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Journal of Mathematical Physics

Super quantum measures on effect algebras with the Riesz decomposition properties

Yongjian Xie, Aili Yang, Fang Ren

Published Oct 1, 2015• 0 citations
Publisher

Abstract

We give one basis of the space of super quantum measures on finite effect algebras with the Riesz decomposition properties (RDP for short). Then we prove that the super quantum measures and quantum interference functions on finite effect algebras with the RDP are determined each other. At last, we investigate the relationships between the super quantum measures and the diagonally positive signed measures on finite effect algebras with the RDP in detail.

DOI
10.1063/1.4933324

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We give one basis of the space of super quantum measures on finite effect algebras with the Riesz decomposition properties (RDP for short). Then we prove that the super quantum measures and quantum interference functions on finite effect algebras with the RDP are determined each other. At last, we investigate the relationships between the super quantum measures and the diagonally positive signed measures on finite effect algebras with the RDP in detail.
An observable on a quantum structure is any σσσ-homomorphism of quantum structures from the Borel σσσ-algebra into the quantum structure. We show that our partial information on an observable known only for all intervals of the form (−∞,t)(-\infty,t)(−∞,t) is sufficient to determine uniquely the whole observable defined on quantum structures like σσσ-MV-algebras, σσσ-effect algebras, Boolean σσσ-algebras, monotone σσσ-complete effect algebras with the Riesz Decomposition Property, the effect algebra of effect operators of a Hilbert space, and a system of functions, and an effect-tribe.
The space POVM H(X) of positive operator-valued probability measures on the Borel sets of a compact (or even locally compact) Hausdorff space X with values in B(H), the algebra of linear operators acting on a d-dimensional Hilbert space H, is studied from the perspectives of classical and nonclassical convexity through a transform Γ that associates any positive operator-valued measure ν with a certain completely positive linear map Γ(ν) of the homogeneous C*-algebra C(X)⊗B(H) into B(H). This association is achieved by using an operator-valued integral in which nonclassical random variables (that is, operator-valued functions) are integrated with respect to positive operator-valued measures and which has the feature that the integral of a random quantum effect is itself a quantum effect. A left inverse Ω for Γ yields an integral representation, along the lines of the classical Riesz representation theorem for linear functionals on C(X), of certain (but not all) unital completely positive linear maps φ:C(X)⊗B(H)→B(H). The extremal and C*-extremal points of POVM H(X) are determined.
Effect algebras were introduced as an abstract algebraic model for Hilbert space effects representing quantum mechanical measurements. We study additional structures on an effect algebra E that enable us to define spectrality and spectral resolutions for elements of E akin to those of self-adjoint operators. These structures, called compression bases, are special families of maps on E , analogous to the set of compressions on operator algebras, order unit spaces or unital abelian groups. Elements of a compression base are in one-to-one correspondence with certain elements of E , called projections. An effect algebra is called spectral if it has a distinguished compression base with two special properties: the projection cover property (i.e., for every element a in E there is a smallest projection majorizing a ), and the so-called b-comparability property, which is an analogue of general comparability in operator algebras or unital abelian groups. It is shown that in a spectral archimedean effect algebra E , every a∈E admits a unique rational spectral resolution and its properties are studied. If in addition E possesses a separating set of states, then every element a∈E is determined by its spectral resolution. It is also proved that for some types of interval effect algebras (with RDP, archimedean divisible), spectrality of E is equivalent to spectrality of its universal group and the corresponding rational spectral resolutions are the same. In particular, for convex archimedean effect algebras, spectral resolutions in E are in agreement with spectral resolutions in the corresponding order unit space.
We review the relationship between positive operator-valued measures (POVMs) in quantum measurement theory and asymptotic morphisms in the C*-algebra E-theory of Connes and Higson. The theory of asymptotic spectral measures, as introduced by Martinez and Trout (CMP 226), is integrally related to positive asymptotic morphisms on locally compact spaces via an asymptotic Riesz Representation Theorem. Examples and applications to quantum physics, including quantum noise models, semiclassical limits, pure spin one-half systems and quantum information processing will also be discussed.
Let K be a convex subset of the state space of a finite-dimensional C*-algebra. We study the properties of channels on K, which are defined as affine maps from K into the state space of another algebra, extending to completely positive maps on the subspace generated by K. We show that each such map is the restriction of a completely positive map on the whole algebra, called a generalized channel. We characterize the set of generalized channels and also the equivalence classes of generalized channels having the same value on K. Moreover, if K contains the tracial state, the set of generalized channels forms again a convex subset of a multipartite state space, this leads to a definition of a generalized supermap, which is a generalized channel with respect to this subset. We prove a decomposition theorem for generalized supermaps and describe the equivalence classes. The set of generalized supermaps having the same value on equivalent generalized channels is also characterized. Special cases include quantum combs and process positive operator valued measures (POVMs).