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arXiv:2108.07925

Sequential Products of Quantum Measurements

Stanley Gudder

Published Jun 6, 2007• 2 citations
arXivHTMLPDF

Abstract

Our basic structure is a finite-dimensional complex Hilbert space $H$. We point out that the set of effects on $H$ form a convex effect algebra. Although the set of operators on $H$ 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.

DOI
10.48550/arxiv.2108.07925

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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.
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.
We present a mathematical framework for quantum mechanics in which the basic entities and operations have physical significance. In this framework the primitive concepts are states and effects and the resulting mathematical structure is a convex effect algebra. We characterize the convex effect algebras that are classical and those that are quantum mechanical. The quantum mechanical ones are those that can be represented on a complex Hilbert space. We next introduce the sequential product of effects to form a convex sequential effect algebra. This product makes it possible to study conditional probabilities and expectations.
This article considers quantum systems described by a finite-dimensional complex Hilbert space HHH. We first define the concept of a finite observable on HHH. We then discuss ways of combining observables in terms of convex combinations, post-processing and sequential products. We also define complementary and coexistent observables. We then introduce finite instruments and their related compatible observables. The previous combinations and relations for observables are extended to instruments and their properties are compared. We present four types of instruments; namely, identity, trivial, Lüders and Kraus instruments. These types are used to illustrate different ways that instruments can act. We next consider joint probabilities for observables and instruments. The article concludes with a discussion of measurement models and the instruments they measure.
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Abstract This article points out that observables and instruments can be combined in many ways that have natural and physical interpretations. We shall mainly concentrate on the mathematical properties of these combinations. After an introduction in section 1, section 2 reviews the basic definitions and observables are considered in section 3. We study parts of observables, post-processing, generalized convex combinations, sequential products and tensor products. These combinations are extended to instruments in section 4. We consider properties of observables measured by combinations of instruments. We introduce four special types of instruments, namely Kraus, Lüders, trivial and semitrivial instruments. We study when these types are closed under various combinations. In this work, we only consider finite-dimensional quantum systems. A few of the results presented here have appeared in the author’s previous articles Gudder (2020 arXiv:2005.08117 [quant-ph], arXiv:2005.13642 [quant-ph], arXiv:2009.07371 [quant-ph]).
This article begins with a study of convex effect-state spaces. We point out that such spaces are equivalent to interval effect algebras that generate an ordered linear space and possess an order-determining set of states. We then discuss operations and instruments on interval effect algebras under the assumption of an unrestrictive condition. Effects measured by operations and sequential products of effects relative to operations are considered. Observables are introduced and coexistence of effects are discussed. We also present properties of sequential products of observables and conditioning of observables related to instruments. The final section is devoted to Holevo instruments. Pure and mixed Holevo operations are defined and extended to instruments. The Holevo sequential product of two observables is defined and the marginals of these products are computed. We define the commutant of two effects and derive its properties. Examples are given that illustrate the properties of previously presented concepts.