A C$^{\ast}$-algebraic approach to quantum measurement (Mathematical aspects of quantum fields and related topics)

全文

(1)170. C^{*}∗ -algebraic A AC ‐algebraic approach approach to to quantum quantum measurement measurement ໊‫ݹ‬԰େֶେֶӃ৘ใՊֶ‫ڀݚ‬Պ 名古屋大学大学院情報科学研究科. Ԭଜ 岡村 ࿨໻ 和弥 ∗*. I ɹ Introduction C^{*}∗ -algebraic AC ‐algebraic approach to quantum measurement theory is proposed in the paper. Here, we treat processes of measurements in the Schr¨ odinger picture, that enables us to totally describe Schrödinger dynamical changes induced by measurements in general quantum systems including those with infinite infinite degrees of freedom. C^{*}∗ -algebraic C ‐algebraic quantum theory can make the best use of noncommutative probability theory C^{*}∗ -algebras by Takesaki [21] and Bichteler [2]. The ex[12] and the duality theorem for C ‐algebras proven by ex‐ ∗ ∗ C^{*} -representations of a given C -algebra is not inevitable usually. istence of unitarily inequivalent umitarily ‐representations ‐algebra This fact is, however, the merit of the theory rather than its difficulty: difficulty: We can naturally introintro‐ C^{*}∗ -algebras. duce macroscopic classical levels in quantum systems described by C The quasi‐algebras. quasi‐ ∗ C^{*}∗ -algebras equivalence of o f^{*} -representations ‐representations of C ‐algebras then takes the place of the unitary umitary equivalence. The theory of operator algebras has been greatly contributed to quantum measurement theory from early days. In 1962, 1962, Nakamura and Umegaki [13] used the notion of conditional expectaexpecta‐ tion [24, 23] to characterize the class of measurements for discrete observables called “the von Neumann-L¨ uders projection postulate”. The importance of operator algebraic methods remains Neumann‐Lüders unchanged and is increasingly recognized now. In Section II, we introduce preliminaries on algebraic quantum theory and sector theory. An ∗ C^{*}∗ -algebras, equivalence relation of o f^{*} -representations ‐representations of C ‐algebras, called the quasi-equivalence, quasi‐equivalence, is essenessen‐ tial for the definition odinger definition of sectors. In Section III, we describe measurements in the Schr¨ Schrödinger C^{*}∗ -‐ picture by completely positive (CP) instruments defined defined on central subspaces of duals of C algebras. Mathematical analysis for CP instruments defined defined on von Neumann algebras is effieffi‐ ciently used. *. II ɹ Algebraic Quantum Theory and Sector Theory To begin with, we give an axiomatic system of algebraic quantum theory. Axiom 1 (Observables and states [17]). All the statistical aspects of of a physical system are ∗ C^{*} registered in a C -probablity space (X , ω). Observables are described registered in a ‐probablity space (\mathcal{X}, \omega) . Observables are described by by self-adjoint self‐adjoint elements elements \mat h c a l { X } \omega of of X .. On the other hand, hand, ω statistically correponds to a physical situation (or an experimental setting). setting). ∗*. okamura@math.cm.is.nagoya-u.ac.jp okamura@math.cm.is.nagoya‐u.ac.jp.

(2) 171 171 \triangle of \omega and a Borel set Δ Axiom 2 (Sector as event [17]). For a state ω probE_{\matX hcal{X},, μ of E \mu_{ω\omega}(Δ) (\triangle) gives the prob‐ \triangle under the situation described by ω. \omega . When available observability that a sector belongs to Δ observ‐ \mathcal{B} ables are restricted, probability is given by μ for some subalgebra B restricted, the coarse-grained coarse‐grained probability \mu_{ω,B \omega,\math(Δ) cal{B} (\triangle) for of of Z \mathcalω{Z}_{\o(X mega}(\mathcal). {X}) .. From now on, we shall introduce the mathematical notions appeared in the above axioms. \mathcal{X} and a state ω C^{*}∗ -algebras C^{*}∗ -algebra \omega We assume that C ω) of a C ‐algebras are unital herein. The pair (X ‐algebra X (\mathcal{,X}, \omega) ∗ \mathcal{X} is called a C C^{*} -probablity on X ‐probablity space. Axiom 11 states that every quantum system is described in the language of noncommutative probability theory (See [12] for an introduction to noncomnoncom‐ mutative probability theory). \mathcal{X} a representation of X \mathcal{X} for simplicity. C^{*}∗ -algebra Here we call a ∗ -representation ‐representation of a C ‐algebra X \mathcal{X.}. B(H) \omega on X (π \mathbb{B}(\mathcal{H}) (\pi_{\omeωga},,\matH hcal{H}_{\ωomeg,a}, \OmeΩga_{\ωomeg)a}) denotes the GNS representation of a positive linear functional ω S of B(H), \mathcal{H} . For any subset S denotes the set of bounded linear operators on a Hilbert space H. \mathbb{B}(\mathcal{H}) , ′ ′ we define = {A ∈ B(H) | ∀B ∈ S, AB define the commutant SS' of SS by SS'=\{A\in \mathbb{B}(\mathcal{H})|\forall B\in S,=ABBA} =BA\} and the double ′′ ′′ ′ ′ commutant SS" of SS by by SS"=(S')'. = (S ) . *. A^{*}∗ -representation \mathcal{X} is called a factor Definition 1 (Factor states). A factor ∗ -representation ‐representation (π, of X ‐representation (\pi, \matH) hcal{H}) of ′′ ′ ′′ \ma t h c a l { X } \omega on of X if the center Z (X ) = π(X ) ∩ π(X ) of π(X ) is trivial, i.e., Z (X ) = C1. of if \matπhcal{Z}_{\pi}(\mathcal{X})=\pi(\mathcal{X})"\cap\pi(\mathcal{X}) ’ of \pi(\mathcal{X}) ” trivial, i. e., \mathπcal{Z}_{\pi}(\mathcal{X})=\mathbb{C}1. A state ω ∗ \ma t h c a l { X } \ma t h c a l { X } \mat h cal { X } . a CC^{*} -algebra factor state on X if H) factor representation of ‐algebra X is called a factor if (π of X . ω ,, \mat (\pi_{\omega} hcal{H}) is a factor *. By the definition, definition, we can understand that each factor state corresponds to a physical situation whose values of order parameters are definite. definite. Here we classify representations and states by the quasi-equivalence quasi‐equivalence and the disjointness of them defined defined as follows. Definition 2 (Quasi-equivalence (Quasi‐equivalence and disjointness [3]). ∗ \mathcal{X.}. ,H ,H (1) Let (π (1) Let (\pi_{11}, \mat hcal{H1}_{1)}) and and (π (\pi_{22}, \mat hcal{H2}_{2)}) be b e^{*} -representations ‐representations of of X ′′ ∗ ′′ (i) (π , H ) and (π , H ) are said to be quasi-equivalent (i) (\pi_{11}, \mathcal{H1}_{1}) and (\pi_{22}, \mathcal{H2}_{2}) are said to be quasi‐equivalent if if π\pi_1{1}(X (\mathcal){X})" is i s^{*} -isomorphic ‐isomorphic to to π\pi_2{2}((X \mathcal{X)}) . (ii) ,H ,H \mathcal1{H}_{1→ }arrow \matH hcal{H}_2{2} such (ii) (π (\pi_{11}, \mat hcal{H1}_{1)}) and and (π (\pi_{22}, \mat hcal{H2}_{2)}) are are said said to to be be disjoint disjoint if if there there is is no no non-zero non‐zero VV :: H such that that X\in \mathcal{X}. π\pi_{2}(X)V=V\pi_{1}(X) (X)V = V π (X) for all X ∈ X . for 2 1 \mathcal{X.}. ω positive linear functionals on (2) Let ω \omega_{11} and \omega_{22} be \omega_{11} and \omega_{22} are (2) Let and ω be positive linearfunctionals on X and ω are said said to to be be quasi-equivalent quasi‐equivalent , H ) and (π , H ) are quasi-equivalent (or disjoint, (or disjoint, disjoint, respectively) if if (π quasi‐equivalent disjoint, respecrespec‐ ω ω ω ω (\pi_{\omega_1{1} , \mathcal{H}_{\omeg1a_{1} ) (\pi_{\omega_2{2} , \mathcal{H}_{\omeg2a_{2} ) tively). tively). \mathcal{X} be a C \mathcal{X} . Let M C^{*}∗ -algebra \mathcal{M} be a von Neumann algebra Let X ‐algebra and (π, (\pi, \matH) hcal{H}) a representation of X \mathcal{K}. M \mathcal{M} on a Hilbert space K. \mathcal{M}∗_{*} denotes the set of ultraweakly continuous linear functionals on M ∗}^{*} \mat h cal { X \mat h cal { M } and SS_{n}(\mathcal (M) denotes that of normal states on M. We define the subset V (π) of X by n . define {M}) V(\pi). VV(\pi)=\{\varphi\in (π) = {ϕ\mathcal{X}^{* ∈ X ∗}|\|exists\rho\in\pi(\mathcal{X})_{* ∃ρ ∈ π(X )′′∗ , ∀X ∈X ϕ(X) \varphi(X)=\rho(\pi(X))\} = ρ(π(X))} }", \forall X\in, \mathcal{X},. (1). and the subset S S_{\πpi}(\(X mathcal{X)}) of S(X S(\mathcal{X)}) by ′′ S\mathcal = {ϕ ∈ S(X | ∃ρ {X}) | ∈ \Sexisnts\rho\i (π(X ), ∀X ρ(π(X))}. π (X {S}_{\pi) }(\mathcal {X})=\{\varphi \in \mathcal) {S}(\mathcal n \mathcal) {S}_{n}(\pi (\mathcal∈{X})"),X\fo,rallϕ(X) X\in \mathcal{X},= \varphi (X)=\rho(\pi(X))\} .. (2). ′′ \mathcal∗∗ {X}^{* } , i.e., C \mathcal{X∗}^{*} is called a central \mathcal{L} of X ∈ Z(X Let CC be a central projection of X C\in \mathcal {Z}(\mathcal{X}")).. A subspace L \mathcal{X∗}^{*} if it has the form subspace of X \mathcal{L}=C\mathcal{X}^{* L = CX ∗} . (3).

(3) 172 \mathcal{X} be a C C^{*}∗ -algebra Proposition 3 (See [22, Chapter III] for example). Let Let X ‐algebra and (π, repre‐ (\pi, \matH) hcal{H}) a repre\mathcal{X.}. sentation of of X ∗∗ \mathcal{X}^{**}such There exists a central projection projection C(π) such that of X C(\pi) of. VV(\pi)=C(\pi)\mathcal{X}^{*}=\{C(\pi)\varphi| (π) = C(π)X ∗ = {C(π)ϕ ϕ ∈ X ∗ } = {ϕ ∈ X ∗ | C(π)ϕ = ϕ}.. \varphi\in|\mathcal{X}^{*}\}= \{\varphi\in \mathcal{X}^{*}| C(\pi)\varphi=\varphi\}. (4). \mathcal{M} be a von Neumann Proposition Let M Neumann algebra on a Proposition 4 4 (See [22, Chapter III] for example). Let ∗∗ C of \mathcal{H} . There exists a central projection C \mathcal{M}^{**}such such that M CM∗}.. \mathcal{M}_{* Hilbert space H. of M ∗ =}=C\mathcal{M}^{*. By the above two propositions, we see that VV(\pi) (π) and M \mathcal{M}∗_{*} are typical central subspaces. Both the quasi-equivalence quasi‐equivalence and the disjointness has equivalent conditions as follows: Proposition Proposition 5 5 ([3], [22, Chapter III, Proposition 2.12]). ∗ \mathcal{X} .. The following , H ) and (π ,H following conditions are equivalent: Let (π b e^{*} -representations ‐representations of of X (\pi_{11}, \mathcal{H1}_{1}) (\pi_{22}, \mat hcal{H2}_{2)}) be (i) ,H ,H (ii) (π1 ) = V (π2 ).. (i) (π (\pi_{11}, \mat hcal{H1}_{1)}) and and (π (\pi_{22}, \mat hcal{H2}_{2)}) are are quasi-equivalent. quasi‐equivalent. (ii) VV(\pi_{1})=V(\pi_{2}) (X ) = S (X ). (iv) C(π ) = C(π ). (iii) S π2 hcal{X}) . 1 2 . (iii) S_{\piπ_1{1}}(\mathcal{X})=S_{\pi_{2}}(\mat (iv) C(\pi_{1})=C(\pi_{2}) Proposition 6 ([3]). ∗ \mathcal{X} .. The following ,H ,H following conditions are equivalent: Let (π Let b e^{*} -representations ‐representations of of X (\pi_{11}, \mat hcal{H1}_{1)}) and (π (\pi_{22}, \mat hcal{H2}_{2)}) be (i) (π , H ) and (π , H ) are disjoint. (ii) V (π ) ∩ V (π 1 2 ) = {0}. (i) (\pi_{11}, \mathcal{H1}_{1}) and (\pi_{22}, \mathcal{H2}_{2}) are disjoint. (ii) V(\pi_{1})\cap V(\pi_{2})=\{0\}. (iv) (iii) 1 )C(π2 ) = 0. (iii) SS_{\πpi_1{1(X } (\mathcal){X}∩ )\cap S S_{\piπ_{22} (X (\mathcal){X})== \empty∅. set . (iv) C(π C(\pi_{1})C(\pi_{2})=0. For any pair of factor states, the following theorem of alternatives holds, which enhances the importance of the quasi-equivalence quasi‐equivalence of factor states. \mathcal{X} are either quasi-equivalent Theorem 7. Two factor factor states ω \omega_{11} and ω \omega_{22} on X quasi‐equivalent or disjoint. disjoint.. This theorem follows from the proposition below. \mathcal{X} are either quasiProposition factor representations of Proposition 8 8 (Dixmier [5, Corollary 5.3.6]). Two factor of X quasi‐ equivalent or else disjoint. disjoint.. We shall define by Ojima in order to present present the define the concept of sector, which is introduced by extension of the superselection theory by Doplicher, Haag and Roberts [6, 7] and Doplicher and Roberts [8, 9, 10] 10] into broken symmetry in a unified unified way (See also [15]). . \mathcal{X.}.\hat{\X mathcal{X} Definition 9 (Sector [16]). A A quasi-equivalent factor state is called a sector of quasi‐equivalent class of of a factor of X \mat h cal { X } . denotes the set of sectors of X . of of A sector corresponds to a “pure phase”’ phase” as a generalization of thermodynamic (pure) phase. differ‐ We can understand that two different factor states in the same sector of course describe different physical physical situations but but they share the same value of order parameters. parameters. In other words, the concept of sector is a higher object than that of states and should be regarded as a generalization of the definition definition of thermodynamic pure phases in thermodynamics into the context of quanquan‐ tum theory and (quantum or noncommutative) probability probability theory. Geometric objects living in theory\wedge . \mathcal{X} of the system. To describe macroscopic classical levels are described via the sector space X “mixed phases”, we shall use the following theorem:.

(4) 173 \mathcal{X} be a C \omega a state C^{*}∗ -algebra Theorem 10 ([3, Theorem 4.1.25 and Proposition 4.2.9]). Let Let X ‐algebra and ω \mathcal{X} .. There is aa one-to-one on X one‐to‐one correspondence between the two sets below: \omega such that (1) (1) The The set set of of barycentric barycentric measures measures μ\mu of of ω such that \in∆t_{\trρiangle}dμ(ρ) and \intE_{E_{\Xmath\∆ are disjoint disjoint \rho d\mu(\rho) and cal{X}\backsρlash \tridμ(ρ) angle} \rho d\mu(\rho) are for any Δ for \triangle\i∈n \matB(E hcal{B}(E_{\mathXcal{X). }) . \mathcal{B} of (2) Neumann subalgebras (2) The The set set of of von von Neumann subalgebras B of Z \mathcalω{Z}_{\o(X mega}(\mathcal). {X}) . ∗ ∞ \mathcal{B} is The above B. →\mu}κ(f)μ\in(f Zω(\mat(X i s^{*} -isomorphic ‐isomorphic to the image of of the map κ \kappa_μ {\mu} : L L^{\infty}(E (E_{\matXhcal,{X}μ) , \mu)\ni∋f\mapstfo\kappa_{ \math)cal{Z∈ } _{\omega} hcal{X)}) defined by . \langle\Omega_{\omega}|\kap a_{\mu}(f)\pi_{\omega}(X)\Omega_{\omefga}\(ρ)ρ(X) rangle= \int f(\rho)\rhodμ(ρ) (X)d\mu(\rho). Ωω |κμ (f )πω (X)Ωω

(5) =. (5). X\in \mathcal{X} for all X ∈ X and ff\in ∈ L∞ (EX , μ). for L^{\infty}(E_{\mathcal{X}}, \mu) .. \omega and deThe measure corresponding to the center Z de‐ \mathcalω{Z}_{\o(X mega}(\mathcal){X}) is called the central measure of ω \omega corresponding to a von noted by μ \mu_{\omega} ω . Furthermore, μ \mu_{\omeω,B ga,\mathcal{B} denotes the barycentric measure of ω \mathcal{B} of Z Neumann subalgebra B \mathcalω{Z}_{\o(X mega}(\mathcal). {X}) . Central measures of states have the following good property for our purpose. purpose.. \mathcal{X} ,, the central measure μ \omega on X \omega is Theorem 11 ([3, Theorem 4.2.10]). For every state ω \mu_{\omega} of ω ω of \ma t h c a l { X } \ma t h c a l { X } of factor states on X . If X is separable, then μ is supported pseudo-supported on the set F F_{\matX hcal{X} offactor pseudo‐supported . lf separable, \mu_{\omega} ω by F F_{\matXhcal{X}... The above theorem states that every state can be always decomposed into mutually disjoint factor by its probfactor states states by its central central measure. measure. This This fact fact allows allows us us to to interpret interpret aa general general state state as as aa prob‐ abilistic mixture of representatives of “sectors as elementary events”. Therefore, we adopted Axiom 2. C^{*}∗ -algebras III ɹ CP instruments defined on C ‐algebras. Due to previous investigations [4, 19, 19, 18], 18], it is valid that we adopt the description of processes of quantum measurement in the Schr¨ o dinger Schrödinger picture by the concept of completely positive (CP) instrument when the observable algebra of the quantum system under consideration is a von Neumann algebra. It is known that there exists its operational characterization (see [18] for instance). C^{*}∗ -algebras, In order to define define CP instruments on C ‐algebras, we have to take transitions among sectors C^{*}∗ -algebra. into account. For the purpose, we use central subspaces of the dual space of a given C ‐algebra. The results presented here are simple extensions of those for the case of CP instruments defined defined on von Neumann algebras. W^{*}∗ -algebras. \mathcal{M} and N \mathcal{N} be W Let M \mathcal{M}∗_{*} ‐algebras. P P(\mat(M hcal{M}_{*∗},\mat, N hcal{N∗}_{)*}) denotes the set of positive linear maps of M \mathcal{M}. into N pairing of M \mathcal{N∗}_{*} . Also, ·, \mathcal{M}∗_{*} and M. \{\cdot, ·

(6) \cdot\} denotes the pairing \mathcal{X} be a C \mathcal{I} is called an instrument C^{*}∗ -algebra Definition 12. Let X F) ‐algebra and (S, space. I (S, \mathcal {F}) a measurable space. for (X following three conditions: for if it satisfies the following (\mathcal, S) {X}, S) if ∗ \ma t h c a l { I } \sigma \mat h cal { F } (1) I is a map of F into Cout X{X}^{*∗ )}) for for some projections in X (1) is a map of into PP(C_{i(C n}\mathcal {X}^{*},,C_{out}\mathcal some non-zero non‐zero σ-finite ‐finite central central projections ∗∗ \mathcal { X}^{* * } . C C_{out} C_{in}, of X . in , C out of.

(7) 174 ∗ (2) ρ, 1

(8) for all n}\mathcal in X{X}^{*}.. (2) I(Δ)ρ, \{\mathcal{I}(\triangl1

(9) e)\rho,= 1\}=\{\rho, 1\rangle for all ρ\rho\i∈n C_{iC ∗ ∗ ∗ (3) ∈ (Cout X ) and mutually \mathcal{F}, C_{\dot{C \imathin } n}\matXhcal{X}^{*,}, M (3) For For every every ρ\rho\in∈ M\in(C_{out}\mathcal{X}^{*})^{*}and mutually disjoint disjoint sequence sequence {Δ of F, \{\triangle_{jj}\}_}{j\inj∈N \mathbb{N} of ∞ . \{ mathcal{I}(\bigcup_{j}\triangle_{j})\rho,M\}=\sum_{j=1}^{\infty}\{ mathcal {I}(\triangle_{j})\rho,M\}. I(∪j Δj )ρ, M

(10) =. I(Δj )ρ, M

(11) ... (6). j=1. for{I}for(\mathcal (X , {S) if I(Δ) is completely positive An instrument I \mathcal X}, S) is said to be completely positive (CP) (CP)if\mathcal{I}(\triangle) \triangle∈ \in \mathcal for all Δ F.{F}. for ∗ \mathcal{I} for (X \mathcal{F} into P When we emphasize that an instrument I Cout X{X}^{*∗ ), in X (\mathcal, S) {X}, S) is a map of F P(C_{i(C n}\mathcal {X}^{*},,C_{out}\mathcal }) , \mathcal{I} is an instrument for (X we say that I , S). in ,{{\iC (\mathcal, {C X}, C_{\dot matout h}}n}, C_{out }, S) . \mathcal{M} be a von Neumann algebra on a Hilbert space H. \mathcal{H} . As seen in Section II, the predual M Let M \mathcal{M}∗_{*} ∗ \mathcal{M} is a central subspace of M \mathcal{M}^{*} . Thus the von Neumann algebraic definition of M defimition of instruments is a special case of the above definition. definition. \mathcal{I} for \mathcal∗{X}* )) ∗ ,, we Cout , S) For , Cin ,C_{in}, \varphi on in X For every every CP CP instrument instrument I for (X (\mathcal{X}, C_{out}, S) and and normal normal state state ϕ on (C (Cın we define define \triangle∈ \in \mathcal the probability measure Iϕ F) by Iϕ(Δ) F.{F}. \Vert \mathcal{I}\varphi\Vert on (S, (S, \mathcal {F}) by \Vert \mathcal{I}g\Vert(\triangle)== \Vert \matI(Δ)ϕ hcal{I}(\triangle)\varphi\Vert for all Δ ∗ ∗ ∗ ∗ \mathcal{I} for (X \mathcal{∗I}^{*} : (C \mathcal{I} is × \mathcal F {→ (C For every instrument I out X{X}^{*)})^{*}\cross in X{X}^{*) (\mathcal, S), {X}, S) , the dual map I (C_{out}\mathcal F}arrow(C_{i n}\mathcal })^{*} of I defined defined by ∗ (7) I(Δ)ρ, =e=\{ ρ, \langle \mathcal{I}(\trianglM e)\rho,

(12) M\rangl \rho,\matIhcal{I}^(M, {*}(M, \trianglΔ)

(13) e)\rangle *. ∗ ∗ ∗ ∗ \triangle∈ \in \mathcal {F} . For every map J for all ρ\rho\in∈C_{in}\mathcal Cin X{X}^{*∗ ,}, M ∈ (Cout X ∗ )∗ and Δ F. ×F (Cn}\mat \mathcal{J} : (C inhX M\in(C_{out}\mathcal{X}^{*})^{*} (C_{outout }\mathX cal{X}^{*}))^{*}\cross \mathcal→ {F}arrow(C_{i cal{X}^{*)})^{*} satisfying the following three conditions, there uniquely exists an instrument (X (\mathcal, S) {X}, S) such that ∗: J \mathcal{= J}=\mathI cal{I}^{*} : ∗ ∗ \triangle\i∈ n \mathcal {F} , the M →\mathcal{J}(M, J (M, Δ) ∈ (Cin X ∗}))^{*∗} is (1) For F, out X ) })^{*∋ (1) For every every Δ the map map (C (C_{out}\mathcal{X}^{* }\ni M\mapsto \triangle)\in(C_{in} \mathcal{X}^{* is normal, normal, positive positive and linear. linear. (2) (1, S) =S)=1. 1. (2) J\mathcal{J}(1, ∗ (3) For every ρ ∈ (Cout X ∗ )∗ and \mathcal{F}, { \imatin h} n}\X mathcal{X}^{,*}, M (3) For every \rho\in∈C_{\dotC M\in(C_{out}\mathcal{X}^{*})^{*} and mutually mutually disjoint disjoint sequence sequence {Δ of F, \{\triangle_{jj}\}_}{j\inj∈N \mathbb{N} of ∞ . \langle\rho,\mathcal{J}(M,\bigcup_{j}\triangle_{j})\rangle=\sum_{j=1} ^{\infty}\langle\rho,\mathcal{J}(M,\triangle_{j})\rangle. ρ, J (M, ∪j Δj )

(14) =. ρ, J (M, Δj )

(15) ... (8). j=1. \mathcal{I} denotes the dual map I \mathcal{∗I}^{*} of an instrument I \mathcal{I} for (X From now on, I (\mathcal, S). {X}, S) . ∗ C^{*} \mat h cal { N } -algebras and M, N von Neumann algebras. X ⊗ Let X , Y be C \ m a t h c a l { Y } \mat h cal { X } \ ot i m es_{ \min}\matY hcal{Y} denotes the injective \mathcal{X}, \mathcal{M}, ‐algebras min ∗ \ma t h c a l { X } W^{*} -tensor \mathcal{M} and N \mathcal{N}. For every tensor product of X and Y, \mathcal{M}\over⊗ line{\otimes}\N mathcal{N} does the W \mathcal{Y} , and M ‐tensor product of M \mathcal{I} for (X : (C X{X}^{*∗ )})^{*∗}\oti⊗ CP instrument I , S), there exists a unital (binormal) CP map Ψ \ P s i _ { \ m a t h c a l { I } , I out minn} (\mathcal{X}, S) (C_{out}\mathcal mes_{\mi ∞ ∗ ∗ L (S, I) → (C X ) such that L^{\infty}(S,\mathcal{I})arrow(C_{\dot{ \imath} nin }\mathcal{X}^{*})^{*}. Ψ (M{I}}(M\oti ⊗ m[χ ])angl= I(M, \PsiI_{\mathcal es[\chi∆_{\tri e}])=\mathcal {I}(M, \triΔ) angle). (9). \triangle∈ \in \mathcal for all M ∈ (Cout X ∗ )∗ and Δ F.{F}. M\in(C_{out}\mathcal{X}^{*})^{*} Here we define define the normal extension property, family of posterior states and measuring propro‐ cess. All of them have played the role of deepening physics and mathematics of instruments defined defined von Neumann algebras [18]. We will see in Theorem 16 16 that their importance is not ∗ C^{*} -algebras. different for the case of instruments defined defined on C ‐algebras..

(16) 175 \mathcal{I} be a CP \mathcal{I} is CP instrument for Definition 13 (The normal extension property). Let Let I for (X (\mathcal, S). {X}, S) . I CP said to have the normal extension property (NEP) if there exists a unital normal CP map Ψ \ o v erline{\Psi_{\mIathcal{I} : if ∞ ∗ ∗ ∗ ∗ that (C (C_{outout }\mathcalX{X}^{*}))^{*}\overl⊗ ine{\otL imes}L^{\(S, infty}(S,\matI) hcal{I})→ arrow(C_ {(C \dot{in\imatXh} n}\mat)hcal{Xsuch }^{*})^{*}such. Ψ \ overliIne{\P|si_(C {\mathcout al{I} }|_{(CX_{out∗}\m)ath∗cal{⊗X}^{*})min ^{*}\otimesL_{\m∞in} L(S,I) ^{\infty}(S,\mathcal= {I})}=\Psi_{Ψ \mathcaIl{I} .. (10). \mathcal{I} be an instrument for Definition 14 (Family of posterior states). Let Let I for (X \varphi a normal (\mathcal, S) {X}, S) and ϕ ∗ ∗ ∗ ∗ \mat h cal { X } * X ) . A family {ϕ } of normal states on (C X ) is called a family postestate on (C in out state on (Cın ) . A family \{\varphis_{s}\s∈S }_{s\in S} of normal states on (C_{out}\mathcal{X}^{* })^{*}is called a family of ofposte‐ rior states with respect to (I, following two conditions: if it satisfies the following (\mathcal{I},ϕ) \varphi) if ∗ (1) function SS\ni∋s\mapsto\varphi_{s}\in s → ϕs ∈C_{out}\mathcal{X}^{*}is Cout X ∗ is weakly (1) The The function weakly Iϕ-measurable. \Vert \mathcal{I}\varphi\Vert ‐measurable. ∗ ∗ X ) and Δ ∈ F, (2) For all M ∈ (C \triangle\in \mathcal{F}, out (2) For all M\in(C_{out}\mathcal{X}^{*})^{*}and I(Δ)ϕ, M

(17) = ϕs , M

(18) dIϕ(s).. (11) *. *. \{\mathcal{I}(\triangle)\varphi, M\rangle=\int_{\triangle}\langle\varphi_{s}, M\rangle d\Vert \mathcal{I}\varphi\Vert(s) ∆. \mathcal{K} , a normal Definition 15 (Measuring process). A 4-tuple = (K, σ, E, a Hilbert space K, 4‐tuple M ofa \mathbb{M}=(\mathcal{K}, \sigma,UE, U)) of state σ\sigma on B(K), \mathcal{H}⊗ \otimes \matK, hcal{K} , is \mathbb{B}(\mathcal{K}) , a spectral measure EE : F \mathcal→ {F}arrow B(K) B(\mathcal{K}) and a unitary operator UU on H ∗ ∗ (M, Δ) | M ∈ (C X ) , Δ ∈ F} ⊂ called a measuring process for for (X , S) if it satisfies {I if M in (\mathcal{X}, S) \{\mathcal{I}_{\mathbb{M} (M, \triangle)|M\in(C_{\dot{ \imath} n}\mathcal{X} ^{*})^{*}, \triangle\in \mathcal{F}\}\subset ∗∗ \mat h cal { M } \ma t h c a l { H } M,, where C \mathcal{X}^{* ,}, H is a Hilbert space on which C_{in} non‐zero σ-finite a‐finite central projection of of X in is a non-zero ∗ ∗ as faithfully represents elements of \mathcal{I}_M {\mathb {M} : B(H) faithfully of (C in X ) })^{* (C_{in}\mathcal{X}^{* }as bounded operators and I B(\mathcal{H})\cross×\mathF cal{F}arrow→ \mathbb{B(H) B}(\mathcal{H}) 1 is defined by by^{1} I\mathcal{I}_{\mathbb{M}}(X, (id ⊗ σ)[U ∗ (X}(X\otimes ⊗ E(Δ))U (12) M (X, Δ) = \triangle)=(id\otimes\sigma)[U^{*. E(\triangle))U]]. \triangle∈ \in \mathcal for all X F.{F}. for X\in \mat∈hbb{B(H) B}(\mathcal{H}) and Δ. The following is the main theorem of this section. \mathcal{X} be a C \mathcal{I} for C^{*}∗ -algebra Theorem 16. Let Let X F) For an instrument I for ‐algebra and (S, space. For (S, \mathcal {F}) a measurable space. (X , S), the following conditions are equivalent: following (\mathcal{X}, S) , \mathcal{I} has (1) NEP. (1) I has the the NEP. ∗ ∗ strongly measurable family {ϕ (2) For every \varphi on (2) For eveノツ normal normal state state ϕ on (C (C_{1}\1matX hcal{X}^{)*})^{*} ,, there there exists exists aastハongly measurablefamily \{\varphis_}{S}\s∈S }_{8\in S} of of posterior states with respect to (I, (\mathcal{I},ϕ). \varphi) . If C C_{\dot{{\imath}}n}=C_{out} If below. in = Cout ,, the above conditions are equivalent to the condition below. \mat h bb{ M } (3) There exists a measuring processes M for (X , S) such that I = }_{\matM hbb{M}.. (3) There exists a measuring processes for (\mathcal{X}, S) such that \mathcal{I}=\mathcal{II. Proof. We can prove the theorem in the same way as [18, Theorems 3.4 and 5.5]. Proof.. \square. ࢀߟจ‫ݙ‬ 参考文献 [1] W. Arveson, An invitation to CC^{*}∗ -algebras, ‐algebras, (Springer, New York, 2012). ∗ C^{*} [2] K. Bichteler, A generalization to the non-separable -algebras, non‐separable case of Takesaki’s duality theorem for C ‐algebras, InIn‐ vent. Math. 9 (1969), 89–98. 89‐98.. [3] O. Bratteli & D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics Vol.1 &D. OperatorAlgebras Vol.1 (2nd printing of 2nd ed.), (Springer, 2002). 1 \mathcal{M} and \mathcal{N} be id\otimes\sigma \mathcal{M}\over⊗ line{\otimes}\N mathcal{N} by Let M be von ⊗ σ :: M by ρ = hcal{N∗}_{*,}, we lLet and N von Neumann Neumann algebras. algebras. For For every every σ\sigma\i∈n \matN we define define id \langle\rh⊗ o\otimes\siσ, gma, X

(19) X\rangle= X\in \mat∈ hcal{M}M \overline{\ot⊗ imes}\matN hcal{N.}. ρ, ⊗ σ)(X)

(20) for all ρ\rho\i∈n \matM hcal{M}_{*∗} and X \{\rho,(id (id\otimes\sigma)(X)\}.

(21) 176 [4] E.B. Davies & J.T. Lewis, An operational approach to quantum probability, Commun. Math. Phys. 17 (1970), &J.T. 239‐260. 239–260. [5] J. Dixmier, CC^{*}∗ -Algebras. ‐Algebras. (North-Holland, (North‐Holland, Amsterdam, 1977). 1977). [6] S. Doplicher, R. Haag & J.E. Roberts, Fields, observables and gauge transformations I & II, Comm. Math. Phys. 13 (1969), 1–23; 1−23; ibid. ibid. 15 (1969), 173–200. 173‐200. [7] S. Doplicher, R. Haag & J.E. Roberts, Local observables and particle statistics, I & II, Comm. Math. Phys. 23 (1971), 199–230; 199−230; ibid. ibid. 35 (1974), 49–85. 49‐85. ∗ C^{*} [8] S. Doplicher & J.E. Roberts, Endomorphism of C -algebras, Doplicher& ‐algebras, cross products and duality for compact groups, Ann. of Math. 130 (1989), 75–119. 75‐119.. [9] S. Doplicher & J.E. Roberts, A new duality theory for compact groups, Invent. Math. 98 (1989), 157–218. Doplicher& 157‐218. [10] S. Doplicher & J.E. Roberts, Why there is a field Doplicher& field algebra with a compact gauge group describing the superssupers‐ election structure in particle physics, Comm. Math. Phys. 131 (1990), 51–107. 51‐107. [11] G.G. Emch, Algebraic methods in statistical mechanics and quantum field field theory, (Wiley-Interscience, (Wiley‐Interscience, New York, 1972). 1972). [12] A. Hora & N. Obata, Quantum probability and spectral analysis of of graphs, Theoretical and Mathematical Physics, (Springer, Berlin, 2007). [13] M. Nakamura & H.Umegaki, On von Neumann’s theory of measurements in quantum statistics, Math. Japon. Nakamura&H.Umegaki, 7 (1962), 151–157. 151‐157. [14] J. von Neumann, Mathematische Grundlagen der Quantenmechanik, (Springer, Berlin, 1932); 1932); Mathematical Foundations of of Quantum Mechanics, (Princeton UP, Princeton, 1955). 1955). [15] I. Ojima, A unified –Order parameters of symmeunified scheme for generalized sectors based on selection criteria ‐Order symme‐ tries and of thermality and physical meanings of adjunctions–, adjunctions‐, Open Sys. Inform. Dyn. 10 (2003), 235–279. 235‐279. [16] I. Ojima, “Micro-Macro “Micro‐Macro Duality in Quantum Physics,”pp.143-161 Physics,”pp.143‐161 in Proc. Intern. Conf. on Stochastic AnalAnal‐ ysis, Classical Cıassical and Quantum, (World (Worıd Scientific, Scientific, 2005), arXiv:math-ph/0502038. arXiv:math‐ph/0502038. [17] I. Ojima, K. Okamura & H. Saigo, Derivation of Born Rule from Algebraic Aıgebraic and Statistical Axioms, Open Sys. Inform. Dyn. 21 (2014), 1450005. 1450005. [18] K. Okamura & M. Ozawa, Measurement theory in local quantum physics, J. Math. Phys. 57 Okamura& 57 (2016), 015209. [19] M. Ozawa, Quantum measuring processes of continuous obsevables, J. Math. Phys. 25 (1984), 79–87. 79‐87. [20] M. Ozawa, Uncertainty relations for noise and disturbance in generalized quantum measurements, Ann. Phys. (N.Y.) 331 (2004), 350–416. 350‐416. C^{*}∗ -algebras, [21] M. Takesaki, A duality in the representation theory of C ‐algebras, Ann. of Math. (2) 85 (1967), 370-382. 370‐382.. [22] M. Takesaki, Theory of Algebras 1, I, (Springer, Berlin, 1979). of Operator OperatorAlgebras 1979). [23] M. Takesaki, Theory of Algebras II, lL (Springer, Berlin, 2003). of Operator OperatorAlgebras [24] H. Umegaki, Conditional expectation in an operator algebra, Tohoku Math. J. 6(2) (1954), 177–181. 177‐181..

(22)