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Stan Gudder

Publications and source records attributed to Stan Gudder.

66 records · Page 4Linked to original sources
arXiv2011

Discrete Quantum Processes

A discrete quantum process is defined as a sequence of local states $ρ_t$, $t=0,1,2,...$, satisfying certain conditions on an $L_2$ Hilbert space $H$. If $ρ=\limρ_t$ exists, then $ρ$ is called a global state for the system. In important cases, the global state does not exist and we must then work with the local states. In a natural way, the local states generate a sequence of quantum measures which in turn define a single quantum measure $μ$ on the algebra of cylinder sets $\cscript$. We consider the problem of extending $μ$ to other physically relevant sets in a systematic way. To this end we show that $μ$ can be properly extended to a quantum measure $\mutilde$ on a "quadratic algebra" containing $\cscript$. We also show that a random variable $f$ can be "quantized" to form a self-adjoint operator $\fhat$ on $H$. We then employ $\fhat$ to define a quantum integral $\int fd\mutilde$. Various examples are given

math-ph↗
arXiv2011

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 $μ(A)=D(A,A)$ on $H$. We show that these operators have certain positivity, additivity and continuity properties. If $ρ$ is a state on $H$, then $D_ρ(A,B)=\rmtr\sqbrac{ρD(A,B)}$ and $μ_ρ(A)=D_ρ(A,A)$ 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 $\int fdμ_ρ=\rmtr (ρ\fhat\,)$. A tail-sum formula is proved for the quantum integral. The paper closes with an example that illustrates some of the theory.

math-ph↗
arXiv2011

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 $μ_n$ on the space of $n$-paths, and the $μ_n$ in turn induce a quantum measure $μ$ on the cylinder sets within the space $Ω$ of untruncated paths. Although $μ$ cannot be extended to a continuous quantum measure on the full $σ$-algebra generated by the cylinder sets, an important question is whether it can be extended to sufficiently many physically relevant subsets of $Ω$ in a systematic way. We begin an investigation of this problem by showing that $μ$ can be extended to a quantum measure on a "quadratic algebra" of subsets of $Ω$ that properly contains the cylinder sets. We also present a new characterization of the quantum integral on the $n$-path space.

quant-ph↗
arXiv2010

Hilbert Space Representations of Decoherence Functionals and Quantum Measures

We show that any decoherence functional $D$ can be represented by a spanning vector-valued measure on a complex Hilbert space. Moreover, this representation is unique up to an isomorphism when the system is finite. We consider the natural map $U$ from the history Hilbert space $K$ to the standard Hilbert space $H$ of the usual quantum formulation. We show that $U$ is an isomorphism from $K$ onto a closed subspace of $H$ and that $U$ is an isomorphism from $K$ onto $H$ if and only if the representation is spanning. We then apply this work to show that a quantum measure has a Hilbert space representation if and only if it is strongly positive. We also discuss classical decoherence functionals, operator-valued measures and quantum operator measures.

quant-ph↗
arXiv2010

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 experiment

quant-ph↗
arXiv2010

Quantum measures and the coevent interpretation

This paper first reviews quantum measure and integration theory. A new representation of the quantum integral is presented. This representation is illustrated by computing some quantum (Lebesgue)${}^2$ integrals. The rest of the paper only considers finite spaces. Anhomomorphic logics are discussed and the classical domain of a coevent is studied. Pure quantum measures and coevents are considered and it is shown that pure quantum measures are strictly contained in the extremal elements for the set of quantum measures bounded above by one. Moreover, we prove that any quantum measure on a finite event space $\ascript$ can be transferred to an ordinary measure on an anhomomorphic logic $\ascript ^*$. In this way, the quantum dynamics on $\ascript$ can be described by a classical dynamics on the larger space $\ascript ^*$.

quant-ph↗
arXiv2009

Quantum integrals and anhomomorphic logics

The full anhomomorphic logic of coevents $\ascript ^*$ is introduced. Atoms of $\ascript ^*$ and embeddings of the event set $\ascript$ into $\ascript ^*$ are discussed. The quantum integral over an event $A$ with respect to a coevent $ϕ$ is defined and its properties are treated. Integrals with respect to various coevents are computed. Reality filters such as preclusivity and regularity of coevents are considered. A quantum measure $μ$ that can be represented as a quantum integral with respect to a coevent $ϕ$ is said to 1-generate $ϕ$. This gives a stronger reality 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 shown that ordinary measures do not 1 or 2-generate coevents except in a few simple cases. Examples are given which show that there are quantum measures that 2-generate but do not 1-generate coevents. Examples also show that there are coevents that are 2-generated but not 1-generated. For simplicity only finite systems are considered.

quant-ph↗
arXiv2009

Examples of quantum integrals

We first consider a method of centering and a change of variable formula for a quantum integral. We then present three types of quantum integrals. The first considers the expectation of the number of heads in $n$ flips of a "quantum coin". The next computes quantum integrals for destructive pairs examples. The last computes quantum integrals for a (Lebesgue)^2 quantum measure. For this last type we prove some quantum counterparts of the fundamental theorem of calculus.

quant-ph↗
arXiv2009

An anhomomorphic logic for quantum mechanics

Although various schemes for anhomomorphic logics for quantum mechanics have been considered in the past we shall mainly concentrate on the quadratic or grade-2 scheme. In this scheme, the grade-2 truth functions are called coevents. We discuss properties of coevents, projections on the space of coevents and the master observable. We show that the set of projections forms an orthomodular poset. We introduce the concept of precluding coevents and show that this is stronger than the previously studied concept of preclusive coevents. Precluding coevents are defined naturally in terms of the master observable. A result that exhibits a duality between preclusive and precluding coevents is given. Some simple examples are presented.

quant-ph↗
arXiv2009

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.

quant-ph↗
arXiv2009

Boolean Inner product Spaces and Boolean Matrices

This article discusses the concept of Boolean spaces endowed with a Boolean valued inner product and their matrices. A natural inner product structure for the space of Boolean n-tuples is introduced. Stochastic boolean vectors and stochastic and unitary Boolean matrices are studied. A dimension theorem for orthonormal bases of a Boolean space is proven. We characterize the invariant stochastic Boolean vectors for a Boolean stochastic matrix and show that they can be used to reduce a unitary matrix. Finally, we obtain a result on powers of stochastic and unitary matrices.

math.RA↗