this formalism in the foundations of quantum theory is, I think, a matter for the physicists to decide; the empirical judgment involved in this application is not in the scope of this paper.If a formalism is to have a reasonable interpretation as a theory of deduction, it seems clear that it must include a relation which, when it holds between two elements of the formalism, may be reasonably interpreted as expressing that one of the elements 'follows from' the other, in •The author wishes to thank Robert Piziak for advice and comments in the course of research for this paper.
QBism interprets quantum theory as a normative discipline for an agent's probability assignments and their revision across possible experience. This paper develops a logical formalization of that picture. A well-formed core datum consists of an admissible prior space, a finite family of actual measurements, Born kernels, and update kernels. For each such datum, we introduce a guarded dynamic language for histories and posterior states and prove a global reduction theorem. We next consider effectively semialgebraic data over an effectively presented real closed field. For data in this class, we translate the fragment without dynamic operators into first-order formulas in the corresponding language of ordered rings, thereby reducing validity to first-order reasoning over real closed fields. Together, the reduction and first-order translation yield a sound and complete recursive calculus and a decision procedure for validity. Finally, assuming a symmetric informationally complete (SIC) reference measurement, we show that quantum theory in finite dimensions realizes the framework through SIC coordinates, POVMs, and quantum instruments. We also prove that the corresponding SIC image satisfies the standard qplex geometry conditions, namely the consistency bounds and the lower polar condition, and that under explicit coefficient field hypotheses the resulting quantum datum is effectively semialgebraic.