In a recent paper, the author introduced an operational description of physical theories where probabilities are replaced by counterfactual statements belonging to a three-valued (i.e. possibilistic) semantic domain. The complete axiomatic of these Generalized possibilistic Theories is generalized and clarified in the present paper. The problem of bipartite experiments is then addressed as the main skill of this paper. An axiomatic for the tensor product of our spaces of states is given and different solutions are explicitly constructed. This description of tensor products of Inf semi-lattices is partly independent from the usual mathematical description of this problem. The nature of the tensor product of orthocomplemented Inf semi-lattices is then also explored. This subject is indeed fundamental for the development of a reconstruction program for quantum theory within our framework. Our analysis constitutes a first step towards this achievement.
In a recent paper, the author introduced an operational description of physical theories where probabilities are replaced by counterfactual statements belonging to a three-valued (i.e. possibilistic) semantic domain. The complete axiomatic of these Generalized possibilistic Theories is generalized and clarified in the present paper. The problem of bipartite experiments is then addressed as the main skill of this paper. An axiomatic for the tensor product of our spaces of states is given and different solutions are explicitly constructed. This description of tensor products of Inf semi-lattices is partly independent from the usual mathematical description of this problem. The nature of the tensor product of orthocomplemented Inf semi-lattices is then also explored. This subject is indeed fundamental for the development of a reconstruction program for quantum theory within our framework. Our analysis constitutes a first step towards this achievement.
Chapter 1: On the existence of quantum representations for two dichotomic measurements. Under which conditions do outcome probabilities of measurements possess a quantum-mechanical model? This kind of problem is solved here for the case of two dichotomic von Neumann measurements which can be applied repeatedly to a quantum system with trivial dynamics. The solution uses methods from the theory of operator algebras and the theory of moment problems. The ensuing conditions reveal surprisingly simple relations between certain quantum-mechanical probabilities. It also shown that generally, none of these relations holds in general probabilistic models. This result might facilitate further experimental discrimination between quantum mechanics and other general probabilistic theories. Chapter 2: Possibilistic Physics. I try to outline a framework for fundamental physics where the concept of probability gets replaced by the concept of possibility. Whereas a probabilistic theory assigns a state-dependent probability value to each outcome of each measurement, a possibilistic theory merely assigns one of the state-dependent labels "possible to occur" or "impossible to occur" to each outcome of each measurement. It is argued that Spekkens' combinatorial toy theory of quantum mechanics is inconsistent in a probabilistic framework, but can be regarded as possibilistic. Then, I introduce the concept of possibilistic local hidden variable models and derive a class of possibilistic Bell inequalities which are violated for the possibilistic Popescu-Rohrlich boxes. The chapter ends with a philosophical discussion on possibilistic vs. probabilistic. It can be argued that, due to better falsifiability properties, a possibilistic theory has higher predictive power than a probabilistic one. Chapter 3: The quantum region for von Neumann measurements with postselection. It is determined under which conditions a probability distribution on a finite set can occur as the outcome distribution of a quantum-mechanical von Neumann measurement with hyphenation given that the scalar product between the initial and the final state is known as well as the success probability of the postselection. An intermediate von Neumann measurement can enhance transition probabilities between states such that the error probability shrinks by a factor of up to 2. Chapter 4: A presentation of the category of stochastic matrices. This chapter gives generators and relations for the strict monoidal category of probabilistic maps on finite cardinals (i.e., stochastic matrices). Chapter 5: Convex Spaces: Definition and Examples. We try to promote convex spaces as an abstract concept of convexity which was introduced by Stone as "barycentric calculus''. A convex space is a set where one can take convex combinations in a consistent way. By identifying the corresponding Lawvere theory as the category from chapter 4 and using the results obtained there, we give a different proof of a result of Swirszcz which shows that convex spaces can be identified with algebras of a finitary version of the Giry monad. After giving an extensive list of examples of convex sets as they appear throughout mathematics and theoretical physics, we note that there also exist convex spaces that cannot be embedded into a vector space: semilattices are a class of examples of purely combinatorial type. In an information-theoretic interpretation, convex subsets of vector spaces are probabilistic, while semilattices are possibilistic. Convex spaces unify these two concepts.
Since the 1980s, there has been a physical problem relating to the detection of multipartite entangled states that helps us to investigate fundamental point of view on interpretations of quantum mechanics. On the other hand, information theories based on the mathematical structure of quantum mechanics, for example, quantum cryptography and quantum computation, have been investigated by researchers whose number has been increasing day by day in the world. This field became to be called quantum information theories. A practical experimental realization of quantum information processing needs that we detect experimentally multipartite entangled states. The number of researchers for multipartite entangled states is increasing rapidly. Especially, Greenberger-Horne-Zeilinger (GHZ) states are useful for quantum information processing information processing using quantum mechanics, which improves the conventional information processing ability which had been based on classical theories. However, practical experimental realization of the theories and controlling quantum states have inevitable difficulties due to the interaction with environments. The requirement in maintaining multipartite entangled states in relation to the time that takes to make a practical experiment is also a problem. Many experimentalists are tackling this problem, and recently, several groups have reported that multipartite entangled states have been physically detected in their laboratories. The main difficulty is how to judge whether or not the observed quantum state is indeed a multipartite entangled state. For this purpose, it is necessary to construct theoretical models to analyze such experimental data. In this thesis, we apply a mathematical structure as to density operators described in a high-dimensional Hilbert space, and we employ mathematical theories to prove if the corresponding density operator represents a multipartite entangled state or not. The main aim for constructing such models is to analyze the raw experimental data. Moreover, this model helps experimentalists in the sense that it allows them to see how to confirm that observed states are indeed the desired state. With mathematical theories developed before this thesis, only limited applications for this purpose were allowed. Some theories cover two-dimensional Hilbert space only. It has not been even discussed how to analyze the fidelity to GHZ states of the observed state. This has led to some doubt of that experimental data might not always confirm three-partite entangled states. However, by using the method of this thesis, statistical check based on such experimental data of the existence of three particle-entangled states is possible under appropriate assumptions. This example symbolizes that our analytical knowledge has improved very much in the last few years. This thesis gives us an advanced mathematical model for analyzing experimental data.
Up to now we have limited ourselves to states of a single particle. In the present chapter we shall introduce the description of two-particle states. Once this case is understood, it will be easy to generalize to any number of particles. States of two (or more) particles lead to very rich configurations called entangled states. A remarkable feature is that two entangled quantum particles, even at arbitrarily large spatial separations, continue to form a single entity and no classical probabilistic model is able to reproduce the correlation between particles. In the first section we shall present the essential mathematical formalism, that of the tensor product. This will permit us in Section 6.2 to describe quantum mixtures using the state operator formalism. Section 6.3 is devoted to the study of important physical consequences like the Bell inequalities and interference experiments involving entangled states, which will lead us to a deeper understanding of quantum physics. Finally, in the last section we shall briefly review applications to measurement theory and quantum information theory. The latter is undergoing rapid development at present and has applications to quantum computing, cryptography, and teleportation. The tensor product of two vector spaces Definition and properties of the tensor product We wish to construct the space of states of two physical systems which we assume initially to be completely independent. Let ℌ N 1 and ℌ M 2 be the spaces of states of the two systems, of dimension N and M , respectively.
We develop a possibilistic semantic for quantum phenomena in an operational perspective. This semantic is based on a Chu duality between preparation processes and yes/no tests, the target space being a three-valued set equipped with an informational interpretation. After having defined the notion of states, we develop a precise axiomatic for the space of states. The 'information principle', proposed as a building block for some quantum axiomatic program, is translated into our framework to emphasize the quantum character of our description. This principle suffices to constrain the space of states to be a locally-boolean qualitative domain. The subset of pure states is then characterized within this domain structure. After having carefully precised the notions of properties and measurements, we explore the notion of compatibility between measurements. The existence of minimally-disturbing measurements is then emphasized as a key axiom to describe the space of yes/no tests. The conditions of existence of such measurements is translated into a simple property on the space of states, and an explicit formula for these measurements is given. Having reduced the space of yes/no tests to this set of minimally-disturbing operations, our Chu space becomes bi-extensional. The space of 'descriptions', associated to families of compatible measurements, inherits a structure of coherence domain. A subset of properties corresponding to 'classical properties' is identified and explored. Endly, the symmetries of the system are characterized as a general sub-class of Chu morphisms. We prove that these symmetries preserve the class of minimally-disturbing measurements and the orthogonality relation between states. Explicit expressions of these symmetries are identified for a large class of them, called 'linear symmetries'.
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.
A generalized probability theory arises naturally from an attempt to formulate a statistical model for quantum mechanics.One soon finds that the probability theory of Kolmogorov is inadequate for the description of a quantum mechanical system mainly because the quantum mechanical events fail to form a er-algebra of subsets of a set.Instead, they have a much less richly endowed algebraic structure which we shall a "logic".We therefore look upon generalized probability as a stochastic theory in which the events form a logic.Besides being useful in the study of quantum mechanics, this theory has a purely mathematical interest since it constructs a stochastic formalism on an algebraic structure and at the same time gives an abstract generalization of Hubert space theory.The first formulation of this theory is due to G. Bodiou [1], [2].More recently, V. S. Varadarajan [8] has emphasized the stochastic properties of an axiomatic formulation of quantum mechanics due to G. W. Mackey [5] and has proved some important results in this theory.The author heartily recommends these references for the background and motivation of this study.In this paper, the main interest centers on the spectral theory of the subject.The spectrum is important physically since it represents the "allowable" values of a quantum mechanical observable.It is also mathematically useful since it introduces topological methods into the formalism.2. Definitions and notation.Let L be a complemented lattice.That is, L is a lattice with first and last elements, 0,1 respectively and a-*a' satisfying the conditions (i) ia'Y = a for all a eL;(ii) a z% b implies b' zi a'; (iii) a V a ' -1 for all a e L.