DOCSLIB.ORG

Sequential Products on Effect Algebras

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Sequential Products on Effect Algebras SEQUENTIAL PRODUCTS ON EFFECT ALGEBRAS Stan Gudder Richard Greechie Department of Mathematics Department of rvIathematics U niversity of Denver Louisiana Tech University Denver Colorado 80208 Rusto且， Louisiana 71272 [email protected] [email protected] Abstract A sequential effect algebra (SEA) is an effect algebra on which a sequential product with natural properties is defined. The propertíes of sequential products on Hi1bert space effect algebras are discussed. For a general SEA, relationships between. sequential independence. coexistence and compatibility are given. It is shown that the sharp elements of a SEA form an orthomodular poset. The sequential center of a SEA is discussed and a characterization of when the sequential center ís isomorphic to a fuzzy set system is presented. It is shown that the existence of a sequential product is a strong restriction that eliminates many effect algebras frorn being SEA's. For example, there are no finite nonboolean SEA's. A measure of sharpness called he sharpness index is studied. The existence of horizontal sums of SEA's is characterized and exarnples of horizontal sums and tensor products are presented. 1 Introd uction Two measurement5 αand b cannot .be performed 5imultaneously in generaL 50 they are frequently executed 5equentially. 飞斗飞~ denote by α 。 b a sequen­ tial measurement in whichαis performed first and b second. \Ve cal1 α 。 b l the sequential product of αand b. \Ve shal1 restrict our attention to yes-no measurements. called effects, which have only two possible results. For gen­ erality, we do not 臼sume that effects are perfectly accurate me臼urements. That is. they may be fuzzy or unsharp. :也5 we shall see, the sharp effects are those that satisfy α 。 α=α. A paradigm situation is an optica1 bench in which abeam of partic1es prepared in a certain state is injected at the left and then impinge first upon a filterαand then upon a filter b. Particles that p臼s through both filters enter a detection device at the right of b. Because of quantum interference. the order of placement of αand b usually makes a difference and we have α 。 b#b 。 α. 1f it happens thatα 。 b = boa we say thatαand b are sequentially independent and writeαI b. 1n recent years quantum effects have been studied within a general a1ge­ braic frame 飞;\'ork called an effect algebra. 1n Section 2 we sha11 summarize the basic de且 nitions concerning effect algebras and the properties of sequential products on Hilbert space effect a1gebras. The simplest of these properties are employed as a.xioms in Section 3 for a sequential effect algebra (SE.-\). A SE:\. is an effect algebra on which a sequential product with natural proper­ ties is defined. \Ve be1ieve that the a.xioms for a SEA are physically motivated and can be tested, for example, in the optical bench situation. Various prop­ erties of a SE人 are proved in Section 3. For instance, relationships between sequential independence, coexistence and compatibility are given. 1t is also sho\\'n that the sharp elements form an orthomodular poset. The sequential center C(E) of a SEA E is the set of elements αε E such thatαI b for every b ε E. 1n Section 4 it is shown that C(E) coincides with the set of sharp central elements which has previously been studied. :\loreover. a characterization is given for when C(E) is isomorphic to a fuzzy set system. Section 5 shows that the existence of a sequentia1 product is a strong restriction that eliminates many effect algebras from being SEA 's. It is shown' that a Boolean algebra admits a unique sequentia1 product and that certain effect algebras admit a sequential product only if they are Boolean. ~Ioreover. it is prO\'ed that if a map preserves the sequential product then it completely preseryes the effect algebra structure of the sharp elements. Section 6 defines the sharpness index of an effect. 1t is demonstrated that if E is isotropically 且 ni 2 tions 8 and 9. The existence of horizontal sums is characterized and some examples of horizontal sums and tensor products are given. 2 Hilbert Space Sequential Products This section summarizes the b臼 ic definitions concerning effect algebr臼 [1 ， 6, 7, 8, 14, 15] and the properties of sequential products on Hilbert space effect algebr出 [2. 3. 10, 12, 13]. If EÐ is a partial binary operation, we write α -L b ifαEÐ b is defined. .\n effect algebra is a system (E, 0, 1, EÐ) where 0, 1 are distinct elements of E and EÐ is a partial binary operation on E that satisfies the following conditions. (E1) If a • b, then b 1. αand bEÐ a =αEÐ b. (E2) If α -L b and c -L (αEÐ b) , then b 1. c ， α 1. (b EÐ c) andαEÐ(bEÐc)= (αEÐ b) EÐ c. (E3) For every αE E there exists a unique a' E E such that α 1. a' and αEÐ a' = 1. (E4) 1fα -L 1, then α=0. 1n the sequel ，飞，yhenever we writeαEÐ b we are implicit!y assuming that α -L b. 飞飞，~e define α ~ b if there exists a c E E such that αEÐ c = b. If such a c E E exists then it is unique and we write c = b e α. It can be shown that (E ， 三，') is a partially ordered set with 0 ~二 α 三 1 for all αεE.α"α. and α 三 b implies b' ~ a'. Moreover, we have α 1. b if and only ifα 三 b'. If α4α we call αan isotropic element and when 0 is the only isotropic element of E, then we call E an orthoalgebra. If the n-fold orthosum a EÐα@ … @α is defined in E we denote this element of E by nα. If there is a largest ηEN such that. na is defined, then ηis the isotropic index of αand if no such n exists. then ηhas isotropic index ∞. An element αε E is sharp ifα ^a' = O. Xotice that ifα 笋 o is sharp then a has isotropic index 1. \Ve say that E is isotropically finite if everyα 笋 o in E has finite isotropic index. 飞飞'e now give some standard examples of effect algebr臼. For a Boolean al­ gebra 8 , defineα 1. bifα ^b = 0 and in this c臼eαEÐb=αvb. Then (8, 0, 1, EÐ) is an effect algebra that happens to be an orthoalgebra. In particular if X is a nonempty set, then (2X ， 白， X , EÐ) is an effect algebra. These effect algebras correspond to classicallogic and set theory. For the function space [0 , 1]''\ 3 on the inten.al [0, 1] ç lR define the functions fo , h by fo(x) = O. fI (x) = 1 for all x εX. For f , 9 ç [0, 1]'\, we define f 1. 9 if f(x) + g(x) 三 1 for all Z εX and in this case (J æ g)(x) = f(x) + g(x). Then ([0, l]X,fo ,h ,æ) is the effect algebra of fuzzy subsets of X. A particularly simple effect algebra is the interval [0, 1] ç R. For α ， b ε[0 ， 1] we defineα 1. b ifα 十 b 三 1 and in this case α æb= α +b. In this section we are mainly concerned with the set ê(H) of all self­ adjoint operators on a Hilbert space H that satisfy 0 三 (Ax ， x) 三位 ， x) for all x E H. For .4., B ε ê(H) we define A 1. B if A + B E ê(H) and in this case .4 EÐ B = A + B. Then (ê(H), O,!, æ) is an effect algebra that we call a Hilbert space effect algebra. This effect algebra is important in studies of the foundations of quantum physics and quantum measurement theory [2 , 3. 4 ，而， 17]. The quantum effects A ε ê (H) correspond to yes-no measurements that may be unsharp. The set of projection operators P(H) on H form an orthoalgebra that is a sub-effect algebra of ê(H). The elements of P( H) correspond to sharp quantum effects. If E and F are effect algebras, we say that rþ: E • F is additive ifα 1. b implies rþ( α) 1.功 (b) and rþ( αæ b) = φ(α) EÐ rþ(b). If rþ: E • F is additive and 0(1) = 1, then φis a morphism. If 4>: E • F is a morphism and φ(α) 1. rþ(b) implies that α 1. b, then φis a monomorphism. .\ surjective monomorphism is an isomorphism. It is easy to see that a morphism d> is an isomorphism if and only if rþ is bijective and φ-1 is a morphism. .\ state on E is a morphism s: E → [O~ 1]. \Ve interpret s( α) as the probability that the effectαis observed (has answer yes) when the system is in the state s. We denote the set of states on E by n(E). A set of states S ç n(E) is order determining if s( α) 三 s(b) for all s ε S implies thatα 三 b. The sequential product on ê(H) is defined by A 0 B = .-1 1/ 2 BA 1/2 1 2 where .-1 / is the unique positive square root of A. [2, 3, 10, 12, 13]. We have that .-1 0 B ε ê (H) because 。三 (.-1 1 / 2 BA 1 / 2 X ， X) = (B 川 2 X ， 川 2X) 三 (A 1/2 ι .4 1 / 2 x) = (Ax, x) 空 (x ， x) for all x E H.
Load more

Recommended publications
  • Dichotomy Between Deterministic and Probabilistic Models in Countably Additive Effectus Theory
    Dichotomy between deterministic and probabilistic models in countably additive effectus theory Kenta Cho National Institute of Informatics, Japan Bas Westerbaan University College London John van de Wetering Radboud University Nijmegen June 6, 2020 such that § hom-sets tf : A Ñ Bu are convex sets, § and the scalars ts : I Ñ I u are the real unit interval r0; 1s Special operations: § States StpAq :“ t! : I Ñ Au § Effects EffpAq :“ tp : A Ñ I u § p ˝ ! is probability that p holds on state ! Generalized Probabilistic Theories GPTs are generalisations of quantum theory. They consist of § systems A; B; C;:::, § the `empty system' I , § operations f : A Ñ B, Special operations: § States StpAq :“ t! : I Ñ Au § Effects EffpAq :“ tp : A Ñ I u § p ˝ ! is probability that p holds on state ! Generalized Probabilistic Theories GPTs are generalisations of quantum theory. They consist of § systems A; B; C;:::, § the `empty system' I , § operations f : A Ñ B, such that § hom-sets tf : A Ñ Bu are convex sets, § and the scalars ts : I Ñ I u are the real unit interval r0; 1s Generalized Probabilistic Theories GPTs are generalisations of quantum theory. They consist of § systems A; B; C;:::, § the `empty system' I , § operations f : A Ñ B, such that § hom-sets tf : A Ñ Bu are convex sets, § and the scalars ts : I Ñ I u are the real unit interval r0; 1s Special operations: § States StpAq :“ t! : I Ñ Au § Effects EffpAq :“ tp : A Ñ I u § p ˝ ! is probability that p holds on state ! Solution: allow more general sets of scalars ts : I Ñ I u.
    [Show full text]
  • Arxiv:Quant-Ph/0611110 V1 10 Nov 2006
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server COORDINATING QUANTUM AGENTS’ PERSPECTIVES: CONVEX OPERATIONAL THEORIES, QUANTUM INFORMATION, AND QUANTUM FOUNDATIONS H. BARNUM1 1Computer and Computational Sciences Division CCS-3, MS B256, Los Alamos National Laboratory, Los Alamos 87545, USA E-mail: [email protected] In this paper, I propose a project of enlisting quantum information science as a source of task- oriented axioms for use in the investigation of operational theories in a general framework capable of encompassing quantum mechanics, classical theory, and more. Whatever else they may be, quantum states of systems are compendia of probabilities for the outcomes of possible operations we may perform on the systems: “operational theories.” I discuss appropriate general frameworks for such theories, in which convexity plays a key role. Such frameworks are appropriate for investigating what things look like from an “inside view,” i.e. for describing perspectival information that one subsystem of the world can have about another. Understanding how such views can combine, and whether an overall “geometric” picture (“outside view”) coordinating them all can be had, even if this picture is very different in nature from the structure of the perspectives within it, is the key to understanding whether we may be able to achieve a unified, “objective” physical view in which quantum mechanics is the appropriate description for certain perspectives, or whether quantum mechanics is truly telling us we must go beyond this “geometric” conception of physics. The nature of information, its flow and processing, as seen from various operational persepectives, is likely to be key to understanding whether and how such coordination and unification can be achieved.
    [Show full text]
  • Arxiv:1606.08229V1 [Math-Ph] 27 Jun 2016 States and Synaptic Algebras
    States and synaptic algebras David J. Foulis∗ Anna Jenˇcov´aand Sylvia Pulmannov´a† Abstract Different versions of the notion of a state have been formulated for var- ious so-called quantum structures. In this paper, we investigate the interplay among states on synaptic algebras and on its sub-structures. A synaptic algebra is a generalization of the partially ordered Jor- dan algebra of all bounded self-adjoint operators on a Hilbert space. The paper culminates with a characterization of extremal states on a commutative generalized Hermitian algebra, a special kind of synaptic algebra. Key Words: synaptic algebra, GH-algebra, Jordan algebra, convex effect algebra, MV-algebra, ℓ-group, order unit normed space, state, extremal state. AMS Classification 81P10, 81Q10 (46B40) 1 Introduction Synaptic algebras, featured in this paper, incorporate several so-called “quan- tum structures.” Quantum structures were originally understood to be math- arXiv:1606.08229v1 [math-ph] 27 Jun 2016 ematical systems that permit a perspicuous representation for at least one of the key ingredients of quantum-mechanical theory, e.g., states, observables, symmetries, properties, and experimentally testable propositions [11, 13, 40]. ∗Emeritus Professor, Department of Mathematics and Statistics, University of Mas- sachusetts, Amherst, MA; Postal Address: 1 Sutton Court, Amherst, MA 01002, USA; [email protected]. †Mathematical Institute, Slovak Academy of Sciences, Stef´anikovaˇ 49, SK-814 73 Bratislava, Slovakia; [email protected]. The second and third authors were sup- ported by grant VEGA No.2/0069/16. 1 In spite of the adjective ‘quantum,’ a variety of mathematical structures aris- ing in classical physics, computer science, psychology, neuroscience, fuzzy logic, fuzzy set theory, and automata theory are now regarded as quantum structures.
    [Show full text]
  • Topics in Many-Valued and Quantum Algebraic Logic
    Topics in many-valued and quantum algebraic logic Weiyun Lu August 2016 A Thesis submitted to the School of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Master of Science in Mathematics1 c Weiyun Lu, Ottawa, Canada, 2016 1The M.Sc. Program is a joint program with Carleton University, administered by the Ottawa- Carleton Institute of Mathematics and Statistics Abstract Introduced by C.C. Chang in the 1950s, MV algebras are to many-valued (Lukasiewicz) logics what boolean algebras are to two-valued logic. More recently, effect algebras were introduced by physicists to describe quantum logic. In this thesis, we begin by investigating how these two structures, introduced decades apart for wildly different reasons, are intimately related in a mathematically precise way. We survey some connections between MV/effect algebras and more traditional algebraic structures. Then, we look at the categorical structure of effect algebras in depth, and in particular see how the partiality of their operations cause things to be vastly more complicated than their totally defined classical analogues. In the final chapter, we discuss coordinatization of MV algebras and prove some new theorems and construct some new concrete examples, connecting these structures up (requiring a detour through the world of effect algebras!) to boolean inverse semigroups. ii Acknowledgements I would like to thank my advisor Prof. Philip Scott, not only for his guidance, mentorship, and contagious passion which have led me to this point, but also for being a friend with whom I've had many genuinely interesting conversations that may or may not have had anything to do with mathematics.
    [Show full text]
  • Divisible Effect Algebras and Interval Effect Algebras
    Comment.Math.Univ.Carolin. 42,2 (2001)219–236 219 Divisible effect algebras and interval effect algebras Sylvia Pulmannova´ Abstract. It is shown that divisible effect algebras are in one-to-one correspondence with unit intervals in partially ordered rational vector spaces. Keywords: effect algebras, divisible effect algebras, words, po-groups Classification: 81P10, 03G12, 06F15 1. Introduction Effect algebras as partial algebraic structures with a partially defined operation ⊕ and constants 0 and 1 have been introduced as an abstraction of the Hilbert- space effects, i.e., self-adjoint operators between 0 and I on a Hilbert space ([10]). Hilbert-space effects play an important role in the foundations of quantum me- chanics and measurement theory ([17], [5]). An equivalent algebraic structure, so-called difference posets (or D-posets) with the partial operation ⊖ have been introduced in [16]. From the structural point of view, effect algebras are a generalization of boolean algebras, MV-algebras, orthomodular lattices, orthomodular posets, orthoalge- bras. For relations among these structures and some other related structures see, e.g., [8]. Besides of quantum mechanics, they found applications in mathematical logic ([7], [6]), fuzzy probability theory ([4]), K-theory of C*- algebras ([18], [19]). A very important subclass of effect algebras are so-called interval effect al- gebras, which are representable as intervals of the positive cone in a partially ordered abelian group. Examples of this kind are, e.g., Hilbert-space effects, MV- algebras, effect algebras with ordering set of states ([2]), effect algebras with the Riesz decomposition property ([20]), convex effect algebras ([14], [15]). We note that a general characterization of interval effect algebras among effect algebras is an open problem.
    [Show full text]
  • Categorical Characterizations of Operator-Valued Measures
    Categorical characterizations of operator-valued measures Frank Roumen Inst. for Mathematics, Astrophysics and Particle Physics (IMAPP) Radboud University Nijmegen [email protected] The most general type of measurement in quantum physics is modeled by a positive operator-valued measure (POVM). Mathematically, a POVM is a generalization of a measure, whose values are not real numbers, but positive operators on a Hilbert space. POVMs can equivalently be viewed as maps between effect algebras or as maps between algebras for the Giry monad. We will show that this equivalence is an instance of a duality between two categories. In the special case of continuous POVMs, we obtain two equivalent representations in terms of morphisms between von Neumann algebras. 1 Introduction The logic governing quantum measurements differs from classical logic, and it is still unknown which mathematical structure is the best description of quantum logic. The first attempt for such a logic was discussed in the famous paper [2], in which Birkhoff and von Neumann propose to use the orthomod- ular lattice of projections on a Hilbert space. However, this approach has been criticized for its lack of generality, see for instance [22] for an overview of experiments that do not fit in the Birkhoff-von Neumann scheme. The operational approach to quantum physics generalizes the approach based on pro- jective measurements. In this approach, all measurements should be formulated in terms of the outcome statistics of experiments. Thus the logical and probabilistic aspects of quantum mechanics are combined into a unified description. The basic concept of operational quantum mechanics is an effect on a Hilbert space, which is a positive operator lying below the identity.
    [Show full text]
  • Interval and Scale Effect Algebras
    ADVANCES IN APPLIED MATHEMATICS 19, 200]215Ž. 1997 ARTICLE NO. AM970535 Interval and Scale Effect Algebras M. K. Bennett² and D. J. Foulis Department of Mathematics and Statistics, Uni¨ersity of Massachusetts, Amherst, Massachusetts Received November 27, 1996 1. EFFECT ALGEBRAS Inwx 6 , we introduced the following definition. 1.1. DEFINITION.Aneffect algebra is a system Ž.E, [ ,0,u consisting of a set E with two special elements 0, u g E, called the zero and the unit, and with a partially defined binary operation [ satisfying the following conditions for all p, q, r g E: Ž.i Commutati¨e Law:If p[qis defined, then q [ p is defined and p [ q s q [ p. Ž.ii Associati¨e Law:If q[rand p [ Žq [ r .are defined, then p[qand Ž.p [ q [ r are defined and p [ Ž.Ž.q [ r s p [ q [ r. Ž.iii Orthosupplement Law: For every p g E there exists a unique q g E such that p [ q is defined and p [ q s u. Ž.iv Zero-One Law:If p[uis defined, then p s 0. The archetypal example of an effect algebra is the standard effect algebra Ž.E,[,z,|for a Hilbert space H, where E consists of all self-adjoint operators A on H such that z F A F |. For A, B g E, A [ B is defined iff A q B g E, in which case A [ B [ A q B.Ž We use the notation [ to mean ``equals by definition.''. If H is the Hilbert space affiliated with a quantum-mechanical system, operators A g E are of significance in repre- senting unsharp measurements or observations on the systemwx 3, 19, 20 .
    [Show full text]
  • Closure Systems Over Effect Algebras Sistemas De Cierre Sobre Algebras
    ISSN-E 1995-9516 Universidad Nacional de Ingeniería COPYRIGHT © (UNI). TODOS LOS DERECHOS RESERVADOS http://revistas.uni.edu.ni/index.php/Nexo https://doi.org/10.5377/nexo.v34i02.11558 Vol. 34, No. 02, pp. 733-743/Junio 2021 Closure systems over effect algebras Sistemas de cierre sobre algebras de efecto Mahdi Ronasi1, Esfandiar Eslami2* 1Department of Mathematics, Kerman Branch, Islamic Azad University, Kerman, Iran. 2Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran. * [email protected] (recibido/received: 03-diciembre-2020; aceptado/accepted: 15-abril-2021) ABSTRACT The present paper is an attempt to introduce the closure systems over effect algebras. At first, we will define closure systems over effect algebras, and for arbitrary set $ U $ and arbitrary subset S of all functions from U to an effect algebra L we will obtain the closure system containing S. Then, we will define the base of this closure system, and for arbitrary subset S of all functions from U to an effect algebra L we will obtain the base of this closure system. Keywords: Closure Systems, Closure operator, Effect Algebra, Base. RESUMEN El presente artículo es un intento de introducir los sistemas de cierre sobre las álgebras de efectos. Primero definiremos sistemas de cierre sobre álgebras de efectos y para el conjunto arbitrario $ U $ y el subconjunto arbitrario S de todas las funciones de U a un álgebra de efectos L obtendremos el sistema de cierre que contiene S. Luego definiremos la base de este sistema de cierre y para un subconjunto arbitrario S de todas las funciones desde U hasta un álgebra de efectos L obtendremos la base de este sistema de cierre.
    [Show full text]
  • Operational Restrictions in General Probabilistic Theories
    Foundations of Physics (2020) 50:850–876 https://doi.org/10.1007/s10701-020-00352-6 Operational Restrictions in General Probabilistic Theories Sergey N. Filippov1,2,3 · Stan Gudder4 · Teiko Heinosaari5 · Leevi Leppäjärvi5 Received: 8 January 2020 / Accepted: 15 June 2020 / Published online: 12 July 2020 © The Author(s) 2020 Abstract The formalism of general probabilistic theories provides a universal paradigm that is suitable for describing various physical systems including classical and quantum ones as particular cases. Contrary to the usual no-restriction hypothesis, the set of accessible meters within a given theory can be limited for different reasons, and this raises a question of what restrictions on meters are operationally relevant. We argue that all operational restrictions must be closed under simulation, where the simulation scheme involves mixing and classical post-processing of meters. We distinguish three classes of such operational restrictions: restrictions on meters originating from restrictions on effects; restrictions on meters that do not restrict the set of effects in any way; and all other restrictions. We fully characterize the first class of restrictions and discuss its connection to convex effect subalgebras. We show that the restrictions belonging to the second class can impose severe physical limitations despite the fact that all effects are accessible, which takes place, e.g., in the unambiguous discrimination of pure quantum states via effectively dichotomic meters. We further demonstrate that there are physically meaningful restrictions that fall into the third class. The presented study of operational restrictions provides a better understanding on how accessible measurements modify general probabilistic theories and quantum theory in particular.
    [Show full text]
  • Remarks on Effect-Tribes
    Kybernetika Sylvia Pulmannová; Elena Vinceková Remarks on effect-tribes Kybernetika, Vol. 51 (2015), No. 5, 739–746 Persistent URL: http://dml.cz/dmlcz/144740 Terms of use: © Institute of Information Theory and Automation AS CR, 2015 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz KYBERNETIKA | VOLUME 5 1 (2015), NUMBER 5, PAGES 739{746 REMARKS ON EFFECT-TRIBES Sylvia Pulmannova´ and Elena Vincekova´ We show that an effect tribe of fuzzy sets T ⊆ [0; 1]X with the property that every f 2 T is B0(T )-measurable, where B0(T ) is the family of subsets of X whose characteristic functions are central elements in T , is a tribe. Moreover, a monotone σ-complete effect algebra with RDP with a Loomis{Sikorski representation (X; T ; h), where the tribe T has the property that every f 2 T is B0(T )-measurable, is a σ-MV-algebra. Keywords: effect-tribe, tribe, monotone σ-complete effect algebra, Riesz decomposition property, MV-algebra Classification: 81P10, 81P15 1. INTRODUCTION Effect algebras were introduced in [8] as an algebraic generalization of the set of all quantum effects, that is, self-adjoint operators between the zero and identity operator on a Hilbert space. The set of quantum effects plays an important role in the theory of quantum measurements, as the most general mathematical model of quantum measure- ments, positive operator valued measures (POVMs), have ranges in the set of quantum effects.
    [Show full text]
  • Generalized Homogeneous, Prelattice and Mv-Effect Algebras
    KYBERNETIKA — VOLUME 41 (2005), NUMBER 2, PAGES 129-142 GENERALIZED HOMOGENEOUS, PRELATTICE AND MV-EFFECT ALGEBRAS ZDENKA RlECANOVA AND IVICA MARINOVA We study unbounded versions of effect algebras. We show a necessary and sufficient condition, when lattice operations of a such generalized effect algebra P are inherited under its embeding as a proper ideal with a special property and closed under the effect sum into an effect algebra. Further we introduce conditions for a generalized homogeneous, prelattice or MV-effect effect algebras. We prove that every prelattice generalized effect algebra P is a union of generalized MV-effect algebras and every generalized homogeneous effect algebra is a union of its maximal sub-generalized effect algebras with hereditary Riesz decomposition property (blocks). Properties of sharp elements, the center and center of compatibility of P are shown. We prove that on every generalized MV-effect algebra there is a bounded orthogonally additive measure. Keywords: effect algebra, generalized effect algebra, generalized MV-effect algebra, prelat­ tice and homogeneous generalized effect algebra AMS Subject Classification: 06D35, 03G12, 03G25, 81P10 1. BASIC DEFINITIONS AND FACTS In 1994, Foulis and Bennett [3] have introduced a new algebraic structure, called an effect algebra. Effects represent unsharp measurements or observations on the quantum mechanical system. For modelling unsharp measurements in Hilbert space, the set of all effects is the set of all selfadjoint operators T on a Hilbert space H with 0 < T < 1. In a general algebraic form an effect algebra is defined as follows: Definition 1.1. A partial algebra (E\ ffi,0,1) is called an effect-algebra if 0,1 are two distinguished elements and © is a partially defined binary operation on P which satisfies the following conditions for any a,b,c€ E: (Ei) b®a = a® 6, if a © 6 is defined, (Eii) (a © b) © c = a © (b © c), if one side is defined, (Eiii) for every a G P there exists a unique b G P such that a © b = 1, (Eiv) if 1 © a is defined then a = 0.
    [Show full text]
  • Algebra of Effects in the Formalism of Quantum Mechanics on Phase Space As an M. V. and a Heyting Effect Algebra∗
    International Journal of Theoretical Physics, Vol. 44, No. 11, November 2005 (C 2005) DOI: 10.1007/s10773-005-0343-7 Algebra of Effects in the Formalism of Quantum Mechanics on Phase Space as an M. V. and a Heyting Effect Algebra∗ Franklin E. Schroeck Jr.1 We prove that the algebra of effects in the phase space formalism of quantum me- chanics forms an M. V. effect algebra and moreover a Heyting effect algebra. It con- tains no nontrivial projections. We equip this algebra with certain nontrivial projec- tions by passing to the limit of the quantum expectation with respect to any density operator. KEY WORDS: M. V. algebra; quantum mechanics on phase space; completion of M. V. algebras. PACS : Primary 02.10.Gd, 03.65.Bz; Secondary 002.20.Qs 1. INTRODUCTION An effect in quantum mechanics on a Hilbert space H is an operator A in H such that 0 ≤ A ≤ I , and two effects, A and B, are added to get A ⊕ B iff 0 ≤ A ⊕ B ≤ I, where A ⊕ B is the operator addition. We wish to specialize this concept to the effects that arise naturally in the formalism of quantum mechanics on phase space (Schroeck, 1996). From this, we will have an immediate gener- alization to H = L2(, dµ) where is any space and dµ is a (group-)invariant measure on . Then we wish to see what axioms are satisfied by this “algebra” of effects. Our plan is to first review the formalism of quantum mechanics on phase space, then to define the algebra of effects within this formalism, and then to show that we in fact have an M.
    [Show full text]