Mathematical structure of Positive Operator Valued Measures and Applications

Transcription

1 DE TTK 1949 Mathematica structure of Positive Operator Vaued Measures and Appications egyetemi doktori (PhD) értekezés Beneduci Roberto Témavezető: Dr. Monár Lajos Debreceni Egyetem Természettudományi Doktori Tanács Matematika- és Számítástudományok Doktori Iskoa Debrecen, 2014.

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3 Ezen értekezést a Debreceni Egyetem Természettudományi Doktori Tanács Matematika- és Számítástudományok Doktori Iskoa Funkcionáanaízis programja keretében készítettem a Debreceni Egyetem természettudományi doktori (PhD) fokozatának enyerése céjábó. Debrecen, Beneduci Roberto jeöt Tanúsítom, hogy Beneduci Roberto doktorjeöt között a fent megnevezett doktori program keretében irányításomma végezte munkáját. Az értekezésben fogat eredményekhez a jeöt önáó akotó tevékenységéve meghatározóan hozzájárut. Az értekezés efogadását javasom. Debrecen, Dr. Monár Lajos témavezető

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5 Mathematica structure of Positive Operator Vaued Measures and Appications Értekezés a doktori (PhD) fokozat megszerzése érdekében a matematika tudományágban. Írta: Beneduci Roberto fizikus. Készüt a Debreceni Egyetem Matematika- és Számítástudományok Doktori Iskoa Funkcionáanaízis doktori programja keretében. A doktori szigorati bizottság: Témavezető: Dr. Monár Lajos enök: Dr tagok: Dr Dr A doktori szigorat időpontja: Az értekezés bíráói: A bíráóbizottság: Dr Dr Dr enök: Dr tagok: Dr Dr Dr Dr Az értekezés védésének időpontja:

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7 Aknowedgement I woud ike to thank Prof. Dr. Lajos Monár for his invauabe support and encouragement. I woud aso ike to thank Dott. Gergo Nagy for his hep in the fina processing of the present dissertation.

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9 Contents 1 Introduction Main properties of POVMs Positive operator vaued measures in the quantum mechanica framework Objectives and resuts of the present work Characterization of commutative POVMs On the separation properties of µ Characterization by means of Feer Markov kernes Characterization of POVMs admitting strong Feer Markov Kernes Absoutey continuous POVMs Unsharp Position Observabe Interpretation of the smearing Appendix On the von Neumann agebra generated by F Sequences of continuous functions POVMs and Naimark s diation Sequences of probabiity measures Sharp versions as projections of Naimark operators Open probems Construction of the universa antismearing function Appendix On the discreteness of the sharp version A A Usefu Lemma On the informationa content of a POVM Informationa content of a quantum observabe On the informationa content of E and F

10 5 Uniform continuity, norm-1 property and ocaization Characterization of uniform continuity On the meaning of uniform continuity Uniform continuity and norm-1 property Anaysis of some reevant physica exampes Bounded position operator Phase observabe Unsharp number observabe Position and Momentum Uniform continuity and ocaization Locaization in phase space Locaization in Configuration Space

11 Chapter 1 Introduction The present dissertation is devoted to the study of positive operator vaued measures (POVMs) which were introduced in the 40 s [73, 74, 71] in order to study sef-adjoint extensions of symmetric operators. A POVM is a σ-additive map F : B(X) F(H) from the Bore σ-agebra of a topoogica space X to the space F(H) of positive operators ess than the identity (effects). This generaizes the concept of spectra resoution of the identity, which is a σ-additive map E : B(R) E(H) from the Bore σ-agebra of the reas to the space of orthogona projections E(H). It is then apparent that the generaization from spectra measures to POVMs is based on the repacement of R and E(H) by a topoogica space X and the space of effects F(H) respectivey. In the 70 s severa schoars [32, 45, 48, 50, 49, 66, 2, 78, 27] used POVMs as the main too in the description of the quantum measurement process and to formuate a genera theory of statistica decision. Moreover POVMs suggested an extension of the concept of quantum observabe which was previousy encoded in the mathematica structure of a spectra measure or, equivaenty, of a sef-adjoint operator. Such an extension turned out to be very fruitfu since it permitted a mathematica representation of the time observabe, the photon ocaization observabe and the phase observabe which resuted to be impossibe in the od framework (spectra measures). Another improvement aowed by POVMs consists in the possibiity of buiding a representation of standard quantum mechanics in a phase space by mapping density operators to positive definite density distribution functions on a sympectic space [78, 83, 27, 85]. Nowdays, POVMs are a standard too in quantum information theory and quantum optics [50, 87, 90]. It was then natura both from the mathematica and the physica viewpoint to ask what are the reationships between POVMs and spectra measures. Four 1

12 possibe answers have been given each one corresponding to a different characterizations of POVMs [73, 48, 2, 31, 10, 11, 22, 53]. Athough the answer given by Naimark [73, 71, 1, 83] is the most powerfu since it refers to genera POVMs and is not imited to the commutative case, it is to be confronted with the probem of the physica interpretation of the extended Hibert space it introduces. As we sha see, if one avoids the commitment with an extended Hibert space, a cear answer to our question can be given in the commutative case [48, 2, 31, 10, 11, 12, 53], the commutative POVMs being the most simiar to the spectra measures. The present dissertation is based on the author s contribution to the formuation of one of the possibe characterizations of commutative POVMs (chapter 2). The anaysis of the reationships between such characterization and Naimark s theorem is the topic of chapter 3. Chapter 4 is devoted to the anaysis of its reevance to the concept of informationa content of an observabe. The ast chapter is devoted to the characterization of the uniform continuity of a genera POVM (not necessariy commutative), to the anaysis of the POVMs with the norm-1 property and to the anaysis of the reevance of norm-1 property and uniform continuity to the ocaization probem in reativistic quantum mechanics. Next we outine the main properties of POVMs, then we show how they emerge in the quantum context and briefy outine the main objectives and resuts of the present work. 1.1 Main properties of POVMs In the present section we reca the main properties of POVMs. We restrict ourseves to the case of POVMs defined on the Bore σ-agebra of a topoogica set X. For a more genera exposition we refer to the book by Berberian [24]. There are other concepts and properties concerning POVMs that, when necessary, wi be introduced in each singe chapter. Definition Let X be a topoogica space and B(X) the Bore σ-agebra on X. A POVM is a map F : B(X) F(H) such that: F ( ) n = F ( n ). n=1 where, { n } is a countabe famiy of disjoint sets in B(X) and the series converges in the weak operator topoogy. It is said to be normaized if n=1 F (X) = 1. 2

13 Definition A POVM is said to be commutative if (1.1) [ F ( 1 ), F ( 2 ) ] = 0 1, 2 B(X). Definition A POVM is said to be orthogona if (1.2) F ( 1 )F ( 2 ) = 0 if 1 2 =. Definition A PVM is an orthogona, normaized POVM. In the case of a PVM E we have 0 = E( )[1 E( )] = E( ) E 2 ( ). Therefore, E( ) is a projection operator for every B(X). We have proved the foowing proposition. Proposition A PVM E on X is a map E : B(X) E(H) from the Bore σ-agebra of B(X) to the space of projection operators on H. Definition A rea PVM E : B(R) F(H) is said to be a spectra measure. Definition The spectrum σ(f ) of a POVM F is the set of points x X such that F ( ) 0, for any open set containing x. The spectrum σ(f ) of a POVM F is a cosed set since its compement X σ(f ) is the union of a the open sets X such that F ( ) = 0. Definition The von Neumann agebra A W (F ) generated by the POVM F is the von Neumann agebra generated by the set {F ( )} B(X). In the foowing we use the symbos w im and u im to denote the imit in the weak operator topoogy and the imit in the uniform operator topoogy respectivey. Definition A POVM is reguar if for any Bore set, w im i F (G j ) = F ( ) = w im i F (O j ) where, {G j } j N, G j, is a decreasing famiy of open sets and {O j } j N, O j, is a increasing famiy of compact sets and the convergence is in the weak operator topoogy. We reca [64] that a topoogica space (X, τ) is second countabe if it has a countabe basis for its topoogy τ; i.e., if there is a countabe subset B of τ such that each member of τ is the union of members of B. 3

14 Proposition A POVM defined on a Hausdorff ocay compact, second countabe space X is reguar. Proof. A ocay compact Hausdorff space is reguar. (See Ref. [69], page 205). By the Urysohn s theorem, a second countabe reguar space is metrizabe (see [69], page 215). Moreover, the second countabiity impies the σ-compactness ([69], page 289). In a metrizabe σ-compact space the ring of Bore sets coincides with the ring of Baire sets (see page 25 in [24]) and the thesis comes from the fact that each Baire POVM is reguar (see Theorem 18 in [24]). We can introduce integration with respect to a POVM. Indeed, for any ψ H, the expression F ( )ψ, ψ defines a probabiity measure and we wi use the symbo d F λ ψ, ψ to mean integration with respect to the measure F ( )ψ, ψ. We sha say that a measurabe function f : N X f(n) R is amost everywhere (a.e.) one-to-one with respect to a POVM F if it is one-to-one on a subset N N such that N N is a nu set with respect to F. We sha say that a function f : X R is bounded with respect to a POVM F, if it is equa amost everywhere to a bounded function g with respect to F, that is, if f = g a.e. with respect to the measure F ( )ψ, ψ, ψ H. For any rea, bounded and measurabe function f and for any POVM F, there is a unique [24] bounded sef-adjoint operator B L s (H) such that (1.3) Bψ, ψ = f(λ)d F λ ψ, ψ, for each ψ H. If equation (1.3) is satisfied, we write B = f(λ)df λ or B = f(λ)f (dλ) equivaenty. By the spectra theorem [35, 79], rea PVMs E (spectra measures) are in a one-to-one correspondence with sef-adjoint operators A, the correspondence being given by A = λde λ. Moreover in this case, a functiona cacuus can be deveoped. Indeed, if f : R R is a measurabe rea-vaued function, we can define the sef-adjoint operator [79] (1.4) f(a) = f(λ)de λ, where E is the PVM corresponding to A. If f is bounded, then f(a) is bounded [79]. In particuar, ( ) i (1.5) E[i] := t i de t = t de t = A i 4

15 and A = t de t is the generator of the von Neumann agebra generated by E. We point out that if F is not projection vaued, equations (1.5) and (1.4) do not hod [59] and, in order to recover the generator of the von Neumann agebra generated by F, we need a the moments of F. In particuar, in the case of a rea commutative POVM F with bounded spectrum and such that F ( ) is discrete for any (see chapter 3 for the detais), we have A = α i F [i], α i 0, i=0 α i < i=0 where, A is a generator of the von Neumann agebra A W (F ). The foowing resut due to Naimark shows that a POVM F in a Hibert space H can aways be interpreted as the restriction to H of a PVM E defined in an extended Hibert space H +. Theorem (Naimark [72, 1, 50, 71]) Let F be a POVM of the Hibert space H. Then, there exist a Hibert space H + H and a PVM E + of the space H + such that F ( ) = P + E + ( ) H where P + is the operator of projection onto H. Naimark s theorem is a powerfu resut on the reationships between PVMs and POVMs but from the physica viewpoint its interpretation is not cear. That is due to the difficuties in interpreting the Hibert space H +. As we sha see in chapter 3, a reationships between the Naimark s extension of F and the sharp version A of F can be estabished in the commutative case. That coud provide new insights in the probem of the interpretation of the Naimark s extension. 1.2 Positive operator vaued measures in the quantum mechanica framework As we have aready seen the set of PVMs is a subset of the set of POVMs. Moreover, rea PVMs (spectra measures) are in one-to-one correspondence with sef-adjoint operators (spectra theorem) [79] and are used in standard quantum mechanics to represent quantum observabes. It was pointed out [2, 27, 32, 50, 78, 82] that POVMs are more suitabe than spectra measures in representing quantum observabes. From a genera theoretica viewpoint, the introduction of POVMs can be justified by anayzing the statistica description of a measurement. Indeed, a measurement procedure can be described as an affine map from the set of states 5

16 S into the set of probabiity measures on B(X). The set of states represents the set of possibe preparation procedures of the system whie the set of probabiity measures represents the statistica distribution of the resuts of the possibe measurements. It was shown [50, 51] that there exists a one-to-one correspondence between POVMs F : B(X) F(H) and affine maps S µ F S ( ) from the set of states S into the set of probabiity measures on B(X). Moreover, this correspondence is determined by the reation µ F S ( ) = T r[sf ( )]. That aows one to interpret the number µ F S ( ) = T r[sf ( )] as the probabiity that the outcomes of a measurement of the observabe F (corresponding to a POVM F ) is in when the physica system is prepared in the state S S. We reca that an anaogous reation hods for standard observabes which are represented by rea PVMs E : B(R) E(H), that is: µ E S ( ) = T r[se( )]. That shows again why the quantum observabes described by POVMs are a generaization of the standard quantum observabes. They are caed generaized observabes or unsharp observabes and, as we aready pointed out, pay a key roe in quantum information theory, quantum optics, quantum estimation theory [45, 50, 66, 27, 83, 90], quantum measurement theory, and in the phase space formuation of quantum mechanics [83, 19, 20]. 1.3 Objectives and resuts of the present work The brief outine we made above raises the probem of giving a cear physica meaning to the POVMs as we as to study the reationships between POVMs and PVMs. As we sha see the answer to the second probem shed ight aso on the first probem. The main aim of the present thesis is to answer the foowing questions: 1) What are the reationships between POVMs and spectra measures? As we sha see a commutative POVM F is the smearing of a spectra measure E (caed the sharp version of F ). Then, the foowing question raises. What are the physica meaning and the mathematica structure of the smearing which connects F to E? (Chapter 2). 2) In chapter 2 we characterize a commutative POVM as the smearing of its sharp version. What are the reationships between this characterization and Naimark s diation theorem? Is there a universa one-to-one function f such that the sharp version A F of any commutative POVM F can be written in the form A F = f(t) df t? (Chapter 3). 6

17 3) Is there any oss of information during the smearing from the sharp version A of F to F? (Chapter 4). 4) Can we give conditions for a POVM to have the norm-1 property? Is the norm-1 property reevant to ocaization observabes? (Chapter 5). Here is a brief ist of the resuts we get. A more detaied overview can be found in the summary at the end of the thesis. 1. Characterization of commutative POVMs [22, 11]: We generaize the resuts I got in Ref.s [22, 11]. In particuar, we prove that each commutative POVM F is the smearing of a spectra measure E reaized by a Feer Markov Kerne. That suggests an interpretation of commutative POVMs as the randomization of rea PVMs. Moreover, we characterize the POVMs whose smearing can be reaized by strong Feer Markov kernes. 2. Anaysis of the reationships between the resuts in item1 and Naimark s diation theorem [16, 17, 18]. We prove that the sefadjoint operator corresponding to the spectra measure E, of which F is the smearing, is the projection of a Naimark s operator [12, 13, 14, 15, 16, 17, 18]. That suggests an interpretation of the Naimark s diation of a commutative POVM. 3. Anaysis of the informationa content of a POVM [15]. We introduce an equivaence reation on the set of observabes which we compare with other we known equivaence reations and prove that it is the ony one for which E is aways equivaent to F [15]. 4. The study of the uniform continuity and norm-1 property of a POVM [21, 23]. We characterize the uniform continuity of a POVM, give a necessary condition for the norm-1 property ( F ( ) = 1, whenever F ( ) 0) and expain its physica meaning and its reevance to the ocaization probem [21]. Then, we prove that severa reevant ocaization observabes cannot have such a property [23]. 7

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19 Chapter 2 Characterization of commutative POVMs The present chapter focuses on the anaysis of commutative POVMs. We reca that the set of PVMs is a subset of the set of commutative POVMs. That suggests to start our anaysis of the mathematica structure of POVMs by studying the reationships between commutative POVMs and PVMs. We prove a strong connection between POVMs and spectra measures, i.e., each commutative POVM F is the smearing (reaized by a Feer Markov kerne) of a spectra measure E. In particuar we generaize the resuts I proved in Ref.s [22, 11] to the case of a POVM F : B(X) F(H) defined on a Hausdorff, ocay compact, second countabe space X. Some previous resuts on commutative POVMs of which the ones we present here are a strengthening are contained in Ref.s [10, 11, 12, 13, 14, 15, 16, 17, 18, 53, 54]. Other important characterizations of POVMs and some anaysis of their reationships can be found in Ref.s [2, 48, 6, 55]. First we introduce the concepts of weak Markov kerne, Markov kerne and smearing. In the foowing the symbos (Λ, B(Λ)) and (X, B(X)) denote measurabe spaces where Λ and X are topoogica spaces and B(Λ) and B(X) the corresponding Bore σ-agebras. Definition A Markov kerne is a map µ : Λ B(X) [0, 1] such that, 1. µ ( ) is a measurabe function for each B(X), 2. µ ( ) (λ) is a probabiity measure for each λ Λ. Definition Let ν be a measure on Λ. A map µ : Λ B(X) R is a weak Markov kerne with respect to ν if: 9

20 1. µ ( ) is a measurabe function for each B(X), 2. for each B(X), 0 µ (λ) 1, ν a.e., 3. µ X (λ) = 1, µ (λ) = 0, ν a.e., 4. for any sequence { i } i N, i j =, µ ( i)(λ) = µ ( i i)(λ), ν a.e. i Definition The map µ : Λ B(X) [0, 1] is a weak Markov kerne with respect to a PVM E : B(Λ) E(H) if it is a weak Markov kerne with respect to each measure ν ψ ( ) := E( ) ψ, ψ, ψ H. In the foowing, by a weak Markov kerne µ we mean a weak Markov kerne with respect to a PVM E. Moreover the function λ µ (λ) wi be denoted indifferenty by µ or µ ( ). Definition A POVM F : B(X) F(H) is said to be a smearing of a POVM E : B(Λ) E(H) if there exists a weak Markov kerne µ : Λ B(X) [0, 1] such that, F ( ) = µ (λ)de λ, B(X). Λ Exampe In the standard formuation of quantum mechanics, the operator Q : L 2 (R) L 2 (R) ψ(x) L 2 (R) Qψ := xψ(x) is used to represent the position observabe. A more reaistic description of the position observabe of a quantum partice is given by a smearing of Q as, for exampe, the optima position POVM where, F Q ( ) = 1 2 π ( µ (x) = 1 2 π e (x y) 2 ) 2 2 dy dex Q = e (x y) dy, B(R) µ (x) de Q x defines a Markov kerne and E Q is the spectra measure corresponding to the position operator Q. In the foowing, the symbo µ is used to denote both Markov kernes and weak Markov kernes. The symbos A and B are used to denote sef-adjoint operators. 10

21 Definition Whenever F, A, and µ are such that F ( ) = µ (A), B(X), we say that (F, A, µ) is a von Neumann tripet. Definition The von Neumann agebra A W (F ) generated by the POVM F is the von Neumann agebra generated by the set {F ( )} B(X). Definition If (F, A, µ) is a von Neumann tripet and A and F generate the same von Neumann agebra then A is named the sharp version of F. 2.1 On the separation properties of µ In the foowing, we assume X to be a Hausdorff, ocay compact, second countabe topoogica space. The symbo S denotes a countabe basis for the topoogy of X. The symbo R(S) denotes the ring generated by S. Notice that by theorem c, page 24, in Ref. [44], R(S) is countabe too. Moreover, R(S) generates the Bore σ-agebra B(X). A weak Markov kerne µ such that (F, A, µ) is a von Neumann tripet, separates the point of Γ σ(a) if the famiy of functions {µ } B(R) separates the points of Γ or, in other words, if the set functions {µ ( ) (λ)} λ Γ are distinct. It is then natura to ask if in genera µ has that property. The foowing theorem answers in the positive. Theorem Let (F, A, µ) be a von Neumann tripet and suppose that A is a sharp version of F. Then, there exists a set Γ σ(a), E A (Γ) = 1, such that the famiy of functions {µ ( )} B(X) separates the points of Γ. Proof. In the foowing, A W (F ) denotes the von Neumann agebra generated by {F ( )} B(X), O 2 := {F ( )} R(S) and A C (O 2 ) is the C -agebra generated by O 2. The von Neumann agebra generated by A C (O 2 ) coincides with A W (F ) (see appendix A). Moreover, A W (F ) = A W (A) since A is the sharp version of F and generates A W (F ). By the Gefand-Naimark theorem [35, 72], there is a * isomorphism φ between A C (O 2 ) and the agebra of continuous functions C(Λ 2 ) where Λ 2 is the spectrum of A C (O 2 ). Moreover, f C(Λ 2 ) φ(f) = f(λ) dẽλ Λ 2 where, Ẽ is the spectra measure from the Bore σ agebra B(Λ 2) to E(H) whose existence is assured by theorem 1, page 895, in Ref. [35]. The Gefand-Naimark isomorphism φ can be extended to a homomorphism between the agebra of the Bore functions on Λ 2 and the von Neumann agebra A W (F ) = A W (A) generated 11

22 by A C (O 2 ) (see Ref. [34], page 360, section 3). Therefore, there is a Bore function h such that (2.1) A = h(λ) dẽλ Λ 2 Let S be a countabe basis for the topoogy of X. Let { i } i N denote an enumeration of the set R(S). Since A C (O 2 ) is the smaest uniform cosed agebra containing {F ( i )} i N, C(Λ 2 ) is the smaest uniform cosed agebra of functions containing {ν i := φ 1 (F ( i ))} i N. In other words {ν i } i N generates C(Λ 2 ). The Stone-Weierstrass theorem [35] assures that {ν i } i N separates the points in Λ 2. On the other hand, the fact that (F, A, µ) is a von Neumann tripet, impies that, for each i R(S), there is a Bore function µ i such that ν i (λ) dẽλ = F ( i ) = µ i (A) = µ i (h(λ)) dẽλ. Λ 2 Λ 2 Then, for each i R(S), there is a set M i Λ 2, Ẽ(M i) = 1, such that (2.2) µ i (h(λ)) = ν i (λ), λ M i. Let M := i=1 M i. Then, and, for each i N, Ẽ(M) = im Ẽ( n i=1m i ) = im n n n Ẽ(M i ) = 1 i=1 (2.3) (µ i h)(λ) = ν i (λ), λ M Λ 2. Since {ν i } i N separates the points in Λ 2, it separates the points in M. Then, equation (2.3) impies that {µ i } i N separates the points in Γ := h(m). Moreover 1, E A (Γ) = E A (h(m)) = Ẽ[h 1 (h(m))] = 1 1 Notice that h(m) is a Bore set. In order to prove that, we first reca that Λ 2 is a Poish space (that is, a compete, separabe, space [63]). Indeed, by theorem 11, page 871, in Ref. [35], it is homeomorphic to a cosed subspace of the Cartesian product i=1 σ(f ( i)), where σ(f ( i)) is a compete separabe metric space, and by theorem 2, page 406, and theorem 6, page 156, in Ref. [64], it is compete and separabe. Moreover, h is measurabe and injective on M. Therefore, Sousin s theorem (see theorem 9 page 440 and Coroary 1 page 442 in Ref. [63]) assures that h(m) is a Bore set. 12

23 where, E A is the spectra measure defined by the reation E A ( ) = Ẽ(h 1 ( )) and such that, A = x de A x whie, h 1 (h(m)) is a Bore set containing M. We have proved that the set of functions {µ i } i N separates the points of Γ and that E A (Γ) = 1. In other words, µ ( ) (λ) µ ( ) (λ ), λ λ, λ, λ Γ. 2.2 Characterization by means of Feer Markov kernes In the present section we give a characterization of the commutative POVMs. First we introduce the concept of strong Markov kerne, i.e., a weak Markov kerne µ ( ) ( ) : Λ B(X) [0, 1] with respect to a PVM E : B(Λ) E(H) such that µ ( ) (λ) is a probabiity measure for each λ Γ Λ, E(Γ) = 1. Then, we prove (theorem 2.2.3) that a POVM F is commutative if and ony if there are a sef-adjoint operator A and a strong Markov kerne µ such that (F, A, µ) is a von Neumann tripet, A is the sharp version of F, and µ is continuous for each R, where R is a ring which generates B(X). It is worth remarking that µ ( ) ( ) : Γ B(X) [0, 1] is a Feer Markov kerne. Therefore, F is commutative if and ony if there exists a bounded sef-adjoint operator A and a Feer Markov kerne µ such that F ( ) = µ (λ) de λ. Γ Moreover, the famiy of functions {µ } R separates the points in Γ (see theorems and 2.2.3). In order to prove the main theorem we need the foowing definitions. Definition Let E : B(Λ) E(H) be a PVM. The map µ ( ) ( ) : Λ B(X) [0, 1] is a strong Markov kerne with respect to E if it is a weak Markov kerne and there exists a set Γ Λ, E(Γ) = 1, such that µ ( ) ( ) : Γ B(X) [0, 1] is a Markov kerne with respect to E. A strong Markov kerne is denoted by the symbo (µ, E, Γ Λ). 13

24 Definition A Feer Markov kerne is a Markov kerne µ ( ) ( ) : Λ B(X) [0, 1] such that the function G(λ) = f(x) dµ x (λ), λ Λ is continuous and bounded whenever f is continuous and bounded. X Theorem A POVM F : B(X) F(H) is commutative if and ony if, there exists a bounded sef-adjoint operator A = λ de λ with spectrum σ(a) [0, 1] and a strong Markov Kerne (µ, E, Γ σ(a)) such that: 1) µ ( ) : σ(a) [0, 1] is continuous for each R(S), 2) F ( ) = Γ µ (λ) de λ, B(X). 3) A W (F ) = A W (A). 4) µ separates the points in Γ. Moreover, µ : Γ B(X) [0, 1] is a Feer Markov kerne. Proof. Let A W (F ) be the von Neumann agebra generated by F. A W (F ) coincides with the von Neumann agebra generated by the set {F ( )} R(S) where, R(S) B(X) is the ring generated by S, the countabe sub-basis for the topoogy of X (see appendix A for the proof). We reca that both S and R(S) are countabe (see theorem c, page 24, in Ref. [44]). Now, we proceed to the proof of the existence of A. Let { i } i N be an enumeration of the set R(S) and O 2 := {F ( )} R(S). Let E (i) denote the spectra measure corresponding to F ( i ) O 2. We have F ( i ) = x de x (i). Therefore, for each i, k N there exists a division { (i,k) j } j=1,...,mi,k of [0, 1] such that (2.4) m i,k x (i,k) j E (i) ( (i,k) j ) F ( i ) 1 k. j=1 By the spectra theorem [35] the von Neumann agebra A W (F ) contains a the projection operators in the spectra resoution of F ( ), B(X). Therefore, the von Neumann agebra A W (D) generated by the set D := {E (i) ( i,k j ), j m i,k, i, k N} is contained in A W (F ) and then (2.5) A W (D) A W (F ) = A W (O 2 ). Moreover, the C -agebra A C (D) generated by D contains the C -agebra A C (O 2 ) generated by O 2 (see equation (2.4)). Summing up the preceding observations, we have A C (O 2 ) A C (D) A W (F ). 14

25 By the doube commutant theorem [56], so that (see equation 2.5), A W (F ) = [A C (O 2 )] [A C (D)] = A W (D) (2.6) A W (D) = A W (F ). By theorem 11, page 871 in Ref. [35], the spectrum Λ of A C (D) is homeomorphic to a cosed subset of i=1 {0, 1}. Let π : Λ i=1 {0, 1} denote the homeomorphism between the two spaces. Now, if we identify Λ with a cosed subset of i=1 {0, 1}, we can prove the existence of a continuous function distinguishing the points of Λ. Indeed, et π(λ) = x := (x 1,..., x n,... ) i=1 {0, 1}. The function f(λ) = is continuous and injective and then it distinguishes the points of Λ. Moreover, since Λ and [0, 1] are Hausdorff, the map f : Λ f(λ) is a homeomorphism. By theorem 1, page 895, in Ref. [35], there exists a spectra measure Ẽ : B(Λ) F(H) such that the map i=1 x i 3 i (2.7) T : C(Λ) B(H) g T (g) = g(λ)dẽλ Λ defines an isometric -isomorphism between A C (D) and C(Λ). The fact that f distinguishes the points of Λ, impies that the sef-adjoint operator A = f(λ) dẽλ Λ is a generator of the von Neumann agebra A W (D) = A W (F ). Indeed, by the Stone-Weierstrass theorem, C(Λ) is singy generated, in particuar f is a generator. Then, the isomorphism between A C (D) and C(Λ) assures that A C (D) is singy generated and that A is a generator. Hence, A W (F ) = A W (D) = [A C (D)] is singy generated. In particuar, A generates A W (F ), i.e., A W (F ) = A W (A). Now, we proceed to the proof of the existence of the weak Markov kerne ν such that (F, A, ν) is a von Neumann tripet. 15

26 By (2.7), for each R(S), there exists a continuous function γ C(Λ) such that F ( ) = γ (λ) dẽλ. Λ Now, we show that, for each R(S), there is a continuous function ν : σ(a) [0, 1] from the spectrum of A to the interva [0, 1] such that ν (f(λ)) = γ (λ), λ Λ, and F ( ) = ν (A). To prove this, et us consider the function ν (t) := (γ f 1 )(t), R(S). It is continuous since it is the composition of continuous functions. Moreover, since ν (f(λ)) = γ (f 1 (f(λ))) = γ (λ). we have, ν (A) = F ( ), R(S). Indeed, by the change of measure principe (page 894, ref. [35]), F ( ) = γ (λ) dẽλ = γ (f 1 (f(λ))) dẽλ Λ Λ = γ (f 1 (t)) de t = ν (t) de t = ν (A) σ(a) where σ(a) = f(λ) is the spectrum of A and E is the spectra measure corresponding to A defined by the reation E( ) = Ẽ(f 1 ( )), B(σ(A)) (see coroary 10, page 902, in Ref. [35]). For each λ σ(a), the map ν ( ) (λ) : R(S) [0, 1] defines an additive set function. Indeed, et R(S) be the disjoint union of the sets 1, 2 R(S). Then, ν ( 1 2)(λ) de λ = F ( 1 2 ) = F ( 1 ) + F ( 1 ) = ν 1 (λ) de λ + ν 2 (λ) de λ σ(a) = [ν 1 (λ) + ν 2 (λ)] de λ so that, by the continuity of the functions ν ( 1)(λ) and ν ( 2)(λ), we get (see theorem 1, page 895, in Ref. [35]) ν ( 1)(λ) + ν ( 2)(λ) = ν ( 1 2)(λ), λ σ(a). 16

27 Now, we extend ν to a B(X). Since A is the generator of A W (F ), for each B(X), there exists a Bore function ω such that. F ( ) = ω (t) de t = (ω f)(λ) dẽλ σ(a) Then, we can consider the map ν : σ(a) B(X) [0, 1] defined as foows { ν (λ) if R(S) (2.8) ν (λ) = ω (λ) if / R(S). Since ν coincides with ν on R(S) it is additive on R(S). In order to prove that ν is a weak Markov kerne, et us consider a set B(X) which is the disjoint union of the sets { i } i N, i B(X). Then, ν ( i=1 i)(x) de x = ν (x)de x = F ( ) = F ( i ) i=1 Λ = ν i (x) de x = so that, by Coroary 9, page 900, in Ref. [35], i=1 i=1 ν i (x) de x ν i (x) = ν (x), i=1 E a.e, which impies that ν : σ(a) B(X) [0, 1] is a weak Markov kerne. In particuar (F, A, ν) is a von Neumann tripet. Now, we proceed to prove the existence of the Markov kerne µ : Γ B(X) [0, 1] such that items 1, 2, and 3 of the theorem are satisfied. Since X is Hausdorff ocay compact second countabe, it is a Poish space (theorem 5.3 in [62]). Then, to each weak Markov kerne ν : σ(a) B(X) [0, 1] such that (F, A, ν) is a von Neumann tripet, there corresponds a Markov kerne φ : σ(a) B(X) [0, 1] such that (F, A, φ) is a von Neumann tripet [53, 54, 11]. Then, for each B(X), ν (λ) de λ = F ( ) = φ (λ) de λ hence, (2.9) φ (λ) = ν (λ), E a.e. 17

28 Now, et { i } i N be an enumeration of R(S). By equation (2.9), for each i N, there is a set N i σ(a), E(N i ) = 0, such that (2.10) φ i (λ) = ν i (λ), λ σ(a) N i. Then, for each i N, (2.11) φ i (λ) = ν i (λ), λ σ(a) N where, N := i=1n i, E(N) = 0. Therefore, for amost a λ σ(a), ν ( ) (λ) is σ-additive on R(S). Now, we can define the map { ν ( ) (λ) λ N µ ( ) (λ) = φ ( ) (λ) λ σ(a) N If we put Γ = σ(a) N, we have that µ ( ) ( ) : Γ B(X) [0, 1] is a Markov kerne. Therefore, µ ( ) ( ) : σ(a) B(X) [0, 1] is a strong Markov kerne. Notice that, for each R(S) and λ σ(a), µ (λ) = ν (λ) so that, µ is continuous for each R(S) and additive on R(S). We aso have, µ (A) = φ (A) = F ( ), R(S). We have proved items 1, 2, and 3. Item 4 comes from theorem It remains to prove that µ is a Feer Markov kerne. By item 1, µ is continuous for each R(S). Notice that for each open set O B(X), there is a countabe famiy of sets i R(S) such that O = i=1 i. Therefore, by theorem 2.2 in Ref. [25], and the continuity of µ for each R(S), im n λ n = λ impies, im n f(t) µ t (λ n ) = f(t) µ t (λ), f C b (X) where, C b (X) is the space of bounded, continuous rea functions. Than, G(λ) := f(t) µt (λ) is continuous whenever f is continuous and µ is a Feer Markov kerne. Finay, we note that F ( ) = µ (A) impies the commutativity of F and that ends the proof. In the proof of theorem we have aso proved the foowing theorem. 18

29 Theorem A POVM F : B(X) F(H) is commutative if and ony if, there exists a bounded sef-adjoint operator A = λ de λ with spectrum σ(a) [0, 1] and a Markov Kerne µ : B(X) σ(a) [0, 1] such that F ( ) = µ (λ) de λ, B(X). σ(a) Proof. We have aready shown the existence of a weak Markov kerne ν : σ(a) B(X) [0, 1] which is additive on the the ring R(S) and such that F ( ) = σ(a) µ (λ) de λ, for each B(X) (see equation 2.8). Moreover, since X is a Poish space, to each weak Markov kerne ν : σ(a) B(X) [0, 1] there corresponds a Markov kerne µ : σ(a) B(X) [0, 1] such that (F, A, µ) is a von Neumann tripet [53, 54, 11]. Finay, we note that the sharp version A of F is unique up to amost everywhere bijections. Theorem [14] Let (F, A; µ) be a von Neumann tripet such that A is the sharp version of F. Than, i) for any for von Neumann tripet ( F, B, µ), there exists a rea function g such that A = g(b), ii) for any von Neumann tripet (F, A, ν) satisfying item i) there exists an amost everywhere one-to-one function h such that A = h(a). 2.3 Characterization of POVMs admitting strong Feer Markov Kernes In the ast section we proved that each commutative POVM admits a strong Markov kerne µ such that µ is a continuous function for each R(S) where, R(S) is a ring which generates the Bore σ-agebra B(X). In the present section we characterize the commutative POVMs for which the Markov kerne µ, whose existence was proved in theorem 2.2.4, is such that µ is continuous for each B(X). Whenever such a Markov kerne exists, we say that the POVM admits a strong Feer Markov kerne. In particuar, we prove that a commutative POVM F admits a strong Feer Markov kerne if and ony if F is uniformy continuous. Definition Let F : B(X) F(H). Let = i=1 i, i j =. If im n i=1 n F ( i ) = F ( ) 19

30 in the uniform operator topoogy then we say that F is uniformy continuous. Notice that the term uniformy continuous derives from the fact that the σ- additivity of F in the uniform operator topoogy is equivaent to the continuity in the uniform operator topoogy. Anaogousy, the σ-additivity of F in the weak operator topoogy is equivaent to the continuity of F in the weak operator topoogy [24]. Definition A Markov kerne µ ( ) ( ) : [0, 1] B(X) [0, 1] is said to be strong Feer if µ is a continuous function for each B(X). Definition We say that a commutative POVM admits a strong Feer Markov kerne if there exists a strong Feer Markov kerne µ such that F ( ) = µ (λ) de λ, where E is the sharp version of F. In order to prove the main theorem of the section we need the foowing emma. Lemma Let F be uniformy continuous. Let µ be a weak Markov kerne and (F, A, µ) a von Neumann tripet. Suppose that µ is continuous for each R(S). Then, for each λ σ(a), µ ( ) (λ) is σ-additive on R(S). Proof. Let, i R(S), i j =, i=1 i =. Then, ( 0 = u im F ( ) F ( n i=1 i ) ) n (µ n = u im (λ) µ i (λ) ) de λ. n By the uniform continuity of F and theorem 1, page 895, in Ref. [35], it foows that, ɛ > 0, there exists a number n N, such that n > n impies, i=1 (2.12) n (µ n µ (λ) µ i (λ) = (λ) µ i (λ) ) de λ i=1 i=1 = F ( ) F ( n i=1 i ) ɛ. By equation (2.12), n µ (λ) µ i (λ) ɛ, λ σ(a). i=1 20

31 Theorem A commutative POVM F : B(X) F(H) admits a strong Feer Markov kerne if and ony if it is uniformy continuous. Proof. Suppose F is uniformy continuous. By theorem 2.2.3, there is a weak Markov kerne µ : σ(a) B(X) [0, 1] such that µ ( ) is continuous for every R(S) and a sef-adjoint operator A such that (F, A, µ) is a von Neumann tripet. By emma 2.3.4, µ is σ-additive on R(S). Therefore Charateodory theorem [65] assures that the map µ : σ(a) R(S) [0, 1] can be extended to a map µ : σ(a) B(X) [0, 1] whose restriction to R(S) coincides with µ. Now we prove that µ is a Markov kerne such that F ( ) = µ (A) and that µ is continuous for each B(X). We proceed by steps. 1) Open sets. Each open set G is the union of a countabe famiy of sets in S, i.e., G = i=1 i, i S. Let us define the set G n := n i=1 i. Therefore, G n G. Moreover, µ Gn is continuous for each n N, and Then, u im n F (G n) = F (G). F (G) = u im F (G i ) = u im i i µ Gi (λ) de λ. By the uniform continuity of F, it foows that, ɛ > 0, there exists a number n N, such that n, m > n impies, (2.13) µ Gn (λ) µ Gm (λ) = [ µ Gn (λ) µ Gm (λ)] de λ By equation (2.13), (2.14) µ Gn (λ) µ Gm (λ) ɛ, λ σ(a). Since µ ( ) (λ) is a probabiity measure, im µ G i (λ) = µ G (λ), i = F (G n ) F (G m ) ɛ. λ σ(a). By equation (2.14), the convergence is uniform and this proves the continuity of µ G. Moreover, F (G) = im F (G i ) = im µ Gi (λ) de λ = µ G (λ) de λ = µ G (A). i i 2) G δ sets. For each G δ set there exists [24] a famiy of open sets {G i } i N, G δ G i, such that i=1 G i = G δ. Then, by the uniform continuity of F, F (G δ ) = F ( i=1g i ) = u im n F ( n i=1g i ) = u im n F ( G n ) 21

32 where, G n := n i=1 G i and G n G δ. By theorem 1, page 895, in Ref. [35], it foows that, ɛ > 0, there exists a number n N, such that n, m > n impies, (2.15) µ Gn (λ) µ Gm (λ) = (µ Gn (λ) µ Gm (λ)) de λ ɛ. Since µ ( ) (λ) is a probabiity measure for each λ σ(a), im µ (λ) = µ Gi Gδ (λ). i By equation (2.15) the convergence is uniform and then µ Gδ is continuous. Moreover, F (G δ ) = im F ( G i ) = im µ i i Gi (λ) de λ = µ Gδ (λ) de λ = µ Gδ (A). 3) Bore sets. We use transfinite induction [63, 33]. Let G 0 be the famiy of open sets in X, ω 1 the first uncountabe ordina and G α, α < ω 1 the Bore hierarchy (see page 236 in Ref. [63]). In particuar, G 1 = G δ, G 2 = G δσ, G 3 = G δσδ,... and G α = ( β<α G β ) σ for each imit ordina α. By means of the same reasoning that we used in items 1 and 2, one can prove the continuity of µ as we as that µ (A) = F ( ) whenever is of the kind G δ,σ, G δσδ.... Anaogousy, if µ is continuous for each G α then, µ is continuous for each in G α+1 and µ (A) = F ( ). Indeed, each set in G α+1 is either the countabe union or the countabe intersection of sets in G α and the reasoning in items 1 and 2 can be used. If α is a imit ordina and µ is continuous for each G β, β < α, then, µ is continuous for each G α = ( β<α G β ) σ and µ (A) = F ( ). Indeed, each set in G α is the countabe union of sets in β<α G β and the reasoning used in item 1 can be used. Therefore, by transfinite induction, µ is continuous for each α<ω1 G α = B(X) [63] and µ (A) = F ( ). In order to prove the second part of the theorem we show that the existence of a strong Feer Markov kerne impies the uniform continuity of F. Suppose that there exists a strong Feer Markov kerne µ such that F ( ) = µ (λ). Since µ is a Markov kerne it is σ-additive. Then, ( im µ (λ) n n µ i (λ) ) = 0, λ σ(a). where,, i B(X), i=1 i =. By hypothesis, n µ (λ) µ i (λ) C(σ(A)), n N. i=1 i=1 22

33 Then, by theorem in appendix B, ( u im µ (λ) n n µ i (λ) ) = 0. By theorem 1, page 895, in Ref. [35], F ( ) = µ, hence n im F ( ) F n ( n i=1 i ) = im µ µ i = 0. n i=1 which proves that F is uniformy continuous. Exampe Let us consider the foowing unsharp position observabe (2.16) Q f ( ) := µ (x) dq x, B(R), µ (x) := [0,1] R i=1 χ (x y) f(y) dy, x [0, 1] where, f is a bounded, continuous function such that f(y) = 0, y / [0, 1] and f(y) dy = 1, [0,1] and Q x is the spectra measure corresponding to the position operator Q : L 2 ([0, 1]) L 2 ([0, 1]) ψ(x) (Qψ)(x) := xψ(x) Notice that, for each B(R), µ : [0, 1] [0, 1] is continuous. Indeed, by the uniform continuity of f, for each ɛ > 0, there is a δ > 0 such that x x δ impies f(x y) f(x y) ɛ, for each y. Therefore, µ (x) µ (x ) = χ (x y) f(y) dy χ (x y) f(y) dy R R = [f(x y) f(x y)] dy ɛ dy 2ɛ [ 1,1] By theorem and the continuity of µ, B(R), Q f is uniformy continuous. That can be proved as foows. Suppose i and f(y) M, y R. Since, for each x [0, 1], µ i (x) = f(x y) dy M dx i 23 ( i ) [ 1,1]

34 we have that, for each ψ H, ψ 2 = 1, ψ, Q f ( i )ψ = µ i (x) ψ 2 (x) dx M [0,1] which proves the uniform continuity of Q f. 2.4 Absoutey continuous POVMs ( i ) [ 1,1] In the present section, we prove that absoutey continuous commutative POVMs admit a strong Feer Markov kerne. Then, we appy the resut to the case of the unsharp position observabe. Definition [82, 83] A POVM F : B(X) F(H) is absoutey continuous with respect to a measure ν : B(X) [0, 1] if there exists a positive number c such that F ( ) c ν( ), for each B(X). Theorem Let F be absoutey continuous with respect to a finite measure ν. Then, F is uniformy continuous. Proof. Suppose i. We have im F ( ) F ( i) = im F ( i ) i i which proves that F is uniformy continuous. c im i ν( i ) = 0. Coroary Let F : B(X) F(H) be absoutey continuous with respect to a finite measure ν. Then, F is commutative if and ony if there exist a sefadjoint operator A and a strong Feer Markov kerne µ : σ(a) B(X) [0, 1] such that: (2.17) F ( ) = µ (A), B(X). Proof. By theorem 2.4.2, F is uniformy continuous. Then, theorem impies the thesis. Exampe Let us consider the unsharp position operator defined as foows. (2.18) Q f ( ) := µ (x) dq x, B(R), µ (x) := [0,1] R χ (x y) f(y) dy, x [0, 1] 24 dx

35 where, f is a positive, bounded, Bore function such that f(x) = 0, x / [0, 1], f(x)dx = 1, [0,1] and Q x is the spectra measure corresponding to the position operator Q : L 2 ([0, 1]) L 2 ([0, 1]) ψ(x) Qψ := xψ(x) Q f is absoutey continuous with respect to the measure ν( ) = M dx. [ 1,1] Indeed, for each ψ H, ψ 2 = 1, ψ, Q f ( )ψ = µ (x) ψ 2 (x) dx M where, the inequaity µ (x) = [0,1] f(x y) dy M [ 1,1] [ 1,1] has been used. Therefore, by theorem 2.4.2, Q f ( ) is uniformy continuous. Moreover, the continuity of f assures the continuity of µ for each B(R) so that µ is a Feer Markov kerne Unsharp Position Observabe In the present subsection, we study an important kind of absoutey continuous POVMs, the unsharp position observabes obtained as the marginas of a covariant phase space observabe. In the foowing H = L 2 (R), Q and P denote position and momentum observabes respectivey and denotes convoution, i.e. (f g)(x) = f(y)g(x y)dy. Let us consider the joint position-momentum POVM [2, 27, 32, 41, 50, 78, 83, 85] F ( ) = U q,p γ Uq,p dq dp where, U q,p = e iqp e ipq and γ = f f, f L 2 (R), f 2 = 1. The margina (2.19) Q f ( ) := F ( R) = 25 dx dx (1 f 2 )(x) dq x, B(R),

36 is an unsharp position observabe. Notice that the map µ (x) := 1 f(x) 2 defines a Markov kerne. Moreover, Q f is absoutey continuous with respect to the Lebesgue measure. Indeed, Q f ( ) = F ( R) = U q,p γ Uq,p dq dp R = dq U q,p γ Uq,p dp R = Q(q) dq 1 dq where, Q(q) = R U q,p γ U q,p dp. Athough Q f is absoutey continuous with respect to the Lebesgue measure on R, it is not uniformy continuous. That does not contradict theorem since the Lebesgue measure on R is not finite. Anyway, Q f is uniformy continuous on each Bore set with finite Lebesgue measure. Now, we show that Q f is not in genera uniformy continuous. We give the detais of the foowing particuar case. Exampe (Optima Phase Space Representation). If we choose f 2 (x) = 1 x 2 2 π e( 2 2 ), R {0}. in (2.19), we get an optima phase space representation of quantum mechanics [78]. In this case, where, Q f ( ) = ( = 1 2 π ) f(x y) 2 ) dy dq x ( (2.20) µ (x) = 1 2 π defines a Markov kerne. e (x y) 2 ) 2 2 dy dq x = µ (x) dq x 26 e (x y) dy

37 In order to prove that Q f is not uniformy continuous we consider the famiy of sets i = (, a i ), im i a i = such that i, and prove that im i Q f ( i ) = 1. For each i N, im µ i (x) = x 1 = im x 2 π im x (, a i x) 1 2 π i e (x y) dy e y2 2 2 dy = 1 2 π Now, we prove that Q f ( i ) = 1, i N. Indeed, if ψ n = χ [ n, n+1] (x), e y2 2 2 dy = 1. (2.21) (2.22) im ψ n, Q f ( i )ψ n = im n n = im n µ i (x) ψ n (x) 2 dx [ n, n+1] µ i (x) dx = 1. Since, for each B(R), Q f ( ) 1, equation (2.21) impies that Q f ( i ) = 1, for each i N. Hence, im i Q f ( i ) = 1 and Q f cannot be uniformy continuous. It is worth noticing that athough Q f is not uniformy continuous, µ is continuous for each interva B(R). Indeed, where, µ (x) µ (x ) = 1 2 π = 1 2 π 1 2 π e (x y) dy e (y) x e (y) dy x e (x y) dy e (y) dy x = {z R z = y x, y }, x = {z R z = y x, y } and, Therefore, x x ɛ impies, µ (x) µ (x ) 1 2 π = ( x x ) ( x x ). e (y) dy 1 2 π dy = 2 π ɛ. 27

38 2.5 Interpretation of the smearing In order to discuss a possibe interpretations of the resuts of the chapter we go back to the exampe of the unsharp position observabe. Let B(R), ψ L 2 ([0, 1]) and (2.23) ψ, Q f ( )ψ := µ (x) d ψ, Q x ψ, µ (x) := [0,1] R χ (x y) f(y) dy, x [0, 1] where, f is a positive, bounded, Bore function such that f(y) = 0, y / [0, 1], and [0,1] f(y)dy = 1, whie Q x is the spectra measure corresponding to the position operator Q : L 2 ([0, 1]) L 2 ([0, 1]) ψ(x) Qψ := xψ(x) We reca that ψ, Q( )ψ is interpreted as the probabiity that a perfecty accurate measurement (sharp measurement) of the position gives a resut in. Then, a possibe interpretation of equation (2.23) is that Q f is a randomization of Q. Indeed [78], the outcomes of the measurement of the position of a partice depend on the measurement imprecision 2 so that, if the sharp vaue of the outcome of the measurement of Q is x then the apparatus produces with probabiity µ (λ) a reading in. It is worth noting that (see exampe 2.3.6) the Markov kerne µ (x) := χ (x y) f(y) dy, x [0, 1] R in equation (2.23) above is such that the function x µ (x) is continuous for each B(R). The continuity of µ means that if two sharp vaues x and x are very cose to each other then, the corresponding random diffusions are very simiar, i.e., the probabiity to get a resut in if the sharp vaue is x is very cose to the probabiity to get a resut in if the sharp vaue is x. That is quite common in important physica appications and seems to be reasonabe from the physica viewpoint. It is then natura to ook for genera conditions which ensure the continuity of λ µ. We have proved that it is aways possibe to choose the Markov kerne µ to be continuous on a ring which generates the Bore σ- agebra B(X). Anyway, that is the most we can do in the genera case. Indeed, 2 There are other possibe interpretations of the randomization. For exampe, it coud be due to the existence of a no-detection probabiity depending on hidden variabes [39]. 28

39 we proved (see theorem 2.3.5) that the continuity for each Bore set is equivaent to the uniform continuity of F which in its turn is equivaent to require that the smearing F ( ) = µ (λ) de λ can be reaized by a strong Feer Markov kerne. It is worth remarking that athough in the genera case the continuity hods ony for a ring of subsets which generates B(X), that is sufficient to prove the weak convergence of µ ( ) (x) to µ ( ) (x ). 2.6 Appendix On the von Neumann agebra generated by F We reca that S B(X) is a countabe basis for the topoogy of X and R(S) is the ring generated by S. Theorem c, page 24, in Ref. [65] ensures the countabiity of R(S). Proof. Let M := {F ( )} B(X), and A W (F ) = A W (M) the von Neumann agebra generated by F. Let G denote the famiy of open subsets of X and O := {F ( ), G}. Since the POVM F is reguar, for each Bore set, there exists a decreasing famiy of open sets G i such that F (G i ) F ( ) strongy. Then, O is dense in M and the von Neumann agebra generated by M coincides with the von Neumann agebra generated by O. Hence, (2.24) A W (F ) = A W (M) = A W (O). Now, we prove that the von Neumann agebra A W (O 1 ) generated by O 1 = {F ( )} R(S) coincides with A W (O). For each open set G, there exists a famiy of sets { i } i N S, such that G = i=1 i. Let G n = n i=1 i. Then, G n G and F (G) = im n F (G n) = im n F ( n i=1 i ). Since the von Neumann agebra generated by O 1 contains F ( n i=1 i) for each n N, it must contain F (G) = im n F ( n i=1 i). Therefore, A W (O) = A W (O 1 ) and, by equations (2.24), (2.25) A W (O 1 ) = A W (O) = A W (F ) which proves that A W (F ) coincides with the von Neumann agebra generated by the set {F ( )} R(S). 29

40 2.6.2 Sequences of continuous functions The foowing theorem is due to Dini. sequences. We give a proof based on the use of Theorem Let {f n (λ)} n N be a non increasing sequence of continuous functions defined on a compact set B [0, 1] with vaues in [0, 1] and such that f n (λ) 0 point-wise. Then, f n (λ) 0 uniformy. Proof. Since f n+1 (λ) f n (λ) for each λ B, we have f n+1 f n. If f n 0 ceary f n (λ) 0 uniformy. Then, suppose f n a > 0. Since f n+1 f n, we have f n a, for each n N. Let λ n be such that f n (λ n ) = f n. Since {λ n } is a bounded sequence of rea numbers, there exists a convergent subsequence {λ nk } k N. Let β be its imit, i.e., β := im k λ nk. The compactness of B assures that β B. Moreover, im k f nk (λ nk ) = a. Let us consider the sequence of numbers f nk (β). We prove that f nk (β) a for each k N. We proceed by contradiction. Suppose that there exists k N such that f n k(β) < a. Then, there exists a neighborhood I(β) of β such that f n k(λ) < a for each λ I(β). Moreover, since λ nk β, there exists N such that k > impies λ nk I(β). Take k > max{ k, }. Then, λ nk I(β) and f nk (λ) f n k(λ), for each λ B. Therefore, f nk (λ nk ) f n k(λ nk ) < a which contradicts the fact that f nk (λ nk ) = f nk a, for each k N. We have proved that f nk (β) a, for each k N. This impies that im k f nk (β) a and contradicts one of the hypothesis of the emma, i.e., im n f n (λ) = 0 for each λ B. 30