If $H_1$ and $H_2$ are finite-dimensional Hilbert spaces, a channel from $H_1$ to $H_2$ is a completely positive, linear map $\mathcal{I}$ that takes the set of states $\mathcal{S}(H_1)$ for $H_1$ to the set of states $\mathcal{S}(H_2)$ for $H_2$. Corresponding to $\mathcal{I}$ there is a unique dual map $\mathcal{I}^*$ from the set of effects $\mathcal{E}(H_2)$ for $H_2$ to the set of effects $\mathcal{E}(H_1)$ for $H_1$. We call $\mathcal{I}^*(b)$ the effect $b$ conditioned by $\mathcal{I}$ and the set $\mathcal{I}^c = \mathcal{I}^*(\mathcal{E}(H_2))$ the conditioned set of $\mathcal{I}$. We point out that $\mathcal{I}^c$ is a convex subeffect algebra of the effect algebra $\mathcal{E}(H_1)$. We extend this definition to the conditioning $\mathcal{I}^*(B)$ for an observable $B$ on $H_2$ and say that an observable $A$ is in $\mathcal{I}^c$ if $A=\mathcal{I}^*(B)$ for some observable $B$. We show that $\mathcal{I}^c$ 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.
If H1 and H2 are finite-dimensional Hilbert spaces, a channel from H1 to H2 is a completely positive, linear map I that takes the set of states S(H1) for H1 to the set of states S(H2) for H2. Corresponding to I there is a unique dual map I∗ from the set of effects E(H2) for H2 to the set of effects E(H1) for H1. We call I∗(b) the effect b conditioned by I and the set Ic=I∗(E(H2)) the conditioned set of I. We point out that Ic is a convex subeffect algebra of the effect algebra E(H1). We extend this definition to the conditioning I∗(B) for an observable B on H2 and say that an observable A is in Ic if A=I∗(B) for some observable B. We show that Ic 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.
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.
Our basic concept is the set E(H) of effects on a finite dimensional complex Hilbert space H. If a,b∈E(H), we define the sequential product a[I]b of a then b. The sequential product depends on the operation I used to measure a. We begin by studying the properties of this sequential product. It is observed that b↦a[I]b is an additive, convex morphism and we show by examples that a↦a[I]b enjoys very few conditions. This is because a measurement of a can interfere with a later measurement of b. 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]b that imply commutativity. We introduce the concept of an effect b given an effect a 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 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.
We introduce the concepts of dual instruments and sub-observables. We show that although a dual instruments measures a unique observable, it determines many sub-observables. We define a unique minimal extension of a sub-observable to an observable and consider sequential products and conditioning of sub-observables. Sub-observable effect algebras are characterized and studied. Moreover, the convexity of these effect algebras is considered. The sequential product of instruments is discussed. These concepts are illustrated with many examples of instruments. In particular, we discuss Lüders, Holero and constant state instruments. Various conjectures for future research are presented.
A measurement model is a framework that describes a quantum measurement process. In this article we restrict attention to MMs on finite-dimensional Hilbert spaces. Suppose we want to measure an observable A whose outcomes Ax are represented by positive operators (effects) on a Hilbert Space H. We call H the base or object system. We interact H with a probe system on another Hilbert space K by means of a quantum channel. The probe system contains a probe (or meter or pointer) observable F whose outcomes Fx are measured by an apparatus that is frequently (but need not be) classical in practice. The MM protocol gives a method for determining the probability of an outcome Ax for any state of H in terms of the outcome Fx. The interaction channel usually entangles this state with an initial probe state of K that can be quite complicated. However, if the channel is nondisturbing in a sense that we describe, then the entanglement is considerably simplified. In this article, we give formulas for observables and instruments measured by nondisturbing MMs. We begin with a general discussion of nondisturbing operators relative to a quantum context. We present two examples that illustrate this theory in terms of unitary nondisturbing channels.
Starting from two Hilbert spaces H1 and H2 of dimension N1 and N2, respectively, one can construct a larger Hilbert space H = H1 ⊗H2, (32.1) a sort of union of the two. The space H is called the tensor product of H1 and H2. State vectors and operators of these spaces will be denoted by an index, (1) or (2). Associated with a pair of state vectors |φ(1)〉 and |χ(2)〉 belongs a state vector |φ(1)〉 ⊗ |χ(2)〉 ≡ |φ(1)〉|χ(2)〉 ≡ |φ(1)χ(2)〉 (32.2) in the combined Hilbert space. Let us choose basis sets {|un(1)〉} and {|uk(2)〉} for H1 and H2. In terms of these one can express the arbitrary vectors |φ(1)〉 and |χ(2)〉 as follows: |φ(1)〉 = ∑ n an|un(1)〉, (32.3) |χ(2)〉 = ∑ k bk|vk(2)〉. (32.4) In the Hilbert space H one may use a basis {|un(1)⊗ |vk(2)〉〉} for expansion of the tensor product in Eq. (32.2). Thus, |φ(1)〉 ⊗ |χ(2)〉 = ∑ n,k anbk|un(1)〉 ⊗ |vk(2)〉. (32.5) The components of a tensor product state vector in H hence are products, anbk, of components of the two state vectors in the Hilbert spaces H1 and H2. The most general state vector in H, |ψ〉, can in the basis {|un(1)〉 ⊗ |vk(2)〉} be expanded as follows: |ψ〉 = ∑ n,k cnk|un(1)〉 ⊗ |vk(2)〉, (32.6) where the double index nk on the expansion coefficient cnk specifies the vector axes in H. Although an arbitrary vector |ψ〉 in H cannot be written as a single vector product of two vectors in H1 and H2 [Eq. (32.5)], |ψ〉 can always be decomposed into a linear combination of tensor product vectors [Eq. (32.6)]. This result relates to the fact that the elements of an arbitrary matrix {cnk} cannot in general be given as the elements {anbk} of a dyadic product of two vectors with components {an} and {bk}. The scalar product in H of two arbitrary basis vectors, |un(1)〉 ⊗ |vk(2)〉 and |un′(1)〉 ⊗ |vk′(2)〉, is given by 〈un′(1)vk′(2)|un(1)vk(2)〉 = 〈un′(1)|un(1)〉〈vk′ (2)|vk(2)〉 = δnn′δkk′ . (32.7) of The last member of Eq. (32.7) is obtained provided each of the bases {|un(1)〉} and |vk(2)〉 is orthonormal in its respective Hilbert space. If this is the case, the basis in H, i.e., {|un〉⊗ |vk(2)〉}, is orthonormal, too.