Quantum measure and integration theory

This article begins with a review of quantum measure spaces. Quantum forms and indefinite inner-product spaces are then discussed. The main part of the paper introduces a quantum integral and derives some of its properties. The quantum integral's form for simple functions is characterized and it is shown that the quantum integral generalizes the Lebesgue integral. A bounded, monotone convergence theorem for quantum integrals is obtained and it is shown that a Radon-Nikodym type theorem does not hold for quantum measures. As an example, a quantum-Lebesgue integral on the real line is considered.Comment: 28 page

Two-Site Quantum Random Walk

We study the measure theory of a two-site quantum random walk. The truncated decoherence functional defines a quantum measure $\mu_n$ on the space of $n$-paths, and the $\mu_n$ in turn induce a quantum measure $\mu$ on the cylinder sets within the space $\Omega$ of untruncated paths. Although $\mu$ cannot be extended to a continuous quantum measure on the full $\sigma$-algebra generated by the cylinder sets, an important question is whether it can be extended to sufficiently many physically relevant subsets of $\Omega$ in a systematic way. We begin an investigation of this problem by showing that $\mu$ can be extended to a quantum measure on a "quadratic algebra" of subsets of $\Omega$ that properly contains the cylinder sets. We also present a new characterization of the quantum integral on the $n$-path space.Comment: 28 page

Quantum measures and integrals

We show that quantum measures and integrals appear naturally in any $L_2$-Hilbert space $H$. We begin by defining a decoherence operator $D(A,B)$ and it's associated $q$-measure operator $\mu (A)=D(A,A)$ on $H$. We show that these operators have certain positivity, additivity and continuity properties. If $\rho$ is a state on $H$, then D_\rho (A,B)=\rmtr\sqbrac{\rho D(A,B)} and $\mu_\rho (A)=D_\rho (A,A)$ have the usual properties of a decoherence functional and $q$-measure, respectively. The quantization of a random variable $f$ is defined to be a certain self-adjoint operator \fhat on $H$. Continuity and additivity properties of the map f\mapsto\fhat are discussed. It is shown that if $f$ is nonnegative, then \fhat is a positive operator. A quantum integral is defined by \int fd\mu_\rho =\rmtr (\rho\fhat\,). A tail-sum formula is proved for the quantum integral. The paper closes with an example that illustrates some of the theory.Comment: 16 page

Uniqueness and order in sequential effect algebras

A sequential effect algebra (SEA) is an effect algebra on which a sequential product is defined. We present examples of effect algebras that admit a unique, many and no sequential product. Some general theorems concerning unique sequential products are proved. We discuss sequentially ordered SEA's in which the order is completely determined by the sequential product. It is demonstrated that intervals in a sequential ordered SEA admit a sequential product

The Universe and The Quantum Computer

It is first pointed out that there is a common mathematical model for the universe and the quantum computer. The former is called the histories approach to quantum mechanics and the latter is called measurement based quantum computation. Although a rigorous concrete model for the universe has not been completed, a quantum measure and integration theory has been developed which may be useful for future progress. In this work we show that the quantum integral is the unique functional satisfying certain basic physical and mathematical principles. Since the set of paths (or trajectories) for a quantum computer is finite, this theory is easier to treat and more developed. We observe that the sum of the quantum measures of the paths is unity and the total interference vanishes. Thus, constructive interference is always balanced by an equal amount of destructive interference. As an example we consider a simplified two-slit experimentComment: 15 pages, IQSA 2010 proceeding

The structure of classical extensions of quantum probability theory

On the basis of a suggestive definition of a classical extension of quantum mechanics in terms of statistical models, we prove that every such classical extension is essentially given by the so-called Misra–Bugajski reduction map. We consider how this map enables one to understand quantum mechanics as a reduced classical statistical theory on the projective Hilbert space as phase space and discuss features of the induced hidden-variable model. Moreover, some relevant technical results on the topology and Borel structure of the projective Hilbert space are reviewed

Models for Discrete Quantum Gravity

We first discuss a framework for discrete quantum processes (DQP). It is shown that the set of q-probability operators is convex and its set of extreme elements is found. The property of consistency for a DQP is studied and the quadratic algebra of suitable sets is introduced. A classical sequential growth process is "quantized" to obtain a model for discrete quantum gravity called a quantum sequential growth process (QSGP). Two methods for constructing concrete examples of QSGP are provided.Comment: 15 pages which include 2 figures which were created using LaTeX and contained in the fil

Spectral representation of infimum of bounded quantum observables

In 2006, Gudder introduced a logic order on bounded quantum observable set $S(H)$. In 2007, Pulmannova and Vincekova proved that for each subset $\cal D$ of $S(H)$, the infimum of $\cal D$ exists with respect to this logic order. In this paper, we present the spectral representation for the infimum of $\cal D$

Two quantum Simpson's paradoxes

The so-called Simpson's "paradox", or Yule-Simpson (YS) effect, occurs in classical statistics when the correlations that are present among different sets of samples are reversed if the sets are combined together, thus ignoring one or more lurking variables. Here we illustrate the occurrence of two analogue effects in quantum measurements. The first, which we term quantum-classical YS effect, may occur with quantum limited measurements and with lurking variables coming from the mixing of states, whereas the second, here referred to as quantum-quantum YS effect, may take place when coherent superpositions of quantum states are allowed. By analyzing quantum measurements on low dimensional systems (qubits and qutrits), we show that the two effects may occur independently, and that the quantum-quantum YS effect is more likely to occur than the corresponding quantum-classical one. We also found that there exist classes of superposition states for which the quantum-classical YS effect cannot occur for any measurement and, at the same time, the quantum-quantum YS effect takes place in a consistent fraction of the possible measurement settings. The occurrence of the effect in the presence of partial coherence is discussed as well as its possible implications for quantum hypothesis testing.Comment: published versio