We first present a realistic framework for quantum probability theory based on the path integral formalism of quantum mechanics and illustrate this framework by constructing a model that describes a quantum particle evolving in a discrete space-time lattice. We then present a finite model for describing the internal dynamics of “elementary particles” and show that this model gives the standard particle classification scheme and successfully predicts particle masses.

Entanglement measures quantify the amount of quantum entanglement that is contained in quantum states. Typically, different entanglement measures do not have to be partially ordered. The presence of a definite partial order between two entanglement measures for all quantum states, however, allows for meaningful conceptualization of sensitivity to entanglement, which will be greater for the entanglement measure that produces the larger numerical values. Here, we have investigated the partial order between the normalized versions of four entanglement measures based on Schmidt (...) decomposition of bipartite pure quantum states, namely, concurrence, tangle, entanglement robustness and Schmidt number. We have shown that among those four measures, the concurrence and the Schmidt number have the highest and the lowest sensitivity to quantum entanglement, respectively. Further, we have demonstrated how these measures could be used to track the dynamics of quantum entanglement in a simple quantum toy model composed of two qutrits. Lastly, we have employed state-dependent entanglement statistics to compute measurable correlations between the outcomes of quantum observables in agreement with the uncertainty principle. The presented results could be helpful in quantum applications that require monitoring of the available quantum resources for sharp identification of temporal points of maximal entanglement or system separability. (shrink)

It is argued that a reformulation of classical measure theory is necessary if the theory is to accurately describe measurements of physical phenomena. The postulates of a generalized measure theory are given and the fundamentals of this theory are developed, and the reader is introduced to some open questions and possible applications. Specifically, generalized measure spaces and integration theory are considered, the partial order structure is studied, and applications to hidden variables and the logic of quantum mechanics are given.

We construct a quantum logic which generates the usual quark states. It follows from this model that quarks can combine only in quark-antiquark pairs and quark (and antiquark) triples. The ground meson and baryon states are also generated and gluons are discussed.

We develop the concept of quantum probability based on ideas of R. Feynman. The general guidelines of quantum probability are translated into rigorous mathematical definitions. We then compare the resulting framework with that of operational statistics. We discuss various relationship between measurements and define quantum stochastic processes. It is shown that quantum probability includes both conventional probability theory and traditional quantum mechanics. Discrete quantum systems, transition amplitudes, and discrete Feynman amplitudes are treated. We close with some examples that illustrate previously (...) defined concepts. (shrink)

AUTHOR: STAN GUDDER (John Evans Professor of Mathematical Physics, University of Denver, USA) -- -/- We consider a discrete scalar, quantum field theory based on a cubic 4-dimensional lattice. We mainly investigate a discrete scattering operator S(x0,r) where x0 and r are positive integers representing time and maximal total energy, respectively. The operator S(x0,r) is used to define transition amplitudes which are then employed to compute transition probabilities. These probabilities are conditioned on the time-energy (x0,r). In order to maintain total (...) unit probability, the transition probabilities need to be reconditioned at each (x0,r). This is roughly analogous to renormalization in standard quantum field theory, except no infinities or singularities are involved. We illustrate this theory with a simple scattering experiment involving a common interaction Hamiltonian. We briefly mention how discreteness of space-time might be tested astronomically. Moreover, these tests may explain the existence of dark energy and dark matter. (shrink)

Fuzzy amplitude densities are employed to obtain probability distributions for measurements that are not perfectly accurate. The resulting quantum probability theory is motivated by the path integral formalism for quantum mechanics. Measurements that are covariant relative to a symmetry group are considered. It is shown that the theory includes traditional as well as stochastic quantum mechanics.

A D-algebra is a generalization of a D-poset in which a partial order is not assumed. However, if a D-algebra is equipped with a natural partial order, then it becomes a D-poset. It is shown that D-algebras and effect algebras are equivalent algebraic structures. This places the partial operation ⊝ for a D-algebra on an equal footing with the partial operation ⊕ for an effect algebra. An axiomatic structure called an effect stale-space is introduced. Such spaces provide an operational interpretation (...) for the partial operations ⊕ and ⊝. Finally, a relationship between effect-state spaces and torsion free interval effect algebras is demonstrated. (shrink)

The concept of an effect test space, which is equivalent to a D-test space of Dvurečenskij and Pulmannová, is introduced. Connections between effect test space. (E-test space, for short) morphisms, and event-morphisms as well as between algebraic E-test spaces and effect algebras, are studied. Bimorphisms and E-test space tensor products are considered. It is shown that any E-test space admits a unique (up to an isomorphism) universal group and that this group, considered as a test group, determines the E-test space (...) uniquely (up to an isomorphism). (shrink)

This article presents and compares various algebraic structures that arise in axiomatic unsharp quantum physics. We begin by stating some basic principles that such an algebraic structure should encompass. Following G. Mackey and G. Ludwig, we first consider a minimal state-effect-probability (minimal SEFP) structure. In order to include partial operations of sum and difference, an additional axiom is postulated and a SEFP structure is obtained. It is then shown that a SEFP structure is equivalent to an effect algebra with an (...) order determining set of states. We also consider σ-SEFP structures and show that these structures distinguish Hilbert space from incomplete inner product spaces. Various types of sharpness are discussed and under what conditions a Brouwer complementation can be defined to obtain a BZ-poset is investigated. In this case it is shown that every effect has a best lower and upper sharp approximation and that the set of all Brouwer sharp effects form an orthoalgebra. (shrink)

The formalism of general probabilistic theories provides a universal paradigm that is suitable for describing various physical systems including classical and quantum ones as particular cases. Contrary to the usual no-restriction hypothesis, the set of accessible meters within a given theory can be limited for different reasons, and this raises a question of what restrictions on meters are operationally relevant. We argue that all operational restrictions must be closed under simulation, where the simulation scheme involves mixing and classical post-processing of (...) meters. We distinguish three classes of such operational restrictions: restrictions on meters originating from restrictions on effects; restrictions on meters that do not restrict the set of effects in any way; and all other restrictions. We fully characterize the first class of restrictions and discuss its connection to convex effect subalgebras. We show that the restrictions belonging to the second class can impose severe physical limitations despite the fact that all effects are accessible, which takes place, e.g., in the unambiguous discrimination of pure quantum states via effectively dichotomic meters. We further demonstrate that there are physically meaningful restrictions that fall into the third class. The presented study of operational restrictions provides a better understanding on how accessible measurements modify general probabilistic theories and quantum theory in particular. (shrink)

We introduce and study the D-model, which reflects the simplest situation in which one wants to calibrate an observable. We discuss the question of representing the statistics of the D-model in the context of an effect algebra.

We present a new approach to measurement theory. Our definition of measurement is motivated by direct laboratory procedures as they are carried out in practice. The theory is developed within the quantum logic framework. This work clarifies an important problem in the quantum logic approach; namely, where the Hilbert space comes from. We consider the relationship between measurements and observables, and present a Hilbert space embedding theorem. We conclude with a discussion of charge systems.

A transient quantum effect is shown to occur for a superposition of stationary states. An alternative to Schrödinger's equation is considered which predicts a transient effect even for energy eigenstates.

This paper develops a theory of quantum automata and their slightly more general versions, q-automata. Quantum languages and η-quantum languages, 0≤η<1, are studied. Functions that can be realized as probability maps for q-automata are characterized. Quantum grammars are discussed and it is shown that quantum languages are precisely those languages that are induced by a quantum grammar. A quantum pumping lemma is employed to show that there are regular languages that are not η-quantum, 0≤η<1.

We show that a quantum system admits hidden variables if and only if there is a rich set of states which satisfy a Bayesian rule. The result is proved using a relationship between Bayesian type states and dispersion-free states. Various examples are presented. In particular, it is shown that for classical systems every state is Bayesian and for traditional Hilbert space quantum systems no state is Bayesian.

State transformations in quantum mechanics are described by completely positive maps which are constructed from quantum channels. We call a finest sharp quantum channel a context. The result of a measurement depends on the context under which it is performed. Each context provides a viewpoint of the quantum system being measured. This gives only a partial picture of the system which may be distorted and in order to obtain a total accurate picture, various contexts need to be employed. We first (...) discuss some basic definitions and results concerning quantum channels. We briefly describe the relationship between this work and ontological models that form the basis for contextuality studies. We then consider properties of channels and contexts. For example, we show that the set of sharp channels can be given a natural partial order in which contexts are the smallest elements. We also study properties of channel maps. The last section considers mutually unbiased contexts. These are related to mutually unbiased bases which have a large current literature. Finally, we connect them to completely random channel maps. (shrink)

We show that an effect algebra E possess an order-determining set of states if and only if E is semiclassical; that is, E is essentially a classical effect algebra. We also show that if E possesses at least one state, then E admits hidden variables in the sense that E is homomorphic to an MV-algebra that reproduces the states of E. Both of these results indicate that we cannot distinguish between a quantum mechanical effect algebra and a classical one. Hereditary (...) properties of sharpness and coexistence are discussed and the existence of {0,1} and dispersion-free states are considered. We then discuss a stronger structure called a sequential effect algebra (SEA) that we believe overcomes some of the inadequacies of an effect algebra. We show that a SEA is semiclassical if and only if it possesses an order-determining set of dispersion-free states. (shrink)

LetB be the set of bounded observables on a quantum logic. A mapJ: B →R is called an expectation functional ifJ is normalized, positive, continuous, and compatibly linear. Two questions are considered. IsJ linear, and isJ an expectation relative to some state? It is shown that the answers are affirmative for hidden variable logics and most Hilbert space logics. An example is given which shows thatJ can be nonlinear on an arbitrary quantum logic.

This article begins with a review of the framework of fuzzy probability theory. The basic structure is given by the σ-effect algebra of effects (fuzzy events) $\mathcal{E}{\text{ }}\left( {\Omega ,\mathcal{A}} \right)$ and the set of probability measures $M_1^ + {\text{ }}\left( {\Omega ,\mathcal{A}} \right)$ on a measurable space $\left( {\Omega ,\mathcal{A}} \right)$ . An observable $X:\mathcal{B} \to {\text{ }}\mathcal{E}{\text{ }}\left( {\Omega ,\mathcal{A}} \right)$ is defined, where $\begin{gathered} X:\mathcal{B} \to {\text{ }}\mathcal{E}{\text{ }}\left( {\Omega ,\mathcal{A}} \right) \\ \left( {\Lambda ,{\text{ }}\mathcal{B}} \right) (...) \\ \end{gathered} $ is the value space of X. It is noted that there exists a one-to-one correspondence between states on $\mathcal{E} \left( {\Omega ,\mathcal{A} } \right)$ and elements of $M_1^ + {\text{ }}\left( {\Omega ,\mathcal{A} } \right)$ and between observables $X:\mathcal{B} \to \mathcal{E} \left( {\Omega ,\mathcal{A} } \right)$ and σ-morphisms from $\mathcal{E} \left( {\Lambda , \mathcal{B}} \right)$ to $\mathcal{E} \left( {\Omega , \mathcal{A}} \right)$ . Various combinations of observables are discussed. These include compositions, products, direct products, and mixtures. Fuzzy stochastic processes are introduced and an application to quantum dynamics is considered. Quantum effects are characterized from among a more general class of effects. An alternative definition of a statistical map $M_1^ + {\text{ }}\left( {\Lambda ,\mathcal{B} } \right)$ is given and it is shown that any statistical map has a unique extension to a statistical operator. Finally, various combinations of statistical maps are discussed and their relationships to the corresponding combinations of observables are derived. (shrink)

Observables on hypergraphs are described by event-valued measures. We first distinguish between finitely additive observables and countably additive ones. We then study the spectrum, compatibility, and functions of observables. Next a relationship between observables and certain functionals on the set of measures M(H) of a hypergraph H is established. We characterize hypergraphs for which every linear functional on M(H) is determined by an observable. We define the concept of an “effect” and show that observables are related to effect-valued measures. Finally, (...) we define “operational” transformations from M(H) to itself and show that they can be described as a certain combination of effects. (shrink)

A discrete quantum mechanics is developed and used to construct models for discrete space-time and for the internal dynamics of elementary particles. This dynamics is given in terms of particles performing a quantum random walk on a multigraph.

Although a few new results are presented, this is mainly a review article on the relationship between finite-dimensional quantum mechanics and finite groups. The main motivation for this discussion is the hidden subgroup problem of quantum computation theory. A unifying role is played by a mathematical structure that we call a Hilbert *-algebra. After reviewing material on unitary representations of finite groups we discuss a generalized quantum Fourier transform. We close with a presentation concerning position-momentum measurements in this framework.

Quantum stochastic models are developed within the framework of a measure entity. An entity is a structure that describes the tests and states of a physical system. A measure entity endows each test with a measure and equips certain sets of states as measurable spaces. A stochastic model consists of measurable realvalued function on the set of states, called a generalized action, together with measures on the measurable state spaces. This structure is then employed to compute quantum probabilities of test (...) outcomes. We characterize those measure entities that are isomorphic to a quantum probability space. We also show that stochastic models provide a phase space description of quantum mechanics and a realistic model of spin. (shrink)

We first define a class of processes which we call regular quantum Markov processes. We next prove some basic results concerning such processes. A method is given for constructing quantum Markov processes using transition amplitude kernels. Finally we show that the Feynman path integral formalism can be clarified by approximating it with a quantum stochastic process.

It is frequently argued that reality and locality are incompatible with the predictions of quantum mechanics. Various investigators have used this as evidence for the existence of hidden variables. However, Bell's inequalities seem to refute this possibility. Since the above arguments are made within the framework of conventional probability theory, we contend that an alternative solution can be found by an extension of this theory. Elaborating on some ideas of I. Pitowski, we show that within the framework of a generalized (...) probability theory, reality, locality, hidden variables, and the predictions of quantum mechanics can be maintained together. Although our principal model in this work is a spin system, there are indications that this program can be extended to more general systems. (shrink)

We present a realistic model in which spin measurements are represented by functions. By employing a simple amplitude density, we derive the usual spin distributions and matrices for the spin-1/2 case. The spin-1 case is also considered. Moreover, we derive the amplitude density itself from deeper principles involving a real-valued influence function.

We summarize a recent search for quantum reality. The full anhomomorphic logic of coevents for an event set is introduced. The quantum integral over an event with respect to a coevent is defined. Reality filters such as preclusivity and regularity of coevents are considered. A quantum measure that can be represented as a quantum integral with respect to a coevent is said to 1-generate that coevent. This gives a stronger filter that may produce a unique coevent called the “actual reality” (...) for a physical system. What we believe to be a more general filter is defined in terms of a double quantum integral and is called 2-generation. It is noted that ordinary measures do not 1 or 2-generate coevents except in a few simple cases. Various general results are stated. (shrink)

We present a survey of a finite-dimensional quark model. We begin with a discussion of measurements on a quantum logic. After making the fundamental assumption that there are three basic colors, the measurement theory provides a natural embedding of the quantum logic into a finite-dimensional Hilbert space. This Hilbert space represents the space of pure quark states. Finite-dimensional quantum mechanics is discussed and the color, and flavor observables are derived. Quark and baryon Hamiltonians are proposed, and a brief description of (...) quark field theory is given. (shrink)

This paper outlines a framework that may provide a mathematically rigorous quantum field theory. The framework relies upon the methods of nonstandard analysis. A theory of nonstandard inner product spaces and operators on these spaces is first developed. This theory is then applied to construct nonstandard Fock spaces which extend the standard Fock spaces. Then a rigorous framework for the field operators of quantum field theory is presented. The results are illustrated for the case of Klein-Gordon fields.

A transition effect matrix (TEM) is a quantum generalization of a classical stochastic matrix. By employing a TEM we obtain a quantum generalization of a classical Markov chain. We first discuss state and operator dynamics for a quantum Markov chain. We then consider various types of TEMs and vector states. In particular, we study invariant, equilibrium and singular vector states and investigate projective, bistochastic, invertible and unitary TEMs.

Various axiomatic models for unsharp quantum measurements are investigated. These include effect spaces (E-spaces), effect test spaces (E-test spaces), effect algebras, and test groups. It is shown that a test group G is the universal group of an E-test space if and only if G is strongly atomistic. It follows that if G is strongly atomistic, then G is an interpolation group. We then demonstrate that if G is an interpolation group, then G is the universal group of an E-space. (...) Finally, it is shown that an E-space is isomorphic to an E-test space if and only if it is strongly atomistic. (shrink)

The article begins with a discussion of sets and fuzzy sets. It is observed that identifying a set with its indicator function makes it clear that a fuzzy set is a direct and natural generalization of a set. Making this identification also provides simplified proofs of various relationships between sets. Connectives for fuzzy sets that generalize those for sets are defined. The fundamentals of ordinary probability theory are reviewed and these ideas are used to motivate fuzzy probability theory. Observables (fuzzy (...) random variables) and their distributions are defined. Some applications of fuzzy probability theory to quantum mechanics and computer science are briefly considered. (shrink)

We define two types of convergence for observables on a quantum logic which we call M-weak and uniform M-weak convergence. These convergence modes correspond to weak convergence of probability measures. They are motivated by the idea that two (in general unbounded) observables are “close” if bounded functions of them are “close.” We show that M-weak and uniform M-weak convergence generalize strong resolvent and norm resolvent convergence for self-adjoint operators on a Hilbert space. Also, these types of convergence strengthen the weak (...) operator convergence and operator norm convergence of bounded self-adjoint operators on a Hilbert space. Finally, we consider spectral perturbation by showing that the spectra of approximating observables approach the spectrum of the limit in a certain sense. (shrink)