A manual of operations is a model for a collection of physical experiments, or a collection of (perhaps overlapping) sample spaces. To understand this theory, it is essential to understand the relations between manuals. Hence, in this dissertation, we study the morphisms (and, more specifically, the interpretations) of manuals. This is followed by an investigation of the categorical properties of the tensor product of two manuals. We define biinterpretations and prove a universal mapping theorem. We then prove that the tensor product is associative, and define, using an inductive limit, an infinite tensor product. In the final chapter, we develop a language and theory that is broad enough to encompass much of the work which has been done under the general title of "quantum logic."
A manual of operations is a model for a collection of physical experiments, or a collection of (perhaps overlapping) sample spaces. To understand this theory, it is essential to understand the relations between manuals. Hence, in this dissertation, we study the morphisms (and, more specifically, the interpretations) of manuals. This is followed by an investigation of the categorical properties of the tensor product of two manuals. We define biinterpretations and prove a universal mapping theorem. We then prove that the tensor product is associative, and define, using an inductive limit, an infinite tensor product. In the final chapter, we develop a language and theory that is broad enough to encompass much of the work which has been done under the general title of "quantum logic."
An original presentation of Categorical Quantum Physics, in the line of Abramsky and Coecke, tries to introduce only objects and assumptions that are clearly relevant to Physics and does not assume compact closure. Adjoint arrows, tensor products and biproducts are the ingredients of this presentation. Tensor products are defined, up to a unitary arrow, by a universal property related to transformations of composite systems, not by assuming a monoidal structure. Entangled states of a tensor product define mixed states on the components of the tensor product. Coproducts that fit the adjoint structure are shown to be defined up to a unitary arrow and to provide biproducts. An abstract no-cloning result is proved.
Quantum mechanics has several deficiencies as a complete theoretical description of the measurement process. Among them is the fact that the quantum mechanical description of correlations between the single measurements of a sequence is quite problematic. A single measurement is defined to be a preparation followed by an observation. In particular, one feels that an infinite sequence of such single measurements which corresponds to the measurement of a question O on a state η where η does not lie entirely in an eigenspace of O should generate a random output sequence. However, quantum mechanics seems to say nothing about this. In this paper, physical theories are defined in such a manner that correlations between single measurements are explicitly included. In particular, a physical theory is considered to be a mapping U, with domain in the set [Qsτ] of infinite instruction strings for carrying out infinite sequences of single measurements and range in the set of probability measures defined on A, the usual σ algebra of subsets of Ω. Ω is the set of all infinite sequences of natural numbers. A fundamental property which any valid physical theory must satisfy is that it agrees with experiment. It is proposed and discussed here that much of the intuitive meaning of agreement between a theory U and experiment with respect to H is given by the statement ∀Qsτ [U(Qsτ) defined⇒E(H,U(Qsτ),ψQsτ)],where U(Qsτ) is the probability measure U associates with the infinite instruction string Qsτ and ψQsτ is the outcome sequence obtained by carrying out Qsτ. E(H, U(Qsτ),ψQsτ) is the statement that all formulas in H with one free sequence variable which are true on Ω almost everywhere with respect to U(Qsτ) are true for ψQsτ. H is a subclass of the class of all formulas in a formal language L. A theorem is proved which states that, if U(Qsτ) corresponds to a nontrivial product probability measure and U H-agrees with experiment, then the outcome sequence ψQsτ is H-random. H-randomness is defined here in terms of the statement E(H, P, φ). Another property of a valid physical theory, which is defined here, is that, for some Qsτ, U(Qsτ) must be determinable on much of AH from ψQsτ. Sufficient conditions for this property to hold are given. AH is the class of all H definable subsets of Ω. Some properties of the statement E(H, P, φ) are given. Among other things, it is proved that, if E(H, P, φ) holds and P is a nontrivial probability measure on A, then φ is not definable in H.
According to quantum mechanics, the pure states of a microsystem are represented by vectors in a Hilbert Space. Sentences of the form, "x є L" (where x is the state vector for a system, L a subspace of the appropriate Hilbert space), may be called Q-propositions: such sentences serve to summarise our information about the results of possible experiments on the system. Quantum logic investigates the relations which hold among the Q-propositions about a given physical sys tem. These logical relations correspond to algebraic relations among the subspaces of Hilbert space. The algebra of this set of subspaces is non-Boolean, and may be regarded either as an orthomodular lattice or as a partial Boolean algebra. With each type of structure we can associate a logic. A general approach to the semantics for such a logic is provided in terms of interpretations of a formal language within an algebraic structure; an interpretation maps sentences of the language homomorphically onto elements of the structure. When the structure in question is a Boolean algera, the resulting logic is classical; here we develop a semantics for the logic associated with partial Boolean algebras. Two systems of proof, based on the natural deduction systems of Gentzen, are shown for this logic. With respect to the given sematics, these calculi are sound and weakly complete. Strong completeness is conjectured. Quantum logic deals with the logical relations between sentences, and so is properly called a logic. However, it is the logic appropriate to a limited class of sentences: proposals that it should replace classical logic wherever the latter is used should be viewed with suspicion.
In this work we present an intuitive construction of the quantum logical axiomatic system provided by George Mackey. The goal of this work is a detailed discussion of the results from the paper 'Physical justification for using the tensor product to describe two quantum systems as one joint system' [1] published by Diederik Aerts and Ingrid Daubechies. This means that we want to show how certain composed physical systems from classical and quantum mechanics should be described logically. To reach this goal, we will, like in [1], discuss a special class of axiomatically defined composed physical systems. With the help of certain results from lattice and c-morphism theory (see [2] and [23]), we will present a detailed proof of the statement, that in the quantum mechanical case, a composed physical system must be described via a tensor product space.
This paper develops some of the work of Foulis, Randall, Aerts, and Piron in the fields of empirical science and quantum logic from an algebraic point of view. More specifically, it begins with three axioms of what is called a 'subtraction algebra,' and generates various theorems associated with properties which are useful in empirical science. After a foundation is established, it moves on to define the term manual. This term is defined as a dominated, atomic, semi-Boolean algebra which satisfies an additional condition called condition M. Several properties of the manual are discussed, and different types of manuals are given: classical semi-classical and non-classical. The paper defines operational complements, operational perspectively, atoms, events, and tests, before moving on to define a logic, and how it is derived from a manual. Properties of the logic are discussed, including a subtraction operation, a partial order, and an ortho complement. Next, a computer program is presented. Its purpose is to take a finite semi-Boolean algebra and decide if the algebra is a manual. This is followed by a brief non-classical probabilistic discussion, which includes topics such as weights, pure states, and dispersion-free states. Aerts' and Piron's work with properties, states, and questions is briefly discussed before moving on to several examples, some of them arising from navigation problems. The examples include the hook, the square, the Wright Triangle, and the free algebra. Empirical techniques are demonstrated on these examples. The examples comprise the bulk of this paper.
The study of quantum systems involving two or more qubits necessitates the development of mathematical tools which capture the interactions between the qubits. This leads to the construction of objects called tensors which are elements of a vector space called a tensor product space. This material is essential since tensors are required for defining the means of data representation in quantum computation. The tensor product is the result of combining two vectors in a particular way designed to capture the interaction between the various parts of the individual vectors. The results of finding tensor products reside in a vector space called the tensor product space, the elements of which are called tensors. The formal construction of a tensor product space is somewhat abstract and can be difficult for the student to understand.