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7 Correlations, Disturbance and Entanglement

7 Correlations, Disturbance and Entanglement

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10.7 Correlations, Disturbance and Entanglement



253



orthogonal to ξ in the span of ϕ, ψ. Then P[ϕ] + P[ψ] = P[ξ] + P[η], and applying

the linear operation IM (X ), one has

Eϕ (X )P[ϕ] + Eψ (X )P[ψ] = Eξ (X )P[ξ] + Eη (X )P[η].

This is a selfadjoint operator of rank not greater than 2, presented in two versions

of spectral decomposition with respect to two distinct orthonormal bases. From the

uniqueness of the spectral decomposition it follows immediately that the spectrum

must be degenerate and the operator a multiple of a rank 2 projection. Hence one

must have Eϕ (X ) = Eψ (X ) = Eξ (X ) = Eη (X ).

This shows that given any vector state ϕ, for all vector states ψ orthogonal to ϕ,

the value of Eψ (X ) is independent of ψ and indeed equal to Eϕ (X ). Furthermore, the

same value is obtained for any vector state ξ not orthogonal to ϕ. Hence for any X ,

there is a constant λ(X ) ∈ [0, 1] such that Eϕ (X ) = λ(X ) for all vector states ϕ. The

measure properties of the map X → λ(X ) follows directly from the corresponding

property of the instrument, which now is IM (X ) P[ϕ] = λ(X )P[ϕ].

Note that this proof does not make use of the complete positivity of the instrument.

The value reproducibility of a repeatable measurement of a discrete observable

means that states for which an outcome has probability equal to 1 will preserve

this property. Ideality is the stronger feature that all sharp properties that commute

with the measured observable and have probability 1 will retain probability 1 in

the state after the measurement. Thus, a Lüders measurement leaves unchanged

all eigenstates of the measured observable. Any non-eigenstate will be changed;

in particular, a vector state is transformed into a mixture of eigenstates. This state

disturbance → I LA (R)( ) due to a Lüders measurement of a sharp observable A can

be witnessed by comparing the statistics B f of some suitable observable B in the final

state f = I LA (R)( ) with the undisturbed statistics B . As the following Theorem

due to Lüders shows, not every observable can be used to detect the disturbance. At

the same time, this gives an operational characterisation of commutativity.

Theorem 10.7 (Lüders [4]) Let A = i ai Pi be a discrete sharp observable and

I A,L its Lüders instrument. Let B be a sharp observable on (Ω, A). Then

[Pi , B(X )] = 0 for all i, X



⇐⇒



I∗A,L (R) ◦ B = B.



(10.42)



Thus, the statistics of any observable B that commutes with A remains unchanged

by a Lüders measurement of A.

Interestingly, commutativity may not be necessary for the statistics of an unsharp

observable to remain unchanged under a Lüders measurement [14].

Proposition 10.6 Let E = {E 1 , E 2 , . . . , E N } be a finite-outcome observable, and

let IE,L be its generalised Lüders instrument, defined by

1/2



IE,L {i} ( ) = E i



1/2



E i , i = 1, 2, . . . , N .



(10.43)



254



10 Measurement



K

Let B = k=1

bk Bk be an effect with a strictly decreasing sequence of eigenvalues

(bk ) ⊂ [0, 1] and spectral projections Bk , k ≤ K , K ∈ N ∪ {∞}. Then



[E i , B] = 0 ∀i = 1, 2, . . . , N



⇐⇒



I∗E,L (R)(B) = B.



(10.44)



If E is a two-outcome observable, E = {E 1 , E 2 }, then (10.44) holds for any effect B.

It is known that I∗E,L (R)(B) = B may hold without E, B commuting if the spectral

condition on B is violated and E has more than two outcomes [15].

Proposition 10.7 Let M = (K, Z, σ, U ) be a measurement scheme with σ = P[φ0 ]

and assume that for all ϕ ∈ H, U (ϕ ⊗ φ0 ) = ϕ ⊗ φ for some unit vectors ϕ ∈ H,

φ ∈ K. Then U acts in one of the following two ways:

(a) U (ϕ ⊗ φ0 ) = V (ϕ) ⊗ φ0 , where V is an isometry in H and φ0 is a fixed unit

vector in K.

(b) U (ϕ ⊗ φ0 ) = ϕ0 ⊗ W ϕ, where W is an isometry from H to K and ϕ0 is a fixed

unit vector in H.

In the first case, the measured observable is trivial, E(X ) = μ(X ) I , with μ(X ) =

φ Z(X )φ , and the associated instrument is given by I(X ) P[ϕ] = μ(X ) P[V ϕ].

In the second case, E is given by E(X ) = W −1 Z(X )W , with the constant instrument

I(X ) P[ϕ] = ϕ E(X )ϕ P[ϕ ].

Proof Let {ϕn : n = 1, 2, . . . } be an orthonormal basis of H. There are systems of

unit vectors ϕn ∈ H, φn ∈ K such that U ϕn ⊗ φ0 = ϕn ⊗ φn . Due to the unitarity of

U , all the vectors ϕn ⊗ φn are mutually orthogonal. We show that one of two cases

(a), (b) must hold:

(a) {ϕn }n∈N is an orthonormal system, all φn are parallel to φ1 ;

(b) {φn }n∈N is an orthonormal system, all ϕn are parallel to ϕ1 .

For two vectors ψ, ξ which are mutually orthogonal, ψ|ξ = 0, we will write

ψ ⊥ ξ. Since U is unitary, this map sends orthogonal vector pairs to orthogonal

pairs. Hence from ϕ1 ⊥ ϕ2 it follows that ϕ1 ⊥ ϕ2 or φ1 ⊥ φ2 . Consider the first

case. Then

U



√1 (ϕ1

2



+ ϕ2 ) ⊗ φ0



= ϕ12 ⊗ φ12 =



√1 ϕ

2 1



⊗ φ1 +



√1 ϕ

2 2



⊗ φ2 ,



where ϕ12 ∈ H, φ12 ∈ K are some unit vectors. Since ϕ1 ⊥ ϕ2 , it follows that φ2 =

cφ1 with some c ∈ C, |c| = 1. So we have

U (ϕ1 + ϕ2 ) ⊗ φ0 = (ϕ1 + cϕ2 ) ⊗ φ1 .

Still considering the case ϕ1 ⊥ ϕ2 , the relation ϕ2 ⊥ ϕ3 implies that ϕ2 ⊥ ϕ3 or

φ2 ⊥ φ3 . Suppose the latter holds. We show that this leads to a contradiction. Indeed

this assumption gives ϕ3 = c ϕ2 and thus



10.7 Correlations, Disturbance and Entanglement



U (ϕ1 + ϕ2 + ϕ3 ) ⊗ φ0 =



255





3ϕ123 ⊗ φ123



= ϕ1 ⊗ φ1 + ϕ2 ⊗ φ2 + ϕ3 ⊗ φ3

= (ϕ1 + cϕ2 ) ⊗ φ1 + ϕ3 ⊗ φ3

where ϕ123 and φ123 are some unit vectors. Recalling that φ2 = cφ1 and, by assumption, φ2 ⊥ φ3 , then φ1 ⊥ φ3 , and we see that ϕ1 + cϕ2 = c ϕ3 for some c = 0.

Upon taking the inner product of both sides with ϕ1 , we get (since ϕ1 ⊥ ϕ2 ) that

ϕ1 |ϕ1 = c ϕ1 |ϕ3 = 0 (since ϕ3 = c ϕ2 ⊥ ϕ1 ). Hence ϕ1 = 0 which is a contradiction.

Thus the assumption is false and we can only have ϕ2 ⊥ ϕ3 . Continuing inductively, we obtain that {ϕi : i ∈ N} is an orthonormal system and all φi = ci φ1 .

Therefore, we obtain possibility (a) in the present case. Linearity then entails that

U (ϕ ⊗ φ0 ) = V (ϕ) ⊗ φ0 for all ϕ ∈ H and some isometric map V .

An analogous consideration can be applied in the second case of φ1 ⊥ φ2 , thus

leading to the possibility (b) and

U (ϕ ⊗ φ0 ) =



ϕi |ϕ U (ϕi ⊗ φ0 ) = ϕ0 ⊗

i



ϕi |ϕ φi = ϕ0 ⊗ W (ϕ)

i



for all ϕ ∈ H and some isometric map W : H → K.

Proposition 10.7 shows that non-entangling measurements are either trivial or

based on a dynamics that swap states. An example of the latter case is a measurement

scheme for which the probe is a copy of the object system and the unitary coupling

is the swap map U : ϕ ⊗ φ → φ ⊗ ϕ. For such maps it has been shown that they

cannot occur as elements of a continuous dynamical group t → Ut [16].



10.8 Appendix

This Appendix gives a proof of Theorem 10.4.

The notion of a repeatable measurement, which dates back to von Neumann’s

1932 book [3], raises the important question of the structure of observables that

admit such measurements. In particular, one may ask whether these observables are

necessarily discrete. This question was finally solved affirmatively by Ozawa [17]

and Łuczak [18] building on important contributions by Stinespring [19], Davies

and Lewis [20] and Davies [13]. In this book measurement schemes are assumed to

be of the form (K, Z, σ, U ) so that their accompanying instruments are completely

positive. Therefore, we may follow the proof of [17] which rests on that property of

an instrument. In [18] the same result is obtained without using complete positivity.

We split the proof of this result into a series of lemmas.



256



10 Measurement



Lemma 10.2 Let M be a repeatable measurement of an observable E and let IM

be the associated instrument. Then for any X, Y ∈ A and A ∈ L(H) the following

properties hold:

(i) I∗M (Ω) E(X )2 = I∗M (Ω) E(X ) ;

(ii) I∗M (X ∩ Y )(A) = I∗M (Y ) AE(X ) = I∗M (Y ) E(X )A ;

(iii) I∗M (X )(A) = I∗M (Ω) AE(X ) = I∗M (Ω) E(X )A .

Proof Property (i) is an immediate consequence of the repeatability of the measurement and we leave it for the reader. To verify (ii), let (K, P, π, V ) constitute a

Stinespring type representation of IM so that for any X ∈ A and A ∈ L(H) we have

IM (X )∗ (A) = V ∗ P(X )π(A)V . Using the repeatability of IM one computes that

π E(X ) V − P(X )V

=



I∗M (Ω)



E(X )



2







π E(X ) V − P(X )V



− I∗M (Ω) E(X ) = 0



so that V ∗ π E(X ) = V ∗ P(X ) and π E(X ) V = P(X )V . Thus for any X, Y ∈ A

and A ∈ L(H),

IM (X ∩ Y )(A) = V ∗ P(X ∩ Y )π(A)V = V ∗ P(X )P(Y )π(A)V

= V ∗ π E(X ) P(Y )π(A)V = V ∗ P(Y )π E(X ) π(A)V

= V ∗ P(Y )π E(X )A V = IM (Y )∗ E(X )A ,

as well as

IM (X ∩ Y )(A) = V ∗ P(X ∩ Y )π(A)V = V ∗ π(A)P(X ∩ Y )V

= V ∗ π(A)P(Y )P(X )V = V ∗ π(A)P(Y )π E(X ) V

= V ∗ P(Y )π AE(X ) V = I∗M (Y ) AE(X ) .

Property (iii) is just a special case of (ii) with Y = Ω.

Let W be the von Neumann algebra generated by the range of E, that is, W =

E(X ) | X ∈ A , and let P be the support projection of the (normal completely

positive) map I∗M (Ω) restricted to W, that is, P is the least projection in W such

that I∗M (Ω)(P) = I .

Lemma 10.3 For any A ∈ W,

I∗M (Ω)(A) = I∗M (Ω)(A P) = I∗M (Ω)(P A) = I∗M (Ω)(P A P).

Proof Using the representation I∗M (Ω)(A) = V ∗ π(A)V we get, for each ϕ,

ψ ∈ H,



10.8 Appendix



257



(I∗M (Ω)(A) − I∗M (Ω)(A P))ϕ ψ



=



I∗M (Ω)(A − A P)ϕ ψ



=



V ∗ π(A − A P)V ϕ ψ



= | π(I − P)V ϕ | π(A)V ψ |





π(I − P)V ϕ



π(A)V ψ = 0,



as π(I − P)V ϕ 2 = V ∗ π(I − P)V ϕ | ϕ = I∗M (Ω)(I − P)ϕ | ϕ = 0. Thus,

I∗M (Ω)(A) = I∗M (Ω)(A P). The other two statements are obtained similarly.

Lemma 10.4 For any A ∈ W, A ≥ 0, if I∗M (Ω)(A) = 0, then P A P = 0.

Proof Let A ∈ W, A ≥ 0, be such that I∗M (Ω)(A) = 0, and let A be the spectral

measure of A. If X ∈ B [0, A ] , then I∗M (Ω) A(X ) = 0 and P ≤ I − A(X ),

i.e., A(X ) ≤ I − P. This implies PA(X )P = 0 and finally P A P = 0.

Lemma 10.5 Let IM be a repeatable instrument of an observable E : A → L(H)

and let P be the support projection of I∗M (Ω). The map X → Π (X ) = PE(X )P

constitutes a projection measure such that Π (X ) = E(X )P = PE(X ) for all X ∈ A

(and Π (Ω) = P).

Proof From Lemma 10.3 we get

I∗M (Ω) Π (X ) = I∗M (Ω) PE(X )P = I∗M (Ω) E(X ) ,

whereas Lemmas 10.2 and 10.3 give

I∗M (Ω) Π (X )2 = I∗M (Ω) PE(X )PE(X )P

= I∗M (Ω) E(X )PE(X )

= I∗M (X ) PE(X ) = I∗M (X )(P)

= I∗M (Ω) PE(X ) = I∗M (Ω) E(X ) .

Therefore, I∗M (Ω) Π (X ) − Π (X )2 = 0, so that by the nondegeneracy of I∗M (Ω)

on PW P (Lemma 10.4) we have Π (X ) = Π (X )2 for all X ∈ A, showing that Π is

a projection measure. A direct computation gives

I∗M (Ω) (Π (X ) − E(X )P)∗ (Π (X ) − E(X )P) = 0,

which, by Lemma 10.4, entails that Π (X ) = E(X )P, and thus also Π (X ) =

Π (X )∗ = PE(X ).

Theorem 10.8 Let (Ω, B(Ω)) be the Borel space associated with a locally compact Hausdorff space Ω. If an observable E : B(Ω) → L(H) admits a repeatable

measurement, then it is discrete.

Proof Let M be a repeatable measurement of E, P the support projection of

I∗M (Ω), and let Π be the projection valued measure of Lemma 10.5 so that E(X ) =



258



10 Measurement



I∗M (X )(I ) = I∗M (Ω)(E(X )) = I∗M (Ω)(PE(X )P) = I∗M (Ω)(Π (X )) for all X ∈

A. Since Π is multiplicative and H is separable there is a countable set X o ⊂ Ω

such that the measure B(Ω) X → Π (X ∩ X o ) ∈ L(Π (X o )(H)) is discrete and

B(Ω) X → Π (X ∩ X oc ) ∈ L(Π (X oc )(H)) is continuous.

Let Q = Π (X oc ) = P − Π (X o ) and define Ψo (A) = QI∗M (Ω)(A)Q for all A ∈

L(QH). Then Ψo (Q) = Q, showing that Ψo is a unital normal positive linear map

L(QH) → L(QH). Therefore, by Proposition 6.1, Ψo is the dual of a trace preserving positive linear map Φo : T (QH) → T (QH). Then for any A ∈ L(QH), and for

all X ∈ B(Ω), T ∈ T (QH),

tr AΠ (X ∩ X oc )Φo ( ) = tr Ψo (AΠ (X ∩ X oc ))

= tr QI∗M (Ω)(AΠ (X ∩ X oc ))Q

= tr QI∗M (Ω)(Π (X ∩ X oc )A)Q

= tr Ψo (Π (X ∩ X oc )A)

= tr Π (X ∩ X oc )AΦo ( )

= tr AΦo ( )Π (X ∩ X oc ) ,

from which it follows that Π (X ∩ X oc )Φo ( ) = Φo ( )Π (X ∩ X oc ) for all X ∈ B(Ω)

and T ∈ T (QH). Since the measure X → Π (X ∩ X oc ) is continuous this implies

that Φo (T ) = 0 for all T ∈ T (QH). That is, Φo , and hence Ψo , is the null map [13,

Theorem 4.3.3]. Therefore, Q = Ψo (Q) = 0.



10.9 Exercises

1.

2.

3.

4.

5.

6.

7.



Prove Proposition 10.1.

Verify Eq. (10.12).

Verify Eq. (10.20).

Verify Eq. (10.25).

Verify Eq. (10.31).

Verify Eqs. (10.32) and (10.33).

Determine the sequential observable obtained from a standard position measurement followed by a (sharp) momentum measurement.

8. Let A, B be commuting selfadjoint operators with closed discrete spectra, and

let M LA , M LB be standard schemes of Lüders measurements of the sharp observables corresponding to A, B. Show that the sequential application of these measurements is order independent and equivalent to their simultaneous application.

9. Consider a measurement scheme (K, Z, σ, U ) and assume that the measured

observable E is a spectral measure. Show that for any X ∈ A the projections

I ⊗ Z(X ) and Vφ Vφ∗ commute with each other.

10. Show that the repeatability condition of Definition 10.3 can be written equivalently in any of the following ways:



10.9 Exercises



259



tr IM (X )2



= tr IM (X )P[ϕ] ;



tr IM (X )IM (X )



= 0;



c



tr IM (Y )IM (X )

tr



f



= 0 for all disjoint X and Y ;



(X )E(X ) = 1 (whenever E (X ) = 0);



E(X )



f



(X ) =



E(Y ∩ X ) =

E(X ) =

I∗M (X )



f



(X );



I∗M (X )



I∗M (X )

c



E(Y ) ;



E(X ) ;



E(X ) = 0;



(10.45)

(10.46)

(10.47)

(10.48)

(10.49)

(10.50)

(10.51)

(10.52)



each condition being valid for any X, Y ∈ A and ∈ S(H).

11. Verify the implication (10.37).

12. Show that a d-ideal measurement of a discrete sharp observable is nondegenerate.

13. Consider a minimal measurement Mm of a discrete sharp observable A =

i ai Pi as given in Example 10.2. Compute the correlation coefficient

cor(A, Z , P[U (ϕ ⊗ φ)]) (as defined in Sect. 9.5). Show that if the generating

vectors {ψi j } form an eigenbasis of A, then cor(A, Z , P[U (ϕ ⊗ φ)]) = 1 for all

initial vector states ϕ for which the correlation coefficient is well defined.

14. Continue the previous exercise and compute the correlations between the final

component states arising from this measurement scheme. Also compute the

correlations between the values of A and Z , that is, the correlations between the

sharp properties Pi and P[φi ].

15. Show that the disturbed momentum P in the standard (approximate) position

measurement, Eq. (10.32) of Sect. 10.4, is of the form P = ν ∗ P, with the probˆ p )|2 dp, where φ ∈ L 2 (R), φ = 1, is the initial

ability measure dν = λ1 |φ(

λ

probe state and λ is the coupling parameter.

16. Consider the von Neumann model of an approximate position measurement,

Sect. 10.4. Compute the correlation cor(Qe , Q p , P[U (ϕ ⊗ φ)]). Assume that the

initial probe state is a Gaussian concentrated at the origin. Study the behaviour

of the correlation coefficient as a function of the width of the Gaussian probe

state φ.

17. Consider a vector state Ψ ∈ H ⊗ K that is not a product state. Use the polar

decomposition of Ψ to construct observables which are strongly correlated in

this state.



References

1. Pellonpää, J.-P., Tukiainen, M.: Minimal normal measurement models of quantum instruments.

arXiv:1509.08886

2. Beltrametti, E., Cassinelli, G., Lahti, P.: Unitary measurements of discrete quantities in quantum

mechanics. J. Math. Phys. 31(1), 91–98 (1990)



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3. von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Die Grundlehren der

mathematischen Wissenschaften, Band 38. Springer, Berlin (1968, 1996). (Reprint of the 1932

original). English translation: Mathematical Foundations of Quantum Mechanics. Princeton

University Press, Princeton (1955, 1996)

4. Lüders, G.: Über die Zustandsänderung durch den Meßprozeß. Ann. Phys. (Leipzig), 443(58):322–328 (1950). English Translation by Kirkpatrick, K.A.: Ann. Phys. (Leipzig) 15(9),

663–670 (2006)

5. Pellonpää, J.-P.: Complete quantum measurements break entanglement. Phys. Lett. A 376(46),

3495–3498 (2012)

6. Pellonpää, J.-P.: Complete measurements of quantum observables. Found. Phys. 44(1), 71–90

(2014)

7. Pellonpää, J.-P.: On coexistence and joint measurability of rank-1 quantum observables. J.

Phys. A 47(5), 052002 (2014)

8. Haapasalo, E., Lahti, P., Schultz, J.: Weak versus approximate values in quantum state determination. Phys. Rev. A 84, 052107 (2011)

9. Aharonov, Y., Albert, D.Z., Vaidman, L.: How the result of a measurement of a component of

the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351–1354 (1988)

10. Dressel, J., Malik, M., Miatto, F.M., Jordan, A.N., Boyd, R.W.: Colloquium: Understanding

quantum weak values: basics and applications. Rev. Mod. Phys. 86(1), 307–316 (2014)

11. Aharonov, Y., Botero, A.: Quantum averages of weak values. Phys. Rev. A 72, 052111 (2005)

12. von Neumann, J.: Die Eindeutigkeit der Schrödingerschen Operatoren. Math. Ann. 104(1),

570–578 (1931)

13. Davies, E.B.: Quantum Theory of Open Systems. Academic Press London, New York (1976)

14. Busch, P., Singh, J.: Lüders theorem for unsharp quantum measurements. Phys. Lett. A 249,

10–12 (1998)

15. Arias, A., Gheondea, A., Gudder, S.: Fixed points of quantum operations. J. Math. Phys. 43(12),

5872–5881 (2002)

16. Busch, P.: The role of entanglement in quantum measurement and information processing. Int.

J. Theor. Phys. 42(5), 937–941 (2003)

17. Ozawa, M.: Quantum measuring processes of continuous observables. J. Math. Phys. 25(1),

79–87 (1984)

18. Łuczak, A.: Instruments on von Neumann algebras. Institute of Mathematics, Łód´z University

(1986)

19. Stinespring, W.F.: Positive functions on C ∗ -algebras. Proc. Am. Math. Soc. 6, 211–216 (1955)

20. Davies, E.B., Lewis, J.T.: An operational approach to quantum probability. Commun. Math.

Phys. 17, 239–260 (1970)



Chapter 11



Joint Measurability



It is one of the key features of quantum mechanics that not all of the observables of

this theory can be measured jointly; in other words, many pairs or larger families of

observables are incompatible. We already observed that measurements necessarily

disturb the system under investigation; here we study the phenomenon that measurements of incompatible observables also influence each other if one attempts to apply

them jointly or sequentially. In order to quantify this irreducible disturbance and to

identify possible ways of mitigating the resulting limitation of measurability, we first

make precise minimal requirements of what constitutes a joint of measurement of

two or more observables. We have already encountered some formulations of this

notion in Sects. 1.2 and 10.3. In Chap. 13 we proceed to present elements of a theory

of approximate joint measurements of incompatible observables.

Insofar as the purpose of a measurement is to determine the values of observables and their distributions, a joint measurement of, say, two observables should

be represented by a suitably defined observable that allows one to account for the

joint occurrence of a pair of outcomes and their probability distribution for each

state. There are several distinct ways of specifying this idea. Some of them are

equivalent, some of different degrees of generality. We start with presenting those

formulations which turn out to be equivalent and then proceed to indicate a few alternatives. We also explore some characterisations of pairs of observables that are jointly

measurable.



11.1 Definitions and Basic Results

The subject of quantum incompatibility and the issue of joint measurability had

a long history in the foundations of quantum mechanics. Accordingly, a variety

of alternative terms were introduced besides joint or simultaneous measurability,

notably compatibility, commensurability and coexistence. We will use the terms

© Springer International Publishing Switzerland 2016

P. Busch et al., Quantum Measurement, Theoretical and Mathematical Physics,

DOI 10.1007/978-3-319-43389-9_11



261



262



11 Joint Measurability



joint measurability and compatibility interchangeably and refer to other notions, like

coexistence, where appropriate. Pairs of observables that are not jointly measurable

will be called incompatible.

In the following we consider observables E1 and E2 with value spaces (Ω1 , A1 )

and (Ω2 , A2 ). We recall that a positive operator bimeasure B : A1 × A2 → L(H)

is a biobservable if the marginal measures X → B1 (X ) = B(X, Ω2 ) and Y →

B2 (Y ) = B(Ω1 , Y ) are observables. If G is an observable with the value space

(Ω1 × Ω2 , A1 ⊗ A2 ) then the effects G1 (X ) = G(X × Ω2 ) and G2 (Y ) = G(Ω1 × Y )

constitute its marginal observables.

Definition 11.1 Let E1 : A1 → L(H) and E2 : A2 → L(H) be any two observables.

(a) E1 and E2 are functions of an observable M, with the value space (Ξ, B), if

there are (measurable) functions f 1 : Ξ → Ω1 and f 2 : Ξ → Ω2 such that for

each X ∈ A1 , Y ∈ A2 ,

E1 (X ) = M( f 1−1 (X )),



E2 (Y ) = M f 2−1 (Y ) .



(b) E1 and E2 are smearings of an observable E, with the value space (Ω, A), if

there are Markov kernels p1 : A1 × Ω → [0, 1] and p2 : A2 × Ω → [0, 1]

such that for each X ∈ A1 , Y ∈ A2 ,

E1 (X ) =



Ω



p1 (X, ω) dE(ω),



E2 (Y ) =



Ω



p2 (Y, ω) dE(ω).



(c) E1 and E2 have a biobservable if there is a biobservable B : A1 × A2 → L(H)

such that B1 = E1 and B2 = E2 .

(d) E1 and E2 have a joint observable if there is an observable G : A1 ⊗A2 → L(H)

such that G1 = E1 and G2 = E2 .

Any two observables that can be measured jointly in the sense of the measurement

theory of Sect. 10.3 are functions of a third observable, and conversely. Clearly, if

E1 and E2 are functions of E, then they are also smearings of E with the kernels

p1 (X, ω) = χ X ( f 1 (ω)) and p2 (Y, ω) = χY ( f 2 (ω)). Conversely, if two observables

are smearings of a third observable it is not entirely obvious, but nevertheless true in

certain circumstances, that they are also functions of an observable (Theorem 11.1).

Biobservables arise in the context of sequential measurements and, more naturally,

in detection schemes with two independent registration devices. Their margins, the

partial observables, arise with ignoring one of the outcomes. If the observables E1

and E2 are functions of a third observable E, then

B (X, Y ) = E ( f 1−1 (X ) ∩ f 2−1 (Y )),

where X ∈ A1 , Y ∈ A2 , ∈ S(H), defines a biobservable B of E1 and E2 . If

observables E1 and E2 have a joint observable E, then they are also functions of E,



11.1 Definitions and Basic Results



263



namely, Ei = E ◦ πi−1 , where πi ((ω1 , ω2 )) = ωi , i = 1, 2. This leaves the question

whether biobservables can be extended to joint observables.

By Theorem 4.2 such an extension is obtained whenever the value spaces (Ωi , Ai )

have the property (D). This is the case, in particular, when the value spaces are

closed or open subspaces of (Rn , B(Rn )) (cf. Proposition 4.9). It then holds that

B(Ω1 ) ⊗ B(Ω2 ) = B(Ω1 × Ω2 ) (see Proposition 4.10).

Theorem 11.1 Assume that the value spaces of the observables E1 : A1 → L(H)

and E2 : A2 → L(H) have the property (D). The following conditions are equivalent:

(i)

(ii)

(iii)

(iv)



E1

E1

E1

E1



and E2

and E2

and E2

and E2



have a biobservable;

have a joint observable;

are functions of a third observable;

are smearings of a third observable.



Proof Theorem 4.2 gives the implication (i) ⇒ (ii). It remains to show that (iv) ⇒

(i). Let (Ω, A) be a measurable space, E : A → L(H) an observable, and p1 and

p2 Markov kernels such that Ω pi (X i , ω)dE(ω) = Ei (X i ) for all X i ∈ Ai where

i = 1, 2. Define a positive operator bimeasure B : A1 × A2 → L(H) by

B(X 1 , X 2 ) =



Ω



p1 (X 1 , ω) p2 (X 2 , ω)dE(ω).



Since E1 (X 1 ) = B(X 1 , Ω2 ) and E2 (X 2 ) = B(Ω1 , X 2 ) for all X i ∈ Ai , it follows

that B is a biobservable for E1 and E2 .

Definition 11.2 Two (or more) observables are said to be jointly measurable or

compatible if they have one and thus all the properties of Theorem 11.1.

We stress that this definition presupposes the assumption that the value spaces of

observables have the property (D).

For compatible observables E1 and E2 , the distributions E1, , E2, can be inferred

from single measurement outcome distributions like M , E , B , G . In addition,

the probabilities E1, (X ), X ∈ A1 , and E2, (Y ), Y ∈ A2 , are bounded from below

by the pair probabilities

M ( f 1−1 (X ) ∩ f 2−1 (Y )),



Ω



p1 (X, ω) p2 (Y, ω) dE (ω), B (X, Y ), G (X × Y ).



By Lemma 4.6 and Proposition 4.8 these lower bounds are optimal if one of the

observables E1 or E2 is sharp. Indeed, in this case we have

M( f 1−1 (X ) ∩ f 2−1 (Y )) =



Ω



p1 (X, ω) p2 (Y, ω) dE(ω) = B(X, Y ) = G(X × Y )



= E1 (X )E2 (Y ) = E2 (Y )E1 (X ) = E1 (X ) ∧ E(H) E2 (Y )



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