Sequential Effect Algebras Research Articles

Spectrality in Convex Sequential Effect Algebras

For convex and sequential effect algebras, we study spectrality in the sense of Foulis. We show that under additional conditions (strong archimedeanity, closedness in norm and a certain monotonicity property of the sequential product), such effect algebra is spectral if and only if every maximal commutative subalgebra is monotone sigma -complete. Two previous results on existence of spectral resolutions in this setting are shown to require stronger assumptions.

Exponential entropy on sequential effect algebras

Exponential entropy on sequential effect algebras

The three types of normal sequential effect algebras

A sequential effect algebra (SEA) is an effect algebra equipped with a sequential product operation modeled after the Lüders product (a,b)↦aba on C∗-algebras. A SEA is called normal when it has all suprema of directed sets, and the sequential product interacts suitably with these suprema. The effects on a Hilbert space and the unit interval of a von Neumann or JBW algebra are examples of normal SEAs that are in addition convex, i.e. possess a suitable action of the real unit interval on the algebra. Complete Boolean algebras form normal SEAs too, which are convex only when 0=1.We show that any normal SEA E splits as a direct sum E=Eb⊕Ec⊕Eac of a complete Boolean algebra Eb, a convex normal SEA Ec, and a newly identified type of normal SEA Eac we dub purely almost-convex.Along the way we show, among other things, that a SEA which contains only idempotents must be a Boolean algebra; and we establish a spectral theorem using which we settle for the class of normal SEAs a problem of Gudder regarding the uniqueness of square roots. After establishing our main result, we propose a simple extra axiom for normal SEAs that excludes the seemingly pathological a-convex SEAs. We conclude the paper by a study of SEAs with an associative sequential product. We find that associativity forces normal SEAs satisfying our new axiom to be commutative, shedding light on the question of why the sequential product in quantum theory should be non-associative.

Rényi Entropy and Rényi Divergence in Sequential Effect Algebra

The aim of this study is to extend the results concerning the Shannon entropy and Kullback–Leibler divergence in sequential effect algebra to the case of Rényi entropy and Rényi divergence. For this purpose, the Rényi entropy of finite partitions in sequential effect algebra and its conditional version are proposed and the basic properties of these entropy measures are derived. In addition, the notion of Rényi divergence of a partition in sequential effect algebra is introduced and the basic properties of this quantity are studied. In particular, it is proved that the Kullback–Leibler divergence and Shannon’s entropy of partitions in a given sequential effect algebra can be obtained as limits of their Rényi divergence and Rényi entropy respectively. Finally, to illustrate the results, some numerical examples are presented.

Commutativity in Jordan operator algebras

While Jordan algebras are commutative, operator commutativity will behave badly for a general non-associative Jordan algebra. When the product operators of two elements commute, the elements are said to operator commute. In some Jordan algebras operator commutation can be badly behaved, for instance having elements a and b operator commute, while a2 and b do not operator commute. In this paper we study JB-algebras, real Jordan algebras which are also Banach spaces in a compatible manner. These include for instance the spaces of self-adjoint elements of unital C⁎-algebras. We show that elements a and b in a JB-algebra operator commute if and only if they generate an associative subalgebra of mutually operator commuting elements, and hence operator commutativity in JB-algebras is as well-behaved as it can be. Letting Ua denote the quadratic operator of a, we also show that positive a and b operator commute if and only if Uab2=Uba2. We use this result to conclude that the unit interval of a JB-algebra is a sequential effect algebra as defined by Gudder and Greechie.

On the properties of spectral effect algebras

The aim of this paper is to show that there can be either only one or uncountably many contexts in any spectral effect algebra, answering a question posed in [S. Gudder, Convex and Sequential Effect Algebras, (2018), arXiv:1802.01265]. We also provide some results on the structure of spectral effect algebras and their state spaces and investigate the direct products and direct convex sums of spectral effect algebras. In the case of spectral effect algebras with sharply determining state space, stronger properties can be proved: the spectral decompositions are essentially unique, the algebra is sharply dominating and the set of its sharp elements is an orthomodular lattice. The article also contains a list of open questions that might provide interesting future research directions.

The Orthogonality of Sequential Products in Sequential Effect Algebras

An effect algebra (EA) is an algebraic structure useful in quantum logic. A sequential effect algebra (SEA) is an EA with a sequential product additionally defined. SEAs are useful in the study of quantum measurement theory and by extension quantum computation. A paper published by Gudder in 2005 posited multiple open problems about sequential effect algebras. Among them was the following question: if a and b are elements of a sequential effect algebra, and a is orthogonal to b, is it the case that the sequential product of a and b is orthogonal to itself? In this paper, we prove his assertion.

Contexts in Convex and Sequential Effect Algebras

A convex sequential effect algebra (COSEA) is an algebraic system with three physically motivated operations, an orthogonal sum, a scalar product and a sequential product. The elements of a COSEA correspond to yes-no measurements and are called effects. In this work we stress the importance of contexts in a COSEA. A context is a finest sharp measurement and an effect will act differently according to the underlying context with which it is measured. Under a change of context, the possible values of an effect do not change but the way these values are obtained may be different. In this paper we discuss direct sums and the center of a COSEA. We also consider conditional probabilities and the spectra of effects. Finally, we characterize COSEA's that are isomorphic to COSEA's of positive operators on a complex Hilbert space. These result in the traditional quantum formalism. All of this work depends heavily on the concept of a context.

An introduction of logical entropy on sequential effect algebra

The main aim of this study was to introduce logical entropy on dynamical systems that their state spaces were sequential effect algebra. In this regard, logical partition was defined on sequential effect algebra and then based on logical partition concept, logical entropy on partitions, conditional logical entropy, and logical entropy on dynamical systems were introduced and their features were analyzed. In addition, it was proved that this entropy is an invariant object under isomorphism relation.

Continuity of the sequential product of sequential quantum effect algebras

In order to study quantum measurement theory, sequential product defined by A∘B = A1/2BA1/2 for any two quantum effects A, B has been introduced. Physically motivated conditions ask the sequential product to be continuous with respect to the strong operator topology. In this paper, we study the continuity problems of the sequential product A∘B = A1/2BA1/2 with respect to other important topologies, such as norm topology, weak operator topology, order topology, and interval topology.

Unified (r, s)-Entropies of Partitions on Sequential Effect Algebras

By using the partitions and refinements of sequential effect algebras (SEAs), we introduce the unified (r, s)-entropies and unified (r, s)-conditional entropies of partitions on SEAs. Several important relations of the unified (r, s)-entropies on SEAs are established, which reflect the noncommutativity of quantum effects.

Remarks on Entropy of Partition on the Sequential Effect Algebras

In this paper, for a given quantum state s on the sequential effect algebra, we introduce the sequential independence of two partitions and refinements with respect to the quantum state s. Under these conditions, we study some interesting properties of partition entropy.

Atomic effect algebras with compression bases

Compression base effect algebras were recently introduced by Gudder [Demonstr. Math. 39, 43 (2006)]. They generalize sequential effect algebras [Rep. Math. Phys. 49, 87 (2002)] and compressible effect algebras [Rep. Math. Phys. 54, 93 (2004)]. The present paper focuses on atomic compression base effect algebras and the consequences of atoms being foci (so-called projections) of the compressions in the compression base. Part of our work generalizes results obtained in atomic sequential effect algebras by Tkadlec [Int. J. Theor. Phys. 47, 185 (2008)]. The notion of projection-atomicity is introduced and studied, and several conditions that force a compression base effect algebra or the set of its projections to be Boolean are found. Finally, we apply some of these results to sequential effect algebras and strengthen a previously established result concerning a sufficient condition for them to be Boolean.

Some Problems for Sequential Effect Algebras

In 2005, Professor Gudder presented 25 open problems of sequential effect algebras, and in this paper, we survey the development of these problems, and we point out two new interesting topics in the sequential effect algebras, too.

Mutual Information and Relative Entropy of Sequential Effect Algebras

In this paper, we introduce and investigate the mutual information and relative entropy on the sequential effect algebra, we also give a comparison of these mutual information and relative entropy with the classical ones by the venn diagrams. Finally, a nice example shows that the entropies of sequential effect algebra depend extremely on the order of its sequential product.

Sequential Product of Quantum Effects: An Overview

This article presents an overview for the theory of sequential products of quantum effects. We first summarize some of the highlights of this relatively recent field of investigation and then provide some new results. We begin by discussing sequential effect algebras which are effect algebras endowed with a sequential product satisfying certain basic conditions. We then consider sequential products of (discrete) quantum measurements. We next treat transition effect matrices (TEMs) and their associated sequential product. A TEM is a matrix whose entries are effects and whose rows form quantum measurements. We show that TEMs can be employed for the study of quantum Markov chains. Finally, we prove some new results concerning TEMs and vector densities.

The nth root of sequential effect algebras

In 2005, Gudder [Int. J. Theor. Phys. 44, 2219 (2005)] presented 25 problems of sequential effect algebras, the 20th problem asked: In a sequential effect algebra, if the square root of some element exists, is it unique? In this paper, we show that for each given positive integer n>1, there is a sequential effect algebra such that the nth root of its some element c is not unique, and the nth root of c is not the kth root of c (k<n). Thus, we answer the problem negatively.

Entropy of Partitions on Sequential Effect Algebras

By using the sequential effect algebra theory, we establish the partitions and refinements of quantum logics and study their entropies.

Effect Algebras Are Not Adequate Models for Quantum Mechanics

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.

The average value inequality in sequential effect algebras

A sequential effect algebra (E, 0, 1, ⊕, ∘) is an effect algebra on which a sequential product ∘ with certain physics properties is defined; in particular, sequential effect algebra is an important model for studying quantum measurement theory. In 2005, Gudder asked the following problem: If a, b ∈, (E, 0, 1, ⊕, ∘) and a⊥b and a ∘ b⊥a ∘ b, is it the case that 2(a ∘ b) ≤ a 2 ⊕ b 2? In this paper, we construct an example to answer the problem negatively.