We develop a characterization of quantum states by means of first order variables and random variables, within a predicative logic with equality, in the framework of basic logic and its definitory equations. We introduce the notion of random first order domain and find a characterization of pure states in predicative logic and mixed states in propositional logic, due to a focusing condition. We discuss the role of first order variables and the related contextuality, in terms of sequents.
We develop a characterization of quantum states by means of first order variables and random variables, within a predicative logic with equality, in the framework of basic logic and its definitory equations. We introduce the notion of random first order domain and find a characterization of pure states in predicative logic and mixed states in propositional logic, due to a focusing condition. We discuss the role of first order variables and the related contextuality, in terms of sequents.
Quantum logic has developed in various directions since its inception.However, many existing logical approaches either focus only on pure states, or use Hilbert-space models in which the abstract relationship between pure states, mixed states, and probabilities is not made explicit.In this work, we introduce a model designed to handle quantum mixed states abstractly.Furthermore, we construct a proof system for the proposed logic and establish its completeness with respect to the intended semantics.
In the last years, quantum computational logics have been developed in order to describe the logical setting of quantum computation [1]. An important challenge is the development of proof theoretical tools for quantum computation, so that the physical process of computation can correspond to a logical process of computation. In particular, the challenge is that quantum computational speed up, due to superposition and entanglement, can find an explanation in terms of logical proofs. The object of our research is a sequent calculus developed in the framework of basic logic, that is a logical platform to study extensional logics, including quantum logics [2, 3]. In basic logic, logical connectives and their rules are the result of importing suitable links at the metalevel into the object level. We suppose that a particular kind of dynamics between the object level and the metalevel is created when considering a quantum system rather than a classical one. This is due to a specific treatment of variables, that, in our view, can create the logical setting for a holistic, rather than compositional, treatment of information, proper of entangled states [4]. Then we develop a predicative sequent calculus. In it, superposition is built by quantifying a propo-sition, associated with a quantum system, on a first order domain determined by the experiments on such system. Then a convenient treatment of variables can create dynamically entangled states in logical
The term proposition usually denotes in quantum mechanics (QM) an element of (standard) quantum logic (QL). Within the orthodox interpretation of QM the propositions of QL cannot be associated with sentences of a language stating properties of individual samples of a physical system, since properties are nonobjective in QM. This makes the interpretation of propositions problematical. The difficulty can be removed by adopting the objective interpretation of QM proposed by one of the authors (semantic realism, or SR, interpretation). In this case, a unified perspective can be adopted for QM and classical mechanics (CM), and a simple first order predicate calculus L(x) with Tarskian semantics can be constructed such that one can associate a physical proposition (i.e., a set of physical states) with every sentence of L(x). The set Pf of all physical propositions is partially ordered and contains a subset PTf of testable physical propositions whose order structure depends on the criteria of testability established by the physical theory. In particular, PTf turns out to be a Boolean lattice in CM, while it can be identified with QL in QM. Hence the propositions of QL can be associated with sentences of L(x), or also with the sentences of a suitable quantum language LTQ(x), and the structure of QL characterizes the notion of testability in QM. One can then show that the notion of quantum truth does not conflict with the classical notion of truth within this perspective. Furthermore, the interpretation of QL propounded here proves to be equivalent to a previous pragmatic interpretation worked out by one of the authors, and can be embodied within a more general perspective which considers states as first order predicates of a broader language with a Kripkean semantics.
Quantum mechanics is basically a mathematical recipe on how to construct physical models. Historically its origin and main domain of application has been in the microscopic regime, although it strictly seen constitutes a general mathematical framework not limited to this regime. Since it is a statistical theory, the meaning and role of probabilities in it need to be defined and understood in order to gain an understanding of the predictions and validity of quantum mechanics. The interpretational problems of quantum mechanics are also connected with the interpretation of the concept of probability. In this thesis the use of probability theory as extended logic, in particular in the way it was presented by E. T. Jaynes, will be central. With this interpretation of probabilities they become a subjective notion, always dependent on one's state of knowledge or the context in which they are assigned, which has consequences on how things are to be viewed, understood and tackled in quantum mechanics. For instance, the statistical operator or density operator, is usually defined in terms of probabilities and therefore also needs to be updated when the probabilities are updated by acquisition of additional data. Furthermore, it is a context dependent notion, meaning, e.g., that two observers will in general assign different statistical operators to the same phenomenon, which is demonstrated in the papers of the thesis. It is also presented an alternative and conceptually clear approach to the problematic notion of "probabilities of probabilities", which is related to such things as probability distributions on statistical operators. In connection to this, we consider concrete numerical applications of Bayesian quantum state assignment methods to a three-level quantum system, where prior knowledge and various kinds of measurement data are encoded into a statistical operator, which can then be used for deriving probabilities of other measurements. The thesis also offers examples of an alternative quantum state assignment technique, using maximum entropy methods, which in some cases are compared with the Bayesian quantum state assignment methods. Finally, the interesting and important problem whether the statistical operator, or more generally quantum mechanics, gives a complete description of "objective physical reality" is considered. A related concern is here the possibility of finding a "local hidden-variable theory" underlying the quantum mechanical description. There have been attempts to prove that such a theory cannot be constructed, where the most well-known impossibility proof claiming to show this was given by J. S. Bell. In connection to this, the thesis presents an idea for an interpretation or alternative approach to quantum mechanics based on the concept of space-time.
A quantum operator is a classical physical variable except that it assumes values in a probability distribution, i.e., a random variable, solved by the Schrödinger equation in the form of a wavefunction. To date, the logical foundation of this construct remains an open question. This paper casts the basic components of quantum mechanics in the framework of general statistics. Our starting point is to define the domain of the wavefunction to be the complex unit circle, thus faithfully observing the periodicity of a wave. We seek to crystallize the meanings of the fundamental elements of quantum mechanics, which often are lost in its formalism, by certain common random variables in ordinary applications, such as random walks, inventory cycles, and human healthy temperatures, all defined on the sample space of a unit circle.
In quantum logic, i.e., within the structure of the Hilbert lattice imposed on all closed linear subspaces of a Hilbert space, the assignment of truth values to quantum propositions (i.e., experimentally verifiable propositions relating to a quantum system) is unambiguously determined by the state of the system. So, if only pure states of the system are considered, can a probability measure mapping the probability space for truth values to the unit interval be assigned to quantum propositions? In other words, is a probability concept contingent or emergent in the logic of quantum propositions? Until this question is answered, the cause of probabilities in quantum theory cannot be completely understood. In the present paper it is shown that the interaction of the quantum system with its environment causes the irreducible randomness in the relation between quantum propositions and truth values.