Central Elements of Atomic Effect Algebras

Various conditions ensuring that an atomic effect algebra is a Boolean algebra are presented. PACS: 02.10.-v.     

Central Elements of Effect Algebras

Central elements of an effect algebra can be characterized by means of a weak form of distributivity and a maximality property. We give examples where both conditions are fulfilled.     

Sharp Elements in Effect Algebras

We show that if for an arbitrary pair of orthogonal sharp elements of an effect algebra E its join exists and is sharp, then the set E_S of all sharp elements of E is a subeffect algebra of E that is an orthomodular poset. Such effect algebras need not be sharply dominating but S-dominating. Further, we show that in every nonproper effect algebra E, E_S is a subeffect algebra that is an orthomodular poset. Moreover, a general theorem for E_S is proved.     

Sharp and Fuzzy Observables on Effect Algebras

Observables on effect algebras and their fuzzy versions obtained by means of confidence measures (Markov kernels) are studied. It is shown that, on effect algebras with the (E)-property, given an observable and a confidence measure, there exists a fuzzy version of the observable. Ordering of observables according to their fuzzy properties is introduced, and some minimality conditions with respect to this ordering are found. Applications of some results of classical theory of experiments are considered. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-032002, grant VEGA 2/6088/26 and Center of Excellence SAS, CEPI I/2/2005.     

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.     

Product Effect Algebras

We introduce a product on an effect algebra. We prove that every product effect algebra with the Riesz decomposition property (RDP), is an interval in an Abelian unital interpolation po-ring, and we show that the category of product effect algebras with the RDP is categorically equivalent with the category of unital Abelian interpolation po-rings. In addition, we show that every product effect algebra with the RDP and with 1 as a product unity is a subdirect product of antilattice product effect algebras with the RDP.     

Pastings of MV-Effect Algebras

We give a method of constructing a lattice effect algebra E with given family of blocks. The given MV-effect algebras are pasted along some common sub-MV-effect algebras in a such manner that there exists an ((o)-continuous) state on the pasting E.     

激发k玻色子q相干态的反聚束效应

构造了激发k玻色子q相干态a+mqz,k,j〉q(k≥3),并用数值计算的方法研究了参数m对反聚束效应的影响.结果表明:反聚束效应明显受到m的调节.对于大x(x=z2),激发(即增加光子)改变了反聚束效应出现的区间.对于小x,当j=0时,激发使原来强烈的聚束效应变为强烈的反聚束效应;当j≠0时,激发后基本上保持着原来强烈的反聚束效应     

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.     

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.     

Atomic Sequential Effect Algebras

Various conditions ensuring that a sequential effect algebra or the set of sharp elements of a sequential effect algebra is a Boolean algebra are presented.     

Subdirect Decompositions of Lattice Effect Algebras

We prove a theorem about subdirect decompositions of lattice effect algebras. Further, we show how, under these decompositions, blocks, sets of sharp elements and centers of those effects algebras are decomposed. As an application we prove a statement about the existence of subadditive state on some block-finite effect algebras.     

Casimir Elements for Some Graded Lie Algebras and Superalgebras

We consider a class of Lie algebras L such that L admits a grading by a finite Abelian group so that each nontrivial homogeneous component is one-dimensional. In particular, this class contains simple Lie algebras of types A, C and D where in C and D cases the rank of L is a power of 2. We give a simple construction of a family of central elements of the universal enveloping algebra U(L). We show that for the A-type Lie algebras the elements coincide with the Gelfand invariants and thus generate the center of U(L). The construction can be extended to Lie superalgebras with the additional assumption that the group grading is compatible with the parity grading.     

激发双参数变形奇偶相干态的反聚束效应

通过在双参数变形奇偶qs相干态上重复作用玻色产生算符,构造了激发奇qs相干态aqs+m|α〉qso和激发偶qs相干态aqs+m|α〉qse,并用数值计算研究了参数m,s和q对反聚束效应的影响.结果显示:(1)当r较小时,对于偶qs相干态,激发可使原来强烈的聚束效应变为强烈的反聚束效应;(2)当q(q≤1)偏离1较远时,随着r2的增大,二阶qs相关函数出现振荡现象(即反聚束效应和聚束效应交替出现),其振幅和周期都随s和q的减小而增大,但不明显受m的调节;(3)当q→1时,二阶相关函数同样出现振荡现象,其振幅和周期不但随着s的减小而增大,而且明显地受到参数m的调节;(4)对于大多数r,二阶qs相关函数对s的敏感度大于对q的敏感度.     

Generalized Effect Algebras of Positive Operators Densely Defined on Hilbert Spaces

Axioms of quantum structures, motivated by properties of some sets of linear operators in Hilbert spaces are studied. Namely, we consider examples of sets of positive linear operators defined on a dense linear subspace D in a (complex) Hilbert space ℋ. Some of these operators may have a physical meaning in quantum mechanics. We prove that the set of all positive linear operators with fixed such D and ℋ form a generalized effect algebra with respect to the usual addition of operators. Some sub-algebras are also mentioned. Moreover, on a set of all positive linear operators densely defined in an infinite dimensional complex Hilbert space, the partial binary operation is defined making this set a generalized effect algebra.     

States on Pseudo Effect Algebras and Integrals

We show that every state on an interval pseudo effect algebra E satisfying an appropriate version of the Riesz Decomposition Property (RDP for short) is an integral through a regular Borel probability measure defined on the Borel σ-algebra of a Choquet simplex K. In particular, if E satisfies the strongest type of RDP, the representing Borel probability measure can be uniquely chosen to have its support in the set of the extreme points of K.