arXiv subjects

Stan Gudder

Publications and source records attributed to Stan Gudder.

At least 19 recordsLinked to original sources

arXiv2026

Quantum Conditional Stochastic Processes

Quantum mechanics contains certain novel mathematical concepts. Among these are complex numbers, Hilbert spaces with their unitary and self-adjoint operators, states represented by complex vectors, superpositions of states, collapse of wave functions, Born's rule for probabilities and others. If we accept that quantum mechanics is probabilistic, then these concepts can be derived and they become secondary. In this work, we begin with what we call a conditional stochastic process, which is based on real numbers and probabilities. As we shall see, such processes are defined by three simple axioms. We then use conditional stochastic processes to derive quantum mechanics by employing a correspondence called a dictionary. We also show that the converse holds. That is, beginning with a quantum system, we employ the dictionary to derive a conditional stochastic process.

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arXiv2025

Geometric Algebras and Fermion Quantum Field Theory

Corresponding to a finite dimensional Hilbert space $H$ with $\dim H=n$, we define a geometric algebra $\gscript (H)$ with $\dim\sqbrac{\gscript (H)}=2^n$. The algebra $\gscript (H)$ is a Hilbert space that contains $H$ as a subspace. We interpret the unit vectors of $H$ as states of individual fermions of the same type and $\gscript (H)$ as a fermion quantum field whose unit vectors represent states of collections of interacting fermions. We discuss creation operators on $\gscript (H)$ and provide their matrix representations. Evolution operators provided by self-adjoint Hamiltonians on $H$ and $\gscript (H)$ are considered. Boson-Fermion quantum fields are constructed. Extensions of operators from $H$ to $\gscript (H)$ are studied. Finally, we present a generalization of our work to infinite dimensional separable Hilbert spaces.

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arXiv2024

Uncertainty of Quantum States

The uncertainty of a quantum state is given by the composition of two components. The first is called the quantum component and is given by the probability distribution of an observable relative to the state. The second is the classical component which is an uncertainty function that is applied to the first component. We characterize uncertainty functions in terms of four axioms. We then study four examples called variance, entropy, geometric and sine uncertainty functions. The final section presents the general theory of state uncertainty.

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arXiv2024

Finite Classical and Quantum Effect Algebras

In this article, we only consider finite effect algebras. We define the concepts of classical and quantum effect algebras and show that an effect algebra $E$ is classical if and only if there exists an observable that measures every effect of $E$. We next consider matrix representations of effect algebras and prove an effect algebra is classical if and only if its matrix representation has precisely one row. We then discuss sum table for effect algebras. Although these are not as concise as matrix representations, they give more immediate information about effect sums which are the basic operations of an effect algebra. We subsequently study states on effect algebras and prove that classical effect algebras are quantum effect algebras. Finally, we consider composites of effect algebras. This allows us to study interacting systems described by effect algebras. We show that two effect algebras are classical if and only if their composite is classical. We point out that scale effect algebras are not the only classical effect algebras and stress the importance of atoms in this work.

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arXiv2024

Finite (quantum) effect algebras

We investigate finite effect algebras and their classification. We show that an effect algebra with $n$ elements has at least $n-2$ and at most $(n-1)(n-2)/2$ nontrivial defined sums. We characterize finite effect algebras with these minimal and maximal number of defined sums. The latter effect algebras are scale effect algebras (i.e., subalgebras of [0,1]), and only those. We prove that there is exactly one scale effect algebra with $n$ elements for every integer $n \geq 2$. We show that a finite effect algebra is quantum effect algebra (i.e. a subeffect algebra of the standard quantum effect algebra) if and only if it has a finite set of order-determining states. Among effect algebras with 2-6 elements, we identify all quantum effect algebras.

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arXiv2024

Quantum Transition Probabilities

Transition probabilities are an important and useful tool in quantum mechanics. However, in their present form, they are limited in scope and only apply to pure quantum states. In this article we extend their applicability to mixed states and to transitions between quantum effects. We also present their dependence on a measured operation or instrument. We begin by defining our concepts on a general quantum effect algebra. These concepts are illustrated using Holevo operations and instruments. We then present transition probabilities in the special case of the Hilbert space formulation of quantum mechanics. We show that for pure states and particular types of operations the transition probabilities reduce to their usual form. We give examples in terms of L\"uders operations and instruments.

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arXiv2024

Interval Effect Algebras and Holevo Instruments

This article begins with a study of convex effect-state spaces. We point out that such spaces are equivalent to interval effect algebras that generate an ordered linear space and possess an order-determining set of states. We then discuss operations and instruments on interval effect algebras under the assumption of an unrestrictive condition. Effects measured by operations and sequential products of effects relative to operations are considered. Observables are introduced and coexistence of effects are discussed. We also present properties of sequential products of observables and conditioning of observables related to instruments. The final section is devoted to Holevo instruments. Pure and mixed Holevo operations are defined and extended to instruments. The Holevo sequential product of two observables is defined and the marginals of these products are computed. We define the commutant of two effects and derive its properties. Examples are given that illustrate the properties of previously presented concepts.

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arXiv2024

Quantum Channel Conditioning and Measurement Models

If $H_1$ and $H_2$ are finite-dimensional Hilbert spaces, a channel from $H_1$ to $H_2$ is a completely positive, linear map $\mathcal{I}$ that takes the set of states $\mathcal{S}(H_1)$ for $H_1$ to the set of states $\mathcal{S}(H_2)$ for $H_2$. Corresponding to $\mathcal{I}$ there is a unique dual map $\mathcal{I}^*$ from the set of effects $\mathcal{E}(H_2)$ for $H_2$ to the set of effects $\mathcal{E}(H_1)$ for $H_1$. We call $\mathcal{I}^*(b)$ the effect $b$ conditioned by $\mathcal{I}$ and the set $\mathcal{I}^c = \mathcal{I}^*(\mathcal{E}(H_2))$ the conditioned set of $\mathcal{I}$. We point out that $\mathcal{I}^c$ is a convex subeffect algebra of the effect algebra $\mathcal{E}(H_1)$. We extend this definition to the conditioning $\mathcal{I}^*(B)$ for an observable $B$ on $H_2$ and say that an observable $A$ is in $\mathcal{I}^c$ if $A=\mathcal{I}^*(B)$ for some observable $B$. We show that $\mathcal{I}^c$ is closed under post-processing and taking parts. We also define the conditioning of instruments by channels. These concepts are illustrated using examples of Holevo instruments and channels. We next discuss measurement models and their corresponding observables and instruments. We show that calculations can be simplified by employing Kraus and Holevo separable channels. Such channels allow one to separate the components of a tensor product.

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arXiv2023

Four Related Combinatorial Problems

This pedagogical article solves an interesting problem in quantum measure theory. Although a quantum measure $\mu$ is a generalization of an ordinary probability measure, $\mu$ need not satisfy the usual additivity condition. Instead, $\mu$ satisfies a grade-2 additivity condition. Besides the quantum measure problem, we present three additional combinatorial problems. These are (1)\enspace A sum of binomial coefficients problem;\enspace (2)\enspace A recurrence relation problem; and\enspace (3)\enspace An interated vector problem. We show that these three problems are equivalent in that they have a common solution. We then show that this solves the original quantum measure problem.

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arXiv2023

Measurement Models with Separable Interaction Channels

Measurement models (MMs) stand at the highest structural level of quantum measurement theory. MMs can be employed to construct instruments which stand at the next level. An instrument is thought of as an apparatus that is used to measure observables and update states. Observables, which are still at the next level, are used to determine probabilities of quantum events. The main ingredient of an MM is an interaction channel $\nu$ between the system being measured and a probe system. For a general $\nu$, the measured observable $A$ need not have an explicit useful form. In this work we introduce a condition for $\nu$ called separability and in this case $A$ has an explicit form. Under the assumption that $\nu$ is separable, we study product MMs and conditioned MMs. We also consider the statistics of MMs and their uncertainty principle. Various concepts are illustrated using examples of L\"uders and Holevo instruments.

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arXiv2023

Multi-Observables and Multi-Instruments

This article introduces the concepts of multi-observables and multi-instruments in quantum mechanics. A multi-observable $A$ (multi-instrument $\mathcal{I}$) has an outcome space of the form $\Omega =\Omega _1\times\cdots\times\Omega _n$ and is denoted by $A_{x_1\cdots x_n}$ ($\mathcal{I}_{x_1\cdots x_n}$) where $(x_1,\ldots ,x_n)\in\Omega$. We also call $A$ ($\mathcal{I}$) an $n$-observable ($n$-instrument) and when $n=2$ we call $A$ ($\mathcal{I}$) a bi-observable (bi-instrument). We point out that bi-observables $A$ ($\mathcal{I}$) and bi-instruments have been considered in past literature, but the more general case appears to be new. In particular, two observables (instruments) have been defined to coexist or be compatible if they possess a joint bi-observable (bi-instrument). We extend this definition to $n$ observables and $n$ instruments by considering joint marginals of $n$-observables and joint reduced marginals of $n$-instruments. We show that a $n$-instrument measures a unique $n$-observable and if a finite umber of instruments coexist, then their measured observables coexist. We prove that there is a close relationship between a nontrivial $n$-observable and its parts. Moreover, a similar result holds for instruments. We next show that a natural definition for the tensor product of a finite number of instruments exist and possess reasonable properties. We then discuss sequential products of a finite number of observables and instruments. We present various examples such as Kraus, Holevo and L\"uders instruments.

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arXiv2022

Entropy of Quantum Measurements

If $a$ is a quantum effect and $\rho$ is a state, we define the $\rho$-entropy $S_a(\rho )$ which gives the amount of uncertainty that a measurement of $a$ provides about $\rho$. The smaller $S_a(\rho )$ is, the more information a measurement of $a$ gives about $\rho$. In Section~2, we provide bounds on $S_a(\rho )$ and show that if $a+b$ is an effect, then $S_{a+b}(\rho )\ge S_a(\rho )+S_b(\rho )$. We then prove a result concerning convex mixtures of effects. We also consider sequential products of effects and their $\rho$-entropies. In Section~3, we employ $S_a(\rho )$ to define the $\rho$-entropy $S_A(\rho )$ for an observable $A$. We show that $S_A(\rho )$ directly provides the $\rho$-entropy $S_\iscript (\rho )$ for an instrument $\iscript$. We establish bounds for $S_A(\rho )$ and prove characterizations for when these bounds are obtained. These give simplified proofs of results given in the literature. We also consider $\rho$-entropies for measurement models, sequential products of observables and coarse-graining of observables. Various examples that illustrate the theory are provided.

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arXiv2022

Dual Instruments and Sequential Products of Observables

We first show that every operation possesses an unique dual operation and measures an unique effect. If $a$ and $b$ are effects and $J$ is an operation that measures $a$, we define the sequential product of $a$ then $b$ relative to $J$. Properties of the sequential product are derived and are illustrated in terms of L\"uders and Holevo operations. We next extend this work to the theory of instruments and observables. We also define the concept of an instrument (observable) conditioned by another instrument (observable). Identity, state-constant and repeatable instruments are considered. Sequential products of finite observables relative to L\"uders and Holevo instruments are studied.

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arXiv2021

Mutually Unbiased Quantum Observables

We begin by defining mutually unbiased (MU) observables on a finite dimensional Hilbert space. We also consider the more general concept of parts of MU observables. The relationships between MU observables, value-complementary observables and two other conditions involving sequential products of observables are discussed. We next present a special motivating case of MU observables called finite position and momentum observables. These are atomic observables related by a finite Fourier transform. Finite position and momentum observables are employed to give examples of parts of MU observables that are value-complementary and those that are not value-complementary. Various open problems involving these concepts are presented. These problems mainly involve extending this work from sharp observables to unsharp observables.

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arXiv2021

Coarse-Graining of Observables

We first define the coarse-graining of probability measures in terms of stochastic kernels. We define when a probability measure is part of another probability measure and say that two probability measures coexist if they are both parts of a single probability measure. We then show that any two probability measures coexist. We extend these concepts to observables and instruments and mention that two observables need not coexist. We define the discretization of an observable as a special case of coarse-graining and show that these have \zeroone stochastic kernels. We next consider finite observables and instruments and show that in these cases, stochastic kernels are replaced by stochastic matrices. We also show that coarse-graining is the same as post-processing in this finite case. We then consider sequential products of observables and discuss the sequential product of a post-processed observable with another observable. We briefly discuss SIC observables and the example of qubit observables.

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arXiv2021

Sequential Products of Quantum Measurements

Our basic structure is a finite-dimensional complex Hilbert space $H$. We point out that the set of effects on $H$ form a convex effect algebra. Although the set of operators on $H$ also form a convex effect algebra, they have a more detailed structure. W introduce sequential products of effect and operations. Although these have already been studied, we introduce the new concept of sequential products of effects with operations and operations with effects. We then consider various special types of operations. After developing properties of these concepts, the results are generalized to include observables and instruments. In particular, sequential products of observables with instruments and instruments with observables are developed. Finally, we consider conditioning and coexistence of observables and instruments.

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arXiv2021

Time Evolution of Quantum Effects

For quantum effects $a$ and $b$ we define the $a$-evolution of $b$ at time $t$ denoted by $b(t\mid a)$. We interpret $b(t\mid a)$ as the influence that $a$ has on $b$ at time $t$ when $a$ occurs, but is not measured at time $t=0$. Using $b(t\mid a)$ we define the time-dependent sequential product $a[t]b$. This is interpreted as an effect that results from first measuring $a$ and then measuring $b$ after a time delay $t$. Various properties of $a[t]b$ are derived and it is shown that $a[t]b$ is constant in time if and only if $a$ and $b$ commute or $a$ is a multiple of a projection. These concepts are extended to observables for a quantum system. The ideas are illustrated with some examples.

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arXiv2020

Operator Isomorphisms on Hilbert Space Tensor Products

This article presents an isomorphism between two operator algebras $L_1$ and $L_2$ where $L_1$ is the set of operators on a space of Hilbert-Schmidt operators and $L_2$ is the set of operators on a tensor product space. We next compare our isomorphism to a well-known result called Choi's isomorphism theorem. The advantage of Choi's isomorphism is that it takes completely positive maps to positive operators. One advantage of our isomorphism is that it applies to infinite dimensional Hilbert spaces, while Choi's isomorphism only holds for finite dimensions. Also, our isomorphism preserves operator products while Choi's does not. We close with a brief discussion on some uses of our isomorphism.

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