Measurement theory in local quantum physics

In this paper, we aim to establish foundations of measurement theory in local quantum physics. For this purpose, we discuss a representation theory of completely positive (CP) instruments on arbitrary von Neumann algebras. We introduce a condition called the normal extension property (NEP) and establish a one-to-one correspondence between CP instruments with the NEP and statistical equivalence classes of measuring processes. We show that every CP instrument on an atomic von Neumann algebra has the NEP, extending the well-known result for type I factors. Moreover, we show that every CP instrument on an injective von Neumann algebra is approximated by CP instruments with the NEP. The concept of posterior states is also discussed to show that the NEP is equivalent to the existence of a strongly measurable family of posterior states for every normal state. Two examples of CP instruments without the NEP are obtained from this result. It is thus concluded that in local quantum physics not every CP instrument represents a measuring process, but in most of physically relevant cases every CP instrument can be realized by a measuring process within arbitrary error limits, as every approximately finite dimensional (AFD) von Neumann algebra on a separable Hilbert space is injective. To conclude the paper, the concept of local measurement in algebraic quantum field theory is examined in our framework. In the setting of the Doplicher-Haag-Roberts and Doplicher-Roberts (DHR-DR) theory describing local excitations, we show that an instrument on a local algebra can be extended to a local instrument on the global algebra if and only if it is a CP instrument with the NEP, provided that the split property holds for the net of local algebras.


Introduction
We aim at establishing measurement theory in local quantum physics. As a first step, we study it on the basis of local state formalism of algebraic quantum field theory (AQFT). AQFT based on the concept of local state is given in the study [34] of one of the authors, K. Okamura (K.O. for short), with his collaborators.
Quantum measurement theory is one area of quantum theory. Quantum information technology and this theory are now developing and having a mutual, good influence on each other. In particular, mathematical theory of quantum measurement makes a great contribution to understanding of measurement. There are two examples: one is the resolution of the dispute on a quantum mechanical analysis on the performance of gravitational wave detectors [39,40,31], and the other is experimental demonstrations of error-disturbance uncertainty relations [23,49,6,24,27,47]. Therefore, mathematical theory of quantum measurement is much worth studying. From the viewpoint of mathematics, we believe that it is very interesting and should be deepened.
Mathematical study of quantum measurement begins with the famous book [33] written by von Neumann. After the studies by Lüders [30], Nakamura and Umegaki [32], and Arveson [3], Davies and Lewis [15] introduced the concept of instruments [15,14] in order to analyze general measurements which do not always satisfy the repeatability hypothesis. Taking this opportunity, the theory greatly developed. At almost the same time, Kraus [28,29] treated completely positive maps in the context of quantum measurement but his interest is only the yes-no measurements. Following these studies, one of the author, M. Ozawa (M.O. for short), established the basis of present quantum measurement theory in [36]. In this paper, completely positive (CP) instruments are introduced in the general setting, and are shown to correpond to statistical equivalence classes of measuring processes in quantum mechanical situations [36,Theorem 5.1]. This result finished the characterization of general measurements in quantum mechanics (see also [41,42] for an axiomatic characterization of generalized quantum measurement). Introductions of families of a posteriori states and of disintegrations and the resolution of Davies-Lewis conjecture in [37,38] also played the crucial role of development of quantum measurement theory.
Here, we develop a representation theory of CP instruments defined on general (σfinite) von Neumann algebras and apply to quantum systems of infinite degrees of freedom, specially, to algebraic quantum field theory (AQFT). We expect that measurement theory in infinite quantum systems is soon needed since quantum information technology is developing very rapidly. We believe that this course of action of the study is very natural.
We discuss a representation theory of CP instruments in section 3: the necessary preliminaries are given in section 2. We introduce a condition called the normal extension property (NEP) for CP instruments to represent physical processes of measurement. Let M be a (σ-finite) von Neumann algebra on a Hilbert space H and (S, F ) a measurable space. In fact, this condition for a CP intrument I for (M, S) is equivalent to the existence of a (faithful) measuring process M = (K, σ, E, U) such that for all ∆ ∈ F and M ∈ M. This equivalence is given in the main theorem of this section. The one-to-one correspondence between CP instruments for (M, S) and statistical equivalence classes of measuring processes for (M, S) follows from the result. This fact is a generalization of [36,Theorem 5.1] to general (σ-finite) von Neumann algebras. The class of CP instruments described by measuring processes is completely characterized in this way.
In section 4, we shall characterize the set CPInst(M, S) of CP instruments for (M, S) by cohomology theory of von Neumann algebras. We begin with a basic result that every CP instrument for (M, S) has NEP if there exists a normal conditional expectation from B(H) into M A CP instrument is said to have the approximately normal extension property (ANEP) if it is approximated by CP instruments with NEP. It is proved that the set CPInst AN (M, S) of CP instruments with ANEP is equal to CPInst(M, S) if M is injective. This beautiful consequence is a natural extension of the basic result mentioned above.
The existence problem of a family of a posteriori states introduced by [37] is discussed in section 5. The main theorem of the section shows that NEP is equivalent to the existence of a strongly F -measurable family of a posteriori states for every normal state. Also, it is proved that a weakly repeatable CP instrument has NEP if and only if it is discrete. Following this result, an important example of CP instruments without NEP is given.
It demonstrates the conceptual superiority of CP instruments in quantum measurement theory.
In the last section, local measurements in DHR-DR theory are developed. We can prove the following theorem under some condition including Haag duality: Theorem 6.1. Suppose that a local net {π 0 (A(O))} O∈K satisfies the split property. Let (Γ, B(Γ)) be a standard Borel space, O a double cone, π a representation of A on H 0 Then the minimal dilation (K, E, V ) of a CP instrument for (B(H 0 ), Γ) extending I satisfies the following intertwining relation: This shows that, for any representation π of A satisfying the DHR selection criterion, every CP instrument for (π(A(O)) ′′ , Γ) described by a measuring process can be extended to a local CP instrument for (π(A)) ′′ , Γ, Λ) for every Λ ∈ K DC ⋐ such that O Λ 1 = O, and that the intertwining relation (2) appears as a mathematical expression of a causal structure of space-time. We are now ready to use both local states discussed in [34] and local CP instruments in order to describe quantum phenomena occurring in local space-time regions.

Preliminaries
Let X and Y be C * algebras and H be a Hilbert space. For any category E, we denote by Ob(E) the set of objects of E. We denote by Rep(X ) by the category of representations of X , Rep(X ; H) by the category of representations of X on H and by Hilb the category of Hilbert spaces. We define two norms · min and · max on the algebraic tensor product X ⊗ alg Y of X and Y by respectively, for every We call the completion X ⊗ min Y (X ⊗ max Y, resp.) of X ⊗ alg Y with respect to the norm · min ( · max , resp.) the minimal (maximal, resp.) tensor product of X and Y. The maximal tensor product X ⊗ max Y has the following property: Proposition 4.7]). Let X and Y be C * -algebras, and H be a Hilbert space. For every (π 1 , π 2 ) ∈ I max (X , Y; H), there exists a representation π of X ⊗ max Y on H such that π(X ⊗ Y ) = π 1 (X)π 2 (Y ) (7) for all X ∈ X and Y ∈ X .
Let M be a von Neumann algebra and Y be a C * -algebra. We denote by Rep n (M; H) the category of normal representations of M on H. We call the completion M ⊗ nor Y of M ⊗ alg Y with respect to the norm · nor defined below the normal tensor product of M and Y: Let M and N be von Neumann algebras. We call the completion M ⊗ bin N of M ⊗ alg N with respect to the norm · bin defined below the binormal tensor product of M and N : for every X = n j=1 M j ⊗ alg N j ∈ M ⊗ alg N , where A C * -algebra X is nuclear if, for every C * -algebra Y, It is known that C * -tensor products with nuclear C * -algebras are unique. A C * -algebra X on a Hilbert space H is injective if there exists a norm one projection from B(H) into X . It is proven in [22] that a von Neumann algebras M is injective if and only if, for every C * -algebra Y, Abelian C * -algebras are both nuclear and injective. A characterization of von Neumann algebras which are nuclear as C * -algebras is given in [  . Let X and Y be C *algebras such that X ⊂ Y, and H be a Hilbert space. For every T ∈ CP(X , B(H)), there exists T ∈ CP(Y, B(H)) such that T (X) = T (X), X ∈ X .
This theorem is known as Arveson extension theorem.
where j E is a CP map of B( We use the following form of Theorem 2.3:

CP Instrument and Quantum Measuring Process
See [42,43] for details of quantum measurement theory around CP instruments and measuring processes. Let M be a von Neumann algebra on a Hilbert space H and (S, F ) be a measurable space. In the paper, we assume that von Neumann algebras are σ-finite. We denote by P n (M) the set of normal positive linear maps from M into itself.
Definition 3.1. An instrument I for (M, S) is a P n (M)-valued measure on (S, F ) satisfying the following three conditions: for all ρ ∈ M * , M ∈ M. We also use the notation I(·, ·) for an instrument I in such a way I for any ∆ ∈ F and M ∈ M.
Proof. Let ν be a finite positive measure on S such that ν ∼ ρ• I for some normal faithful state ρ on M. By Lemma 2.1, there exists a POVM κ I in π T (M)′ such that for all M ∈ M and f ∈ L ∞ (S, ν). Let (E, K, W ) be the minimal Stinespring representation of κ I . By Theorem 2.1, there exists a nondegenerate normal representation π of M on K such that for all M ∈ M and f ∈ L ∞ (S, ν). We denote W V T by V , which is seen to be an isometry.
Let M and N be von Neumann algebras. For every σ ∈ N * , the map id ⊗ σ :

For a faithful measuring process
Let I be a CP instrument for (M, S) and ν a finite positive measure on S such that ν ∼ ρ•I for some normal faithful state ρ on M. By Lemma 3.1, the nuclearity of L ∞ (S, ν) and Proposition 2.1, there exists a binormal representation π : Since L ∞ (S, ν) is a nuclear C * -algebra, it holds that and M ⊗ min L ∞ (S, ν) is a dense C * -subalgebra of a von Neumann algebra M ⊗ L ∞ (S, ν). By Theorem 2.2, there then exists a CP map Ψ I : M ⊗ L ∞ (S, ν) → B(H) such that Ψ I | M⊗ bin L ∞ (S,ν) = Ψ I , which is usually nonnormal. (1) I has normal extension property (NEP) if there exists a unital normal CP map Ψ I : We denote by CPInst NE (M, S) the set of CP instruments for (M, S) with NEP.
We gave the name "normal extension property" in the light of unique extension property [5] used in operator system theory. Let I be a CP instrument for (M, S) with NEP and Ψ I : for all ∆ ∈ F and M ∈ M.
Proof. Let ν be a finite positive measure on S such that ν ∼ ρ• I for some normal faithful state ρ on M.
be a faithful measuring process for (M, S) such that for all ∆ ∈ F and M ∈ M. Then Ψ M satisfies Ψ M | M⊗ bin L ∞ (S,ν) = Ψ I . (i) ⇒ (iii) By assumption, there exists a unital normal CP map Ψ I : M ⊗ L ∞ (S, ν) → M such that Ψ I | M⊗ bin L ∞ (S,ν) = Ψ I . There then exists a minimal Stinespring representation (π, L 1 , W 1 ) of Ψ I , i.e., Furthermore, by Theorem 2.3, there exist a Hilbert space L 2 , a projection E of (M ⊗ L ∞ (S, ν)) ′ ⊗B(L 2 ) and an isometry where a normal CP map j E : A CP instrument I for (B(H), S) is defined by for every X ∈ B(H) and ∆ ∈ F . For every M ∈ M and ∆ ∈ F , it is seen that for every M ∈ M and ∆ ∈ F . (iii) ⇒ (iv) Let I be a CP instrument for (B(H), S) such that I(∆)M = I(∆)M for all ∆ ∈ F and M ∈ M. We denote I(S) by T . By Corollary 2.1, a normal representation of B(H) is unitarily equivalent to the representation id ⊗ 1 L , where L is a Hilbert space. Therefore, there exist a Hilbert space L 1 and a unitary operator W 1 : for all X ∈ B(H). By Lemma 2.1, there exists a positive contractive map κ I : for all X ∈ B(H) and f ∈ L ∞ (S, ν), and that, if f ∈ L ∞ (S, ν) + satisfies κ µ (f ) = 0 then f = 0. Let (E 0 , L 2 , W 2 ) be the minimal Stinespring representation of κ I . Then E 0 is a normal faithful representation of L ∞ (S, ν) on L 2 . Denote W 2 W 1 V T by V . It holds that for all X ∈ B(H) and f ∈ L ∞ (S, ν). Let L 3 be an infinite-dimensional Hilbert space, η 3 be a unit vector of L 3 and η 2 be a unit vector of L 2 . We define an isometry U 0 from H ⊗ Cη 2 ⊗ Cη 3 to H ⊗ L 2 ⊗ L 3 by for all ξ ∈ H. Since it holds that there is a unitary operator U on H ⊗ L 2 ⊗ L 3 , which is an extension of U 0 . We then define a Hilbert space K by K = L 2 ⊗ L 3 , a state σ on B(K) by for all Y ∈ B(K), and a spectral measure E : F → B(K) by for all ∆ ∈ F . It is checked that for all X ∈ B(H) and ∆ ∈ F , and, by the definition of E, there exists a normal faithful representation E of L ∞ (S, ν) on K such that E(χ ∆ ) = E(∆) for all ∆ ∈ F . Thus there exists a faithful measuring process M = (K, σ, E, U) for (M, S) such that for all X ∈ B(H) and ∆ ∈ F . Therefore, for every M ∈ M and ∆ ∈ F , it holds that for all ∆ ∈ F and M ∈ M.
Definition 3.5 (Statistical equivalence class of measuring processes [36]). Two measuring processes M 1 = (K 1 , σ 1 , E 1 , U 1 ) and M 2 = (K 2 , σ 2 , E 2 , U 2 ) for (M, S) are said to be statistically equivalent if, for all M ∈ M and ∆ ∈ F , We  Proof. The proof of the equivalence between (1) and (2) where ω {xn},{yn} = n x n |(·)y n is a normal state on M and {z n } n∈N , {w n } n∈N ⊂ H such that n z n 2 < ∞, n w n 2 < ∞, I(∆) * ω {xn},{yn} = n z n |(·)w n .    [12,51]. Therefore, the second statement of Theorem 4.2 always holds for physically realizable von Neumann algebras. We may understand the physical necessity of CP instruments without NEP by the discussion in the next section.

Existence of A Family of A Posteriori States and Its Consequences
Let M be a von Neumann algebra on a Hilbert space H and (S, F ) a measurable space.  Our interest in this section is the existence of a posteriori states.  The next is an example that not all CP instruments defined on injective von Neumann algebras have NEP, and is strongly related to Theorems 5.2 and 5.3.  By [37,Theorem 4.5], there exists a strongly F -measurable family { ρ I s } s∈S of a posteriori states with respect to ( I, ρ). It is then obvious that the function s → ρ I s | M is ρ • I-measurable, and that, for all ∆ ∈ F and M ∈ M, Thus {ρ I s := ρ I s | M } s∈S is a family of a posteriori states with respect to (I, ρ). The strong F -measurability of {ρ I s } s∈S follows from that of { ρ I s } s∈S . Theorem 5.2. Let M be a von Neumann algebra on a Hilbert space H, (S, F ) a measurable space, and I a CP instrument for (M, S). The following conditions are equivalent: (1) I has NEP.
(2) For every normal state ρ ∈ M * ,1 , there exists a strongly F -measurable family {ρ I s } s∈S of a posteriori states with respect to (I, ρ).
We show that the map M * ,1 ∋ ρ → ρ ∈ (M ⊗ L ∞ (S, ϕ • I)) * ,1 is affine. Let ρ 1 , ρ 2 ∈ M * ,1 , 0 ≤ α ≤ 1, and denote αρ 1 + (1 − α)ρ 2 by ρ. For every M ∈ M and ∆ ∈ F , Since L ∞ (S, ϕ•I) = span · {χ ∆ | ∆ ∈ F } and M⊗L ∞ (S, ϕ•I) = M ⊗ alg L ∞ (S, ϕ • I) uw , and since ρ, ρ 1 and ρ 2 are normal, it holds that For any ρ ∈ M * ,+ , we define ρ ∈ (M ⊗ L ∞ (S, ϕ • I)) * ,+ by It is easily checked that the map M * ,+ ∋ ρ → ρ ∈ (M ⊗ L ∞ (S, ϕ • I)) * ,+ is also affine. Furthermore, we define the map M * ∋ ρ → ρ ∈ (M ⊗ L ∞ (S, ϕ • I)) * by, for all ρ ∈ M * by where ρ 1 , ρ 2 , ρ 3 , ρ 4 ∈ M * ,+ such that ρ = ρ 1 − ρ 2 + i(ρ 3 − ρ 4 ). By a similar discussion in [41,42], ρ does not depend on the choice of ρ 1 , ρ 2 , ρ 3 , ρ 4 ∈ M * ,+ such that ρ = ρ 1 − ρ 2 + i(ρ 3 − ρ 4 ). We now define a unital ultraweakly continuous linear map Ψ I : For every n ∈ N and C * -algebra X , we denote by M n (X ) that of n × n matrices with entries from X . For every n ∈ N, since M n (M ⊗ bin L ∞ (S, ϕ • I)) = M n (M ⊗ min L ∞ (S, ϕ • I)) is a dense C * -subalgebra of M n (M⊗L ∞ (S, ϕ•I)), for all X ∈ M n (M⊗L ∞ (S, ϕ•I)) + , there exists a net {X α } of M n (M ⊗ bin L ∞ (Γ, ϕ • I)) + such that X α → uw X. By a similar discussion appeared in Lemma 3.2, it holds that Ψ (n) Davies-Lewis conjecture was resolved in [36,Theorem 6.6], and completely in [37,Theorem 5.1]. We can strengthen the former result by using the latter one and the method in this section: Proof. For every normal state ρ ∈ M * ,1 , since I is discrete, a strongly F -measurable proper family {ρ I s } s∈S of a posteriori states with respect to (I, ρ) is given by We shall show another example of a CP instrument without NEP in addition to Example 5.1. Let N be a AFD von Neumann algebra of type II 1 on a separable Hilbert space H, A a self-adjoint element of N with continuous spectrum, and E a conditional expectation from N into {A} ′ ∩ N . We define a CP instrument I A for (N , R) by for all N ∈ N and ∆ ∈ B(R), where E A is the spectral measure of A. By the property of conditional expectation, I A precisely measures the spectrum of A and is weakly repeatable. Hence it does not have NEP but has ANEP. This is the reason why we cannot ignore CP instruments without NEP. We conclude that all CP instruments defined on von Neumann algebras describing physical systems are physically realizable and are approximated by measuring processes.

DHR-DR Theory and Local Measurement
First, we give assumptions of algebraic quantum field theory (AQFT). We refer readers to [2,25] for standard reference on AQFT. In the setting of AQFT, it is assumed that all physically realizable states on A and representations of A are locally normal, i.e., normal on A(O) for all O ∈ K.

(Vacuum state and representation).
A vacuum state ω 0 is a P ↑ + -invariant locally normal pure state on A. We denote by (π 0 , H 0 , U, Ω) the GNS representation of (A, P ↑ + , α, ω 0 ). In addition, it is assumed that the spectrum of the generator P = (P µ ) of the translation part of U is contained in the closed future lightcone V + .
For every O ∈ K, we denote by O the closure of O and define the causal } of K consisting of double cones. Furthermore, we adopt the following notations: For a representation π of A on H 0 , this criterion is reduced to the form Next, we give the definition of strictly local CP instrument. This definition is a generalization of that of Halvorson [26] to general representations satisfying the DHR selection criterion.
In DHR-DR theory, it is usually assumed that all factor representations satisfying the DHR selection criterion are quasi-equivalent to irreducible ones. By this assumption and the categorical analysis by Doplicher and Roberts [20,21], all representations satisfying the DHR selection criterion generate type I von Neumann algebras with separable discrete center. There then exists a normal conditional expactation E π : B(H 0 ) → π(A) ′′ for any representation π on H 0 satisfying the DHR selection criterion. Definition 6.2 (Minimal dilation [44]). For a CP instrument for (B(H), S), the triplet (K, E, V ) is called a minimal dilation of I if K is a Hilbert space, E : F → B(K) is a spectral measure and V is an isometry from H into H ⊗ K such that for all X ∈ B(H), and The following proposition holds: Proposition 6.1. Let (Γ, B(Γ)) be a standard Borel space, O a double cone, and π a representation of A on H 0 such that π| A(O ′ ) ∼ = π 0 | A(O ′ ) . Then every strictly local CP instrument I for (π(A) ′′ , Γ, O) has NEP, and the minimal dilation (K, E, V ) of the E πcanonical extension I of I satisfies the following intertwining relation: for all A ∈ π(A(O ′ )) ′′ .
A typical example of strictly local CP instruments is a von Neumann model of an observable affiliated to π(A(O)) ′′ . Even if a CP instrument I for (π(A(O)) ′′ , Γ) has NEP, there does not always exist a strictly local CP instrument I for (π(A) ′′ , Γ, O) such that I(∆)A = I(∆)A for all A ∈ π(A(O)) ′′ . In contrast to this fact, we can show that every CP instrument I for (π(A(O)) ′′ , Γ) with NEP is extended into a local CP instrument I for (π(A) ′′ , Γ, Λ) defined as follows, where Λ ∈ K DC