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arXiv:2009.07371 · Symmetry

Parts and Composites of Quantum Systems

Stanley Gudder

Published Jun 8, 2021• 6 citations
arXivHTMLPDF

Abstract

We consider three types of entities for quantum measurements. In order of generality, these types are observables, instruments and measurement models. If α and β are entities, we define what it means for α to be a part of β. This relationship is essentially equivalent to α being a function of β and in this case β can be employed to measure α. We then use the concept to define the coexistence of entities and study its properties. A crucial role is played by a map α^ which takes an entity of a certain type to one of a lower type. For example, if I is an instrument, then I^ is the unique observable measured by I. Composite systems are discussed next. These are constructed by taking the tensor product of the Hilbert spaces of the systems being combined. Composites of the three types of measurements and their parts are studied. Reductions in types to their local components are discussed. We also consider sequential products of measurements. Specific examples of Lüders, Kraus and trivial instruments are used to illustrate various concepts. We only consider finite-dimensional systems in this article. Finally, we mention the role of symmetry representations for groups using quantum channels.

DOI
10.3390/sym13061031

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We consider three types of entities for quantum measurements. In order of generality, these types are observables, instruments and measurement models. If α and β are entities, we define what it means for α to be a part of β. This relationship is essentially equivalent to α being a function of β and in this case β can be employed to measure α. We then use the concept to define the coexistence of entities and study its properties. A crucial role is played by a map α^ which takes an entity of a certain type to one of a lower type. For example, if I is an instrument, then I^ is the unique observable measured by I. Composite systems are discussed next. These are constructed by taking the tensor product of the Hilbert spaces of the systems being combined. Composites of the three types of measurements and their parts are studied. Reductions in types to their local components are discussed. We also consider sequential products of measurements. Specific examples of Lüders, Kraus and trivial instruments are used to illustrate various concepts. We only consider finite-dimensional systems in this article. Finally, we mention the role of symmetry representations for groups using quantum channels.
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.
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 introduces the concepts of multi-observables and multi-instruments in quantum mechanics. A multi-observable AAA (multi-instrument I\mathcal{I}I) has an outcome space of the form Ω=Ω1×⋯×ΩnΩ=Ω_1\times\cdots\timesΩ_nΩ=Ω1​×⋯×Ωn​ and is denoted by Ax1⋯xnA_{x_1\cdots x_n}Ax1​⋯xn​​ (Ix1⋯xn\mathcal{I}_{x_1\cdots x_n}Ix1​⋯xn​​) where (x1,…,xn)∈Ω(x_1,\ldots ,x_n)\inΩ(x1​,…,xn​)∈Ω. We also call AAA (I\mathcal{I}I) an nnn-observable (nnn-instrument) and when n=2n=2n=2 we call AAA (I\mathcal{I}I) a bi-observable (bi-instrument). We point out that bi-observables AAA (I\mathcal{I}I) and bi-instruments have been considered in past literature, but the more general case appears to be new. In particular, two observables (instruments) have been defined to coexist or be compatible if they possess a joint bi-observable (bi-instrument). We extend this definition to nnn observables and nnn instruments by considering joint marginals of nnn-observables and joint reduced marginals of nnn-instruments. We show that a nnn-instrument measures a unique nnn-observable and if a finite umber of instruments coexist, then their measured observables coexist. We prove that there is a close relationship between a nontrivial nnn-observable and its parts. Moreover, a similar result holds for instruments. We next show that a natural definition for the tensor product of a finite number of instruments exist and possess reasonable properties. We then discuss sequential products of a finite number of observables and instruments. We present various examples such as Kraus, Holevo and Lüders instruments.
Until recently, a quantum instrument was defined to be a completely positive operation-valued measure from the set of states on a Hilbert space to itself. In the last few years, this definition has been generalized to such measures between sets of states from different Hilbert spaces called the input and output Hilbert spaces. This article presents a theory of such instruments. Ways that instruments can be combined such as convex combinations, post-processing, sequential products, tensor products and conditioning are studied. We also consider marginal, reduced instruments and how these are used to define coexistence (compatibility) of instruments. Finally, we present a brief introduction to quantum measurement models where the generalization of instruments is essential. Many of the concepts of the theory are illustrated by examples. In particular, we discuss Holevo and Kraus instruments. Quanta 2023; 12: 27–40.
Observables and instruments have played significant roles in recent studies on the foundations of quantum mechanics. Sequential products of effects and conditioned observables have also been introduced. After an introduction in Section~1, we review these concepts in Section~2. Moreover, it is shown how these ideas can be unified within the framework of measurement models. In Section~3, we illustrate these concepts and their relationships for the simple example of a qubit Hilbert space. Conditioned observables and their distributions are studied in Section~4. Section~5 considers joint probabilities of observables. We introduce a definition for joint probabilities and discuss why we consider this to be superior to the standard definition.
If H1H_1H1​ and H2H_2H2​ are finite-dimensional Hilbert spaces, a channel from H1H_1H1​ to H2H_2H2​ is a completely positive, linear map I\mathcal{I}I that takes the set of states S(H1)\mathcal{S}(H_1)S(H1​) for H1H_1H1​ to the set of states S(H2)\mathcal{S}(H_2)S(H2​) for H2H_2H2​. Corresponding to I\mathcal{I}I there is a unique dual map I∗\mathcal{I}^*I∗ from the set of effects E(H2)\mathcal{E}(H_2)E(H2​) for H2H_2H2​ to the set of effects E(H1)\mathcal{E}(H_1)E(H1​) for H1H_1H1​. We call I∗(b)\mathcal{I}^*(b)I∗(b) the effect bbb conditioned by I\mathcal{I}I and the set Ic=I∗(E(H2))\mathcal{I}^c = \mathcal{I}^*(\mathcal{E}(H_2))Ic=I∗(E(H2​)) the conditioned set of I\mathcal{I}I. We point out that Ic\mathcal{I}^cIc is a convex subeffect algebra of the effect algebra E(H1)\mathcal{E}(H_1)E(H1​). We extend this definition to the conditioning I∗(B)\mathcal{I}^*(B)I∗(B) for an observable BBB on H2H_2H2​ and say that an observable AAA is in Ic\mathcal{I}^cIc if A=I∗(B)A=\mathcal{I}^*(B)A=I∗(B) for some observable BBB. We show that Ic\mathcal{I}^cIc is closed under post-processing and taking parts. We also define the conditioning of instruments by channels. These concepts are illustrated using examples of Holevo instruments and channels. We next discuss measurement models and their corresponding observables and instruments. We show that calculations can be simplified by employing Kraus and Holevo separable channels. Such channels allow one to separate the components of a tensor product.