7 Correlations, Disturbance and Entanglement
Tải bản đầy đủ - 0trang
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
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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)
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3495–3498 (2012)
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Phys. A 47(5), 052002 (2014)
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the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351–1354 (1988)
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570–578 (1931)
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10–12 (1998)
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5872–5881 (2002)
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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 )