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arXiv:2312.13003 · International Journal of Theoretical Physics

Spectrality in Convex Sequential Effect Algebras

Anna Jenčová, Sylvia Pulmannová

Published Aug 25, 2023• 1 citation
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Abstract

Abstract 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 $$\sigma $$ σ -complete. Two previous results on existence of spectral resolutions in this setting are shown to require stronger assumptions.

DOI
10.1007/s10773-023-05431-8

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Abstract 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 σ\sigma σ σ -complete. Two previous results on existence of spectral resolutions in this setting are shown to require stronger assumptions.
It has already been established that the properties required of an abstract sequential product as introduced by Gudder and Greechie are not enough to characterise the standard sequential product a◦b=aba on an operator algebra. We give three additional properties, each of which characterises the standard sequential product on either a von Neumann algebra or a Euclidean Jordan algebra. These properties are (1) invariance under the application of unital order isomorphisms, (2) symmetry of the sequential product with respect to a certain inner product, and (3) preservation of invertibility of the effects. To give these characterisations, we first have to study convex σ-sequential effect algebras. We show that these objects correspond to unit intervals of spectral order unit spaces with a homogeneous positive cone.
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.
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.
Our basic structure is a finite-dimensional complex Hilbert space HHH. We point out that the set of effects on HHH form a convex effect algebra. Although the set of operators on HHH 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.
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.
Our basic concept is the set E(H)\mathcal{E}(H)E(H) of effects on a finite dimensional complex Hilbert space HHH. If a,b∈E(H)a,b\in\mathcal{E}(H)a,b∈E(H), we define the sequential product a[I]ba[\mathcal{I}]ba[I]b of aaa then bbb. The sequential product depends on the operation I\mathcal{I}I used to measure aaa. We begin by studying the properties of this sequential product. It is observed that b↦a[I]bb\mapsto a[\mathcal{I}]bb↦a[I]b is an additive, convex morphism and we show by examples that a↦a[I]ba\mapsto a[\mathcal{I}]ba↦a[I]b enjoys very few conditions. This is because a measurement of aaa can interfere with a later measurement of bbb. We study sequential products relative to Kraus, Lüders and Holevo operations and find properties that characterize these operations. We consider repeatable effects and conditions on a[I]ba[\mathcal{I}]ba[I]b that imply commutativity. We introduce the concept of an effect bbb given an effect aaa and study its properties. We next extend the sequential product to observables and instruments and develop statistical properties of real-valued observables. This is accomplished by employing corresponding stochastic operators. Finally, we introduce an uncertainty principle for conditioned observables.