aa r X i v : . [ qu a n t - ph ] S e p MEASURING PROCESSES AND REPEATABILITYHYPOTHESIS M ASANAO O ZAWA
Department of Mathematics, College of General EducationNagoya University, Chikusa-ku, Nagoya 464-8601, Japan
Abstract
Srinivas [Commun. Math. Phys. 71 (1980), 131–158] proposeda postulate in quantum mechanics that extends the von Neumann-L ¨uders collapse postulate to observables with continuous spectrum.His collapse postulate does not determine a unique state change, butdepends on a particular choice of an invariant mean. To clear thephysical significance of employing different invariant means, we con-struct different measuring processes of the same observable satisfyingthe Srinivas collapse postulate corresponding to any given invariantmeans. Our construction extends the von Neumann type measuringprocess with the meter being the position observable to the one withthe apparatus prepared in a non-normal state. It is shown that thegiven invariant mean corresponds to the momentum distribution ofthe apparatus in the initial state, which is determined as a non-normalstate, called a Dirac state, such that the momentum distribution is thegiven invariant mean and that the position distribution is the Diracmeasure.
The problem of extending the von Neumann-L¨uders collapse postulate [4,5]to observables with continuous spectrum is one of the major problems ofthe quantum theory of measurement. Recently. Srinivas [11] posed a setof postulates which gave an answer to this problem. However. it does notseem to be a complete solution. The following two problems remain.(1) The Srinivas collapse postulate is not consistent with the σ -additivityof probability distributions and it requires ad hoc treatment of calculus ofprobability and expectation. How can we improve his set of postulates inorder to retain the consistency with the σ -additivity of probability?12) His collapse postulate depends on a particular choice of an invariantmean. What is the physical significance of employing different invariantmeans? Can we characterize the various different ways of measuring thesame observable [11;p.149]?The purpose of this paper is to resolve the second question by con-structing different measuring processes of the same observable satisfyingthe Srinivas collapse postulate corresponding to the given invariant means.In our construction, the pointer position of the apparatus is the position ob-servable and the given invariant mean corresponds to the momentum distri-bution at the initial state of the apparatus. Thus the choice of the invariantmean characterizes the state preparation of the apparatus.For the general theory of quantum measurements of continuous observ-ables, we shall refer to Davies [1], Holevo [3] and Ozawa [6–10]. Theentire discussion including the solution of the first question above will bepublished elsewhere. In this paper, we shall deal with quantum systems with finite degrees of free-dom. In the conventional formulation, the states of a system are representedby density operators on a separable Hilbert space H and the observables arerepresented by self-adjoint operators on H . In this formulation, however, asshown in [7;Theorem 6.6], we cannot construct measuring processes satis-fying the repeatability hypothesis, which follows from the Srinivas postu-lates; hence some generalization of the framework of quantum mechanicsis necessary. We adopt the formulation that the states of a system are rep-resented by norm one positive linear functionals on the algebra L ( H ) ofbounded operators on H ; states corresponding to density operators will becalled normal states . For any state σ and compatible observables X, Y weshall denote by Pr { X ∈ dx, Y ∈ dy k σ } the joint distribution of the out-comes of the simultaneous measurement of X and Y . Our basic assumptionis that Pr { X ∈ dx, Y ∈ dy k σ } is a σ -additive probability distribution on R uniquely determined by the relation Z R f ( x, y ) Pr { X ∈ dx, Y ∈ dy k σ } = h f ( X, Y ) , σ i , (2.1)for all f ∈ C ( R ) , where R = R ∪ { + ∞} ∪ {−∞} and C ( R ) standsfor the space of continuous functions on R . If σ is a normal state, Eq.(2.1)is reduced to the usual statistical formula. Apart from classical probabilitytheory, we can consider another type of joint distributions in quantum me-chanics. Let h X, Y i be an ordered pair of any observables. We shall denote2y Pr { X ∈ dx ; Y ∈ dy k σ } the joint distribution of the outcomes of thesuccessive measurement of X and Y , performed in this order, in the initialstate σ . Let η be a fixed invariant mean on the space CB ( R ) of continuousbounded functions on R . Let X be an observable. Denote by E Xη the normone projection from L ( H ) onto { X ( B ); B ∈ B ( R ) } ′ such thatTr [ E Xη [ A ] ρ ] = η u Tr [ e iuX Ae − iuX ρ ] , (2.2)for all normal state ρ and A ∈ L ( H ) , where B ( R ) stands for the Borel σ -field of R , X ( B ) stands for the spectral projection of X for B ∈ B ( R ) , and ′ stands for the operation making the commutant in L ( H ) . Then by a slightmodification, the Srinivas collapse postulate asserts the following relationfor the successive measurement of X and any bounded observable Y : Z R y Pr { X ∈ B ; Y ∈ dy k ρ } = Tr [ X ( B ) E Xη [ Y ] ρ ] , (2.3)for all normal state ρ and B ∈ B ( R ) . Obviously, this relation impliesthe following generalized Born statistical formula [11]: If X and Y arecompatible then Pr { X ∈ B ; Y ∈ C k ρ } = Tr { X ( B ) Y ( C ) ρ } , (2.4)for all normal state ρ and B, C ∈ B ( R ) . Our purpose is to construct a mea-suring process of X which satisfies the Srinivas collapse postulate Eq.(2.3).Throughout this paper, we shall fix an invariant mean η which is, by atechnical reason, a topological invariant mean on CB ( R ) (cf. [2; p.24]). In this section, we shall consider a quantum system with a single degree offreedom. Denote by Q the position observable and by P the momentumobservable. A state δ on L ( L ( R )) is called an η -Dirac state if it satisfiesthe following conditions (D1)–(D2):(D1) For each f ∈ CB ( R ) , h f ( Q ) , δ i = f (0) .(D2) For each f ∈ CB ( R ) , h f ( P ) , δ i = η ( f ) . Lemma 3.1
For any f ∈ CB ( R ) , E Qη ( f ( P )) = η ( f )1 . roof. Let ξ be a unit vector in L ( R ) . Let g ( p ) = | ξ ( − p ) | . Then g isa density function on R . For any f ∈ CB ( R ) , we have h ξ |E Qη ( f ( P )) | ξ i = η u h ξ | e iuQ f ( P ) e − iuQ | ξ i = η u h ξ | f ( P + u | ξ i = η u Z R f ( p + u ) | ξ ( p ) | dp = η u ( f ∗ g )( u ) = η ( f ) , where f ∗ g stands for the convolution of f and g . It follows that E Qη ( f ( P )) = η ( f )1 . QED
Theorem 3.2
For every topologically invariant mean η , there exists an η -Dirac state.Proof. Let φ be a state on { Q ( B ); B ∈ B ( R ) } ′ such that h f ( Q ) , φ i = f (0) for all f ∈ CB ( R ) and δ a state on L ( L ( R )) such that h A, δ i = hE Qη ( A ) , φ i for all A ∈ L ( L ( R )) . Then by Lemma 3.1, δ is obviously an η -Dirac state. QED
Let X be an observable of a quantum system I described by a Hilbert space H . We consider the following measuring process of X by an apparatus sys-tem II. The apparatus system II is a system with a single degree of freedomdescribed by the Hilbert space K = L ( R ) . Thus the composite system I+IIis described by the Hilbert space H ⊗ K , which will be identified with theHilbert space L ( R ; H ) of all norm square integrable H -valued functionson R by the Schr¨odinger representation of K . The pointer position of theapparatus system is the position observable Q . The interaction between themeasured system I and the apparatus system II is given by the followingHamiltonian: H int = λ ( X ⊗ P ) , (4.1)where P is the momentum of the apparatus. The strength λ of the interactionis assumed to be sufficiently large that other terms in the Hamiltonian canbe ignored. Hence the Schr¨odinger equation will be ( h = 2 π ) ∂∂t Ψ t ( q ) = − λ X ⊗ ∂∂t ! Ψ t ( q ) , (4.2)in the q-representation, where Ψ t ∈ H ⊗ K . The measurement is carriedout by the interaction during a finite time interval from t = 0 to t = 1 /λ .4he outcome of this measurement is obtained by the measurement of Q attime t = 1 /λ . The statistics of this measurement depends on the initiallyprepared state σ of the apparatus. According to [7;Theorem 6.6], if σ isa normal state then this measurement cannot satisfy Eq. (2.3). Now weassume that the initial state of the apparatus is an η -Dirac state δ and weshall call this measuring process as a canonical measuring process of X with preparation δ .In order to obtain the solution of Eq. (4.2), assume the initial condition Ψ = ψ ⊗ α, (4.3)where ψ ∈ H and α ∈ K . The solution of the Schr¨odinger equation is givenby Ψ t = e − itλ ( X ⊗ P ) ψ ⊗ α, (4.4)and hence for any ψ ∈ H and β ∈ K , we have Z R h φ ⊗ β ( q ) | Ψ t ( q ) i dq = Z R e − itλxp h φ ⊗ β | X ( dx ) ⊗ P ( dp ) | ψ ⊗ α i = Z R h β | e − itλxP | α ih φ | X ( dx ) | ψ i = Z R ( Z R β ( q ) ∗ α ( q − tλx ) dq ) h φ | X ( dx ) | ψ i = Z R α ( q − tλx ) h β ( q ) φ | X ( dx ) | ψ i dq = Z R h φ ⊗ β ( q ) | α ( q − tλX ) | ψ i dq. It follows that Ψ t ( q ) = α ( q − tλX ) ψ. (4.5)For t = 1 /λ , we have Ψ /λ ( q ) = α ( q − X ) ψ. (4.6) Theorem 4.1
For any f ∈ L ∞ ( R ) , we have U ∗ t (1 ⊗ f ( Q )) U t = f ( tλ ( X ⊗
1) + 1 ⊗ Q ) , (4.7) where U t = e − itλ ( X ⊗ P ) . roof. By Eq. (4.5), for any ψ ∈ H and α ∈ K , we have h ψ ⊗ α | U ∗ t (1 ⊗ f ( Q )) U t | ψ ⊗ α i = Z R f ( q ) h ψ | α ( q − tλX ) ∗ α ( q − tλX ) | ψ i dq = Z R f ( q ) | α ( q − tλx ) | dq h ψ | X ( dx ) | ψ i = Z R f ( q + tλx ) | α ( q ) | dq h ψ | X ( dx ) | ψ i = h ψ ⊗ α | f ( tλ ( X ⊗
1) + 1 ⊗ Q ) | ψ ⊗ α i . Thus the assertion holds.
QED
Suppose that the state of the measured system I at t = 0 is a normal state ρ . We shall denote by ρ ⊗ δ the state at t = 0 of the composite systemI+II, which is defined by the relation h T, ρ ⊗ δ i = hE ρ ( T ) , δ i for all T ∈L ( H ) ⊗ L ( K ) , where E ρ : L ( H ) ⊗ L ( K ) → L ( K ) is a normal completelypositive map such that E ρ ( A ⊗ A ) = Tr [ A ρ ] A for all A ∈ L ( H ) . Thusletting U = e − i ( X ⊗ P ) , the state at t = 1 /λ of the composite system I+II is U ( ρ ⊗ δ ) U ∗ .Let Y be a bounded observable of the system I. By the argument similarwith [7; § .3], the joint distribution Pr { X ∈ dx ; Y ∈ dy k ρ } of the outcomesof the successive measurement of X and Y coincides with the joint distri-bution Pr { Q ∈ dq, Y ∈ dy k U ( ρ ⊗ δ ) U ∗ } of the simultaneous measurementof the pointer position Q and Y at time t = 1 /λ , i.e., Pr { X ∈ dx ; Y ∈ dy k ρ } = Pr { Q ∈ dx, Y ∈ dy k U ( ρ ⊗ δ ) U ∗ } . (5.1)The rest of this section will be devoted to proving Eq. (2.3) for thismeasuring process. Denote by E δ the completely positive map E δ : L ( H ) ⊗L ( K ) → L ( H ) defined by Tr [ E δ [ T ] ρ ] = h T, ρ ⊗ δ i for all normal state ρ and T ∈ L ( H ) ⊗ L ( K ) . From Eq.(2.1), for any f, g ∈ C ( R ) , we have Z R f ( x ) g ( y ) Pr { [ Q ∈ dx, Y ∈ dy k U ( ρ ⊗ δ ) U ∗ } = h g ( Y ) ⊗ f ( Q ) , U ( ρ ⊗ δ ) U ∗ i = Tr [ E δ [ U ∗ ( g ( Y ) ⊗ f ( Q )) U ] ρ ] (5.2)for all normal state ρ . Lemma 5.1
Let X be an observable of the system I. Then for any f ∈ CB ( R ) , E δ [ f ( X ⊗ , ⊗ Q )] = f ( X, . roof. Let ψ ∈ H . For any α ∈ K , we have h α |E | ψ ih ψ | [ f ( X ⊗ , ⊗ Q )] | α i = h ψ ⊗ α | f ( X ⊗ , ⊗ Q ) | ψ ⊗ α i = Z R Z R f ( x, q ) h ψ | X ( dx ) | ψ ih α | Q ( dq ) | α i = h α | F ( Q ) | α i , where F ( q ) = R R f ( x, q ) h ψ | X ( dx ) | ψ i . Thus E | ψ ih ψ | [ f ( X ⊗ , ⊗ Q ] = F ( Q ) . It is easy to see that F ∈ CB ( R ) and hence h F ( Q ) , δ i = F (0) by(D1). We see that h F ( Q ) , δ i = hE | ψ ih ψ | [ f ( X ⊗ , ⊗ Q )] , δ i = h ψ |E | δ ih δ | [ f ( X ⊗ , ⊗ Q )] | ψ i , and F (0) = h ψ | f ( X, | ψ i . It follows that E δ [ f ( X ⊗ , ⊗ Q )] = f ( X, . QED
Theorem 5.2
For any f ∈ CB ( R ) , we have E δ [ U ∗ (1 ⊗ f ( Q )) U ] = f ( X ) . Proof.
From Theorem 4.1, U ∗ (1 ⊗ f ( Q )) U = f ( X ⊗ ⊗ Q ) andhence the assertion follows from applying Lemma 5.1 to g ∈ CB ( R ) suchthat g ( x, y ) = f ( x + y ) . QED
Theorem 5.3
For any Y ∈ L ( H ) , we have E δ [ U ∗ ( Y ⊗ U ] = E Xη ( Y ) . (5.3) Proof.
Let ψ ∈ H . For any α ∈ K , we have h α |E | ψ ih ψ | [ U ∗ ( Y ⊗ U ] | α i = h ψ ⊗ α | U ∗ ( Y ⊗ U | ψ ⊗ α i = Z R h α ( p ) ψ | e ipX Y e − ipX | α ( p ) ψ i dp = Z R h ψ | e ipX Y e − ipX | ψ i h α | E P ( dp ) | α i = h α | F ( P ) | α i , F ( p ) = h ψ | e ipX Y e − ipX | ψ i . Consequently, E | ψ ih ψ | [ U ∗ ( Y ⊗ U ] = F ( P ) . Since F ∈ CB ( R ) , we have from (D2), h F ( P ) , δ i = η ( F ) . We seethat h F ( P ) , δ i = hE | ψ ih ψ | [ U ∗ ( Y ⊗ U ] , δ i = h ψ |E δ [ U ∗ ( Y ⊗ I ) U ] | ψ i , and η ( F ) = η p h ψ | e ipX Y e − ipX | ψ i = h ψ |E Xη ( Y ) | ψ i . Thus, E δ [ U ∗ ( Y ⊗ U ] = E Xη ( Y ) . QED
Theorem 5.4
Let Y ∈ L ( H ) and f ∈ CB ( R ) . Then we have E δ [ U ∗ ( Y ⊗ f ( Q )) U ] = f ( X ) E Xη [ Y ] . (5.4) Proof.
By the Stinespring theorem [12], there is a Hilbert space W , anisometry V : H ⊗ K → W and a ∗ -representation π : L ( H ) ⊗ L ( K ) →L ( W ) such that E δ [ U ∗ AU ] = V ∗ π ( A ) V for all A ∈ L ( H ) ⊗ L ( K ) . ByTheorem 5.2, V ∗ π (1 ⊗ f ( Q )) V = f ( X ) . Thus by easy computations, ( π (1 ⊗ f ( Q )) V − V f ( X )) ∗ ( π (1 ⊗ f ( Q )) V − V f ( X )) = 0 . It follows that π (1 ⊗ f ( Q )) V = V f ( X ) , and hence from Theorem 5.3, wehave E δ [ U ∗ ( Y ⊗ f ( Q )) U ] = V ∗ π ( Y ⊗ f ( Q )) V = V ∗ π ( Y ⊗ V f ( X )= E Xη [ Y ] f ( X ) = f ( X ) E Xη [ Y ] . QED
Now we can prove that the canonical measuring process of X withpreparation δ , where δ is an η -Dirac state, satisfies the Srinivas collapsepostulate for the given invariant mean η . Theorem 5.5
For any bounded observable Y ∈ L ( H ) and B ∈ B ( R ) , wehave Z R y Pr { X ∈ B ; Y ∈ dy k ρ } = Tr [ X ( B ) E Xη [ Y ] ρ ] for all normal state ρ . roof. Denote by C ( R ) the space of continuous functions on R van-ishing at infinity. Let Y be a bounded observable and ρ a normal state. FromEqs. (5.1) and (5.2) and from Theorem 5.4, for any f, g ∈ C ( R ) we have Z R f ( x ) g ( y ) Pr { X ∈ dx ; Y ∈ dy k ρ } = Tr [ f ( X ) E Xη [ g ( Y )] ρ ] . By the bounded convergence theorem and the normality of the state ρ , theset of all Borel functions f satisfying the above equality is closed underbounded pointwise convergence and contains C ( R ) . Thus the equalityholds for all bounded Borel functions f . Since Y is bounded, there is afunction h ∈ C ( R ) such that h ( y ) = y on the spectrum of Y. Let f = χ B and g = h . We have f ( X ) = X ( B ) and f ( Y ) = Y so that we obtain thedesired equality. QED