4 Excursion: Compatibility of Three Qubit Effects
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330
14 Qubits
examples of constructions of joint observables for more than two observables, thus
establishing some sufficient conditions for their joint measurability. Here we briefly
look at three unbiased qubit observables with the generating effects
E1,± = 21 (I ± e1 · σ),
E2,± = 21 (I ± e2 · σ),
E3,± =
1
(I
2
(14.39)
± e3 · σ).
We introduce operators
Gijk =
1
8
αijk I + g ijk · σ , i, j, k ∈ {+, −},
(14.40)
where
αijk = 1 + ije1 · e2 + ike1 · e3 + jke2 · e3
g ijk = ie1 + je2 + ke3
i, j, k ∈ {+, −}.
(14.41)
It is readily verified that these operators satisfy the marginality relations E1,± =
j,k G±,j,k , etc. Thus, the three observables are compatible if all eight operators
Gijk ≥ 0, that is, g ijk ≤ αijk , which is equivalent to
0 ≤ 1 − e1
2
− e2
2
− e3
2
+ ( g ijk − 1)2 .
(14.42)
Inequality (14.42), in turn, can be expressed in terms of the unsharpness fuzz(Eν,λ ) =
1 − eν 2 as:
2−
g ijk − 1
2
≤ fuzz(E1,i ) + fuzz(E2,j ) + fuzz(E3,k ).
(14.43)
With the choice eν 2 ≤ 13 the right hand side is ≥ 2 and the inequality is satisfied
for all possible directions of the vectors eν . We thus have the following.
Proposition 14.3 Three unbiased dichotomic qubit observables E1 , E2 , E3 with
effects E1,i , E2,j , E3,k as given in (14.39) are compatible if
fuzz(E1,i ) + fuzz(E2,j ) + fuzz(E3,k ) ≥ 2.
(14.44)
This condition
√ is satisfied if each effect has fuzziness no less than 2/3, or equivalently,
eν ≤ 1/ 3.
The set of operators Gijk of (14.40) forms a joint observable for E1 , E2 , E3 if
and only if (14.43) holds. If e1 , e2 , e3 have equal length e and
√ are orthogonal, the
operators Gijk cannot form a joint observable unless e ≤ 1/ 3.
We note that the construction (14.41) does not give a joint observable in the trivial
case where e1 = e2 = e3 and these three vectors have length 1: in that case we have
α++− = 0 but g ++− = e1 , so that G++− is not an effect.
14.4 Excursion: Compatibility of Three Qubit Effects
331
Remark 14.5 The method of intersecting spheres reviewed in the previous section
for pairs of unbiased qubit effects was extended in [9] to the case of a triple of qubit
observables with effects Eν,± = 21 (1 ± xν )I ± eν , ν = 1, 2, 3, where positivity is
equivalent to |xν | + eν ≤ 1. revealing the following as a necessary condition for
their joint measurability: there exists a vector g such that
− e1 − e2 − e3 − g + e1 + e2 − e3 − g
+ e1 − e2 + e3 − g + − e1 + e2 − e3 − g ≤ 4.
(14.45)
Geometrically, this is the condition that the point with position vector g must lie
in the intersection of four solid spheres centred at points A, B, C, D with position
vectors −(e1 + e2 + e3 ), e1 + e2 − e3 , −e1 + e2 + e3 , and +e1 − e2 + e3 and radii
that add to 4. Recalling that the Fermat-Torricelli point associated with a set of points
in R3 is defined as the point for which the sum of distances from all points in the set
is minimised, the above necessary joint measurability condition can be phrased by
saying that the Fermat-Torricelli point of A, B, C, D must have its sum of distances
no greater than 4.
In the case of observables {Eν,± } with orthogonal vectors, this condition reduces
to (14.44), which is therefore necessary and sufficient if the effects Eν,± are unbiased.
Finally, in [10] it is shown that an unbiased triple of qubit effects is jointly measurable
if and only if the Fermat-Torricelli point of the associated points A, B, C, D satisfies
(14.45).
Remark 14.6 It has been shown in [9] that measurement realisations for joint measurements of compatible pairs of triples of qubit effects can be obtained by an adaptation of the Arthurs-Kelly model [11], which was originally formulated for position
and momentum measurements (see Sect. 19.1).
14.5 Approximate Joint Measurements of Qubit
Observables
Let A and B be any two incompatible sharp qubit observables. To be specific,
and without loss of generality, we assume that cos θ = a · b > 0, where a, b are
the unit vectors associated with A, B, respectively. We shall identify their optimal approximate joint measurements among the jointly measurable pairs of simple (unsharp) qubit observables (C, D). We will see that optimal approximations
can be found among pairs of unbiased observables C = C± = 21 (I ± c · σ) ,
D = D± = 21 (I ± d · σ) ; recall that these are jointly measurable exactly when
the effects C+ , D+ are compatible, which is the case if and only if
c + d + c − d ≤ 2.
332
14 Qubits
We note that any dichotomic qubit observable can be obtained as a smearing of
some sharp qubit observable. Indeed, let E = E± = 21 ((1 ± x)I ± e · σ) , where the
positivity of the effects E± is equivalent to |x| + e ≤ 1. Using the spectral measure
Eeˆ ·σ = 21 (I ± eˆ · σ) , one observes that E = p Eeˆ ·σ , where p is the Markov kernel
p(+, +) = (1 + x) + e , p(−, +) = 1 − p(+, +),
p(+, −) = (1 + x) − e , p(−, −) = 1 − p(+, −).
Error Measure for Qubit Approximations
We shall use the metrics D1 , D∞ as well as the Wasserstein-2 distance Δ2 to quantify
the degree of approximation of the measurements.
For simple observables E = {E+ , E− }, F = {F+ , F− }, the distances D1 (E, F) and
D∞ (E, F) are readily computed. Observing that E+ − F+ = F− − E− one gets
D1 (E, F) = D∞ (E, F) = E+ − F+ = E− − F− = 21 |e0 − f0 | +
1
2
e−f .
By the result quoted in Remark 13.2 for the finite valued probability measures, the
Wasserstein-2 distance is known to be controlled by the distance defined by total
variation norm. Here we demonstrate this connection showing that Δ2 (E, F)2 =
4D∞ (E, F) [12].
For comparison, we note that for C = p Ecˆ ·σ one has D∞ (C, Ecˆ ·σ ) = 21 c0 −
1 + 21 1 − c .
In contrast to the metrics D1 , D∞ , the distance Δ2 depends explicitly on the
value space and on the metrics chosen. While the observables to be approximated
are ±1-valued, the approximators could a priori be allowed to have different values
(although we choose them to be dichotomic). Suppose that target observable E, say, is
±1-valued, but the approximating observable F has the values a± (with a+ > a− ). In
order to calculate their 2-distance, we need to minimise the quantity (cf. Eq. (13.2))
γ
Δ2 (E , F )2 =
(x − y)2 dγ(x, y),
where γ is any coupling for E , F . The 2-distance should vanish when the probabilities of E and F coincide for their corresponding values, ±1 ↔ a± .
Any coupling γ ∈ Γ (E , F ) is given by four positive numbers,
(1, a+ ) → γ++ = γ,
(1, a− ) → γ+− = E (+1) − γ,
(−1, a+ ) → γ−+ = F (a+ ) − γ,
(−1, a− ) → γ−− = 1 − E (+1) − F (a+ ) + γ.
14.5 Approximate Joint Measurements of Qubit Observables
333
It is then straightforward to obtain
γ
Δ2 (E , F )2 = (1 + a− )2 − 4γ(a+ − a− ) − 4E (+1)a−
+ F (a+ ) (1 + a+ )2 − (1 + a− )2 .
To minimise this quantity, γ must be chosen as large as allowed by the positivity
constraints (given that a+ − a− > 0), hence γ = min{E (+1), F (a+ )}. Now it is
easy to see that the minimum, Δ2 E , F , can only vanish for E (+1) = F (a+ ) if
a+ = 1 and a− = −1.
Thus we assume that the approximating observables are also ±1-valued. We then
obtain
Δ2 E , F
2
= 4|E (+1) − F (+1)| = 2 e0 − f0 + n · (e − f ) .
By maximising this over all states
Δ2 E, F
2
=
n
one has the worst-case error estimate
= 2 |e0 − f0 | + e − f
= 4D∞ E, F .
(14.46)
Optimal Approximations and Measurement Uncertainty Relation
We consider now the problem of finding optimal approximate joint measurements
of two incompatible sharp qubit observables A, B with outcome sets {−1, +1}. As
discussed above, we choose the approximators to be dichotomic observables with
the same outcome spaces. Hence the joint observables will have four outcomes. We
choose the measure D∞ to quantify the error as the distance between A, B and the
corresponding margins C, D of the joint observable in question.
We call a point (D1 , D2 ) ∈ [0, 1] × [0, 1] admissible if D1 = D∞ C, A and D2 =
D∞ D, B for some jointly measurable qubit observables C and D. Not all points
in the square [0, 1] × [0, 1] are admissible; for instance the point (0, 0) is not an
admissible point since this would mean that C = A and D = B, which is impossible
since C, D are compatible but A, B are not. We shall characterise the region of
admissible points. The search for admissible points (D1 , D2 ) is narrowed down by
the following simple observation. Let C = {c0 I, (1 − c0 )I} be a trivial observable.
Then D∞ C, A = max{c0 , 1 − c0 } and therefore
D∞ C, A
c0 ∈ [0, 1] =
1
,1
2
.
Thus, approximations by trivial observables will never give distances below 21 . Furthermore, since such a C is jointly measurable with any observable D, and since
D∞ D, B can assume any value in [0, 1], it follows that all points in the set
[0, 1] × [0, 1] \ [0, 21 ) × [0, 21 ) are trivially admissible. We will therefore concentrate
on the characterisation of admissible points (D1 , D2 ) in the region [0, 21 ] × [0, 21 ].
For the remainder of this Chapter, the approximators for the sharp target observables A, B will be understood to be
334
14 Qubits
C=
D=
1
(c I
2 0
1
(d I
2 0
+ c · σ), 21 ((2 − c0 )I − c · σ) ,
+ d · σ), 21 ((2 − d0 )I − d · σ) ,
respectively. We shall also use the associated unbiased approximators
C(1) =
1
(I
2
± c · σ) , D(1) =
1
(I
2
± d · σ) .
The next two results will be proved in the Appendix.
Lemma 14.3 Any admissible point (D1 , D2 ) ∈ [0, 21 ] × [0, 21 ] has a realisation of
the type D1 = D∞ C(1) , A , D2 = D∞ D(1) , B , where c and d are in the plane
spanned by a and b.
Lemma 14.4 The set of admissible points is a closed convex set which is reflection
symmetric with respect to the axis D1 = D2 ; that is, with every admissible point
(D1 , D2 ) the point (D2 , D1 ) is also admissible. Thus, the segment of the boundary
curve defined as the graph of the function
D1 → D2∗ (D1 ) = min D2 : (D1 , D2 ) is admissible
(14.47)
is convex, symmetric and belongs to the set of admissible points.
Example 14.1 If D1 = D∞ C(1) , A = 0 (i.e. c = a), then the joint measurability
requirement implies that c and d are parallel and thus,
D∞ D(1) , B =
1
2
d−b ≥
1
2
1 − (a · b)2 =
1
2
sin θ.
The lower bound is attained when d = cos θ a = (a · b)a. We conclude that
0, 21 sin θ and 21 sin θ, 0 are points in the boundary of the admissible
region.
We next determine the boundary point with D1 = D2 = D0 . Due to the convexity
of the admissible region and its reflection symmetry with respect to the line D1 = D2 ,
it follows immediately that the admissible region is bounded below tightly by the
straight line D1 + D2 = 2D0 . This situation is sketched in Fig. 14.1. Determination
of the value of D0 yields the following result.
Proposition 14.4 Any admissible point (D1 , D2 ) = D∞ C, A , D∞ D, B
fies the error trade-off relation
D∞ C, A + D∞ D, B ≥ 2D0 ,
satis-
(14.48)
14.5 Approximate Joint Measurements of Qubit Observables
Fig. 14.1 The admissible
region (shaded area) and the
line D1 + D2 = 2D0 (thick
line). The dashed line is the
symmetry axis D1 = D2
335
D2
sin θ
2
sinθ
2
0
D1
where the lower bound is
1
2D0 = √
2 2
1
a+b + a−b −2 = √
2
1 + 2 [A+ , B+ ] − 1 .
The point D0 , D0 is admissible.
Before we prove this result, we comment on its significance. The second expression
given for the lower bound 2D0 is a monotonic function of the commutator of the
two projections defining the sharp observables being approximated. So, it is the
degree of noncommutativity of these observables that limits the accuracy of their
joint approximation. The first form of 2D0 is proportional to the expression a +
b + a − b − 2, which appears in the qubit compatibility condition (14.34c). For
projections (here a, b are unit vectors), this expression is always positive unless a, b
are collinear, in which case its value is zero. Thus, the quantity 2D0 is once again a
measure of incompatibility of the two sharp observables.
Proof The equality of the two expressions for the bound is due to the identities
a + b + a − b = 2 cos 2θ + sin
θ
2
√
= 2 1 + sin θ,
sin θ = a × b = 2 [A+ , B+ ] .
Consider the set of all jointly measurable unbiased observables C(1) , D(1) such that
c, d have equal fixed distance from a, b, respectively: c − a = d − b = 2D (so
that D∞ C(1) , A = D∞ D(1) , A = D). If (c, d) is not symmetric under reflection
with respect to the line through a + b, denote by c¯ and d¯ the mirror images of d and
¯ = 1 (I ± c¯ · σ) and
c, respectively. Then, if C, D are jointly measurable, so are C
2
336
14 Qubits
¯ = 1 (I ± d¯ · σ) as the condition (14.34c) is invariant under reflections. Due to
D
2
Proposition 11.4, the observables
1 (1)
C
2
1 (1)
D
2
ˆ (1) =
¯ (1) = C
+ 21 C
ˆ (1) =
¯ (1) = D
+ 21 D
1
(I
2
1
(I
2
± 21 (c + c¯ ) · σ) ,
¯ · σ)
± 21 (d + d)
are jointly measurable. It is clear from their definitions that the vectors (c + c¯ ) and
¯ are mirror images of each other. As c, d have equal distance 1 D from a, b,
(d + d)
2
respectively, this means that c and c¯ have equal distance 2D from a. It follows that
the distance from a to (c + c¯ )/2 is less than 2D (or = 2D if c = c¯ ). We conclude
that if c, d are not mirror images of each other, there is a pair of jointly measurable
ˆ (1) , D
ˆ (1) ) with smaller (and equal) distances from A, B and
observables (namely, C
mirror symmetric vectors. This shows that the minimal equal distance approximations
of A, B by means of jointly measurable observables occur among the unbiased pairs
C(1) , D(1) with c, d mirror symmetric with respect to a + b.
If coordinates are chosen such that a = (sin 2θ , cos 2θ ), b = (− sin 2θ , cos 2θ ), then
let a symmetric pair c, d be given by c = (u, v) and d = (−u, v), with u, v > 0. For
such pairs, the joint measurability condition for C, D assumes the form u + v ≤ 1.
It follows that the shortest (equal) distances 2D of c, d from a, b are assumed when
u + v = 1 and a − c is perpendicular to the line u + v = 1. But this distance 2D is
equal to the distance of the lines u + v = 1 and u + v = cos 2θ + sin 2θ , hence
2D =
√1
2
cos 2θ + sin 2θ − 1 = 2D0 .
This completes the proof.
The approximations C(1) and D(1) leading to the boundary point (D0 , D0 ) are
generally not among the smearings of A and B. Indeed, let us denote by 2D0c the
smallest distance achieved under the assumptions that D0c = D∞ C , A = 21 (1 −
c ) = D∞ D , B = 21 (1 − d ) where C , D are jointly measurable and (unbiased) smearings of A, B, respectively, with c = c a and d = d b. If the vectors a
and b are orthogonal, then D0c = D0 . However, if 0 < θ < π2 , then
D0c
1
=
1−
2
√
1 − sin θ
cos θ
> D0 .
(14.49)
The approximating vectors c, d and c , d for a, b are illustrated in Fig. 14.2. We
conclude that to attain the best jointly measurable approximations of two sharp qubit
observables, we are forced to seek approximating observables beyond their smeared
versions.
The inequality (14.48) is tight only in the point (D1 , D2 ) = (D0 , D0 ), for which the
two approximation errors are equal. Using the function D2∗ defined in Eq. (14.47), the
14.5 Approximate Joint Measurements of Qubit Observables
337
Fig. 14.2 The vectors c, d correspond to the optimal compatible approximations C(1) and D(1) ,
and the vectors c , d correspond to the closest (unbiased) compatible smearings of A, B. The latter
are clearly suboptimal joint approximations
optimal, tight measurement uncertainty relation for two simple sharp qubit observables can be expressed as
D∞ D, B ≥ D2∗ D∞ C, A .
(14.50)
The function D2∗ can be completely—albeit implicitly—characterised [13], and it has
been shown that this optimal error bound is governed by an interplay between the
incompatibility of A and B and the degrees of unsharpness of C, D [14]. The explicit
form of this trade-off and its derivation are rather involved and we sketch here only
the solution for the case a · b = 0.
We note first that the compatibility constraint c + d + c − d ≤ 2 can be
described geometrically as the condition that for given c = 0, the vector d must
lie within a closed ellipsoid with boundary given by the limiting case of equality in
this inequality; the ellipsoid has its semi-major axis along c and lies within the unit
ball centred at the origin, touching the surface at ±ˆc. Equivalently, one may consider
d = 0 fixed and find c constrained within a similar ellipsoid with axis along d. It
is clear that in any optimising constellation, the end points of c, d must lie on the
surfaces of their respective constraining ellipsoids; to be as close to a, b, respectively,
they must lie within the plane spanned by the latter vectors. Moreover, the endpoint
of the vector c (d) will lie on the surface of a ball centred at a (b), which is thus
tangent to the constraining ellipsoid and has radius equal to the distance c − a
( d − b ).
Given the orthogonality of a, b, it is not hard to see that these conditions are met
when c = ca, d = db with c, d > 0, where the compatibility condition is met if and
only if c2 + d 2 = 1, in agreement with inequality (14.34a). In that case, the optimal
errors are
D∞ C, A = 21 (1 − c), D∞ D, B = 21 (1 − d),
338
14 Qubits
and so
2D∞ (C, A) − 1
2
+ 2D∞ (D, B) − 1
2
= 1.
For general unbiased compatible approximators C, D, one thus obtains the qubit
measurement uncertainty relation
D∞ (C, A) −
1 2
2
+ D∞ (D, B) −
1 2
2
≤ 41 .
(14.51)
This shows that the admissible region is lower-bounded within [0, 21 ] × [0, 21 ] by the
segment of the circle with radius 21 centred at ( 21 , 21 ), in line with Fig. 14.1.
Finally we note that there is a close connection between the measurement
uncertainty relation just found for observables A, B with orthogonal vectors a, b
and the preparation uncertainty relation (14.14) [12]. If we define the state 0 =
1
(I + (c + d) · σ) with orthogonal vectors c = ca, d = db satisfying c2 + d 2 ≤ 1,
2
we find that
Δ(A, 0 )2 = 1 − a · (c + d)
2
= 1 − c2 , Δ(B, 0 )2 = 1 − b · (c + d)
2
= 1 − d2,
and so, in agreement with (14.14),
Δ(A, )2 + Δ(B, )2 = 2 − c2 − d 2 ≥ 1.
The connection with the measurement uncertainty relation (14.51) becomes apparent if we observe that both relations are equivalent to c2 + d 2 ≤ 1. This condition
ensures both the conditions for 0 to be a state and for c, d to define compatible qubit
observables C, D. We can shed more light on this connection by recalling that a joint
observable of the kind (14.38) can be given for C, D, which here assumes the form
Gkl = 14 (I + (kc + ld) · σ);
the positivity of these operators is equivalent to c2 + d 2 ≤ 1, hence again to the
positivity of 0 .
There is also a close analogy with a similar connection between preparation and
measurement uncertainty in the case of phase space measurements [12]. As we have
seen, a covariant phase space observable is generated by a positive trace-one operator
T by application of the Weyl operators (and integration over phase space cells). In
the present case we can similarly define a (projective) representation of the shift
group on the discrete ‘phase space Z2 × Z2 (already encountered in Example 13.5).
Thus, let a = i, b = j, c = k be a right-handed orthogonal triple of unit vectors;
consider X = a · σ = σ1 as the ‘position’, Y = b · σ = σ2 as the ‘momentum’, and
put Z = c · σ = σ3 . A shift of the values of X alone is generated by the unitary and
selfadjoint operator Y as YXY = −X, and similarly a shift of the values of Y alone is
generated by X as XYX = −Y . Then XY = iZ, which generates simultaneous shifts,
ZXZ = −Z, ZYZ = −Y . Finally X 2 = Y 2 = Z 2 = I.
14.5 Approximate Joint Measurements of Qubit Observables
339
We see that the effects of the above joint observable can then be generated from
= G++ as G+− = 21 X 0 X, G−+ = 21 Y 0 Y , G−− = 21 Z 0 Z. In this way one can
say that the measurement error relation for the margins C, D is reduced to the preparation uncertainty relation for A, B in the state 0 .
1
2 0
Remark 14.7 The steps performed in this section towards obtaining the qubit measurement uncertainty relation (14.48) are illustrated in an interactive demonstration
available at the Wolfram Demonstrations Project web page with the URL http://
demonstrations.wolfram.com/HeisenbergTypeUncertaintyRelationForQubits/.
Qubit Measurement Uncertainty in Terms of Error Bar Width
In Proposition 12.4 and Theorem 13.4 we have seen versions of preparation and
measurement uncertainty relations for discrete observables formulated in terms of
overall width and error bar width, respectively. To illustrate that these measures yield
nontrivial relations even in the simplest discrete case of two-point value spaces, we
specify these relations to the case of the two sharp qubit observables Z = Eσ3 and
X = Eσ1 . The value spaces are ΩZ = ΩX = {1, −1}, and we equip them with the
discrete metric d. (We note that the choice of the values does not affect the results
below.)
Let ε1 , ε2 ∈ [0, 1] with ε1 + ε2 ≤ 1, and let be any state. The overall widths
Wε1 (Z ) and Wε2 (X ) then satisfy the following inequality.
max Od z, Wε1 (Z )
z∈ΩZ
· max Od x, Wε2 (X )
x∈ΩX
≥ 2(1 − ε1 − ε2 )2 .
(14.52)
For the choice ε1 = ε2 = 0, the value of the bound is 2. Suppose Z (say) is sharply
localised, so that one of the outcomes has probability 1; then
maxz∈ΩZ Od z, Wε1 (Z ) = 1. It follows that maxx∈ΩX Od y, Wε2 (X ) = 2; this
means that X cannot be sharply localised in the eigenstates of Z, and vice versa.
On the other hand, if one stipulates
max Od z, Wε1 (Z )
z∈ΩZ
= max Od y, Wε2 (X )
x∈ΩX
= 1,
√
then the above inequality entails that one must accept ε1 + ε2 ≥ 1 − 1/ 2. This
simply reproduces the Landau–Pollak relation (12.16) (which can, of course, also be
verified by direct calculation in the present case):
1
max Z {z} + max X {x} ≤ 1 + √ .
z∈ΩZ
x∈ΩX
2
Let M be an observable on ΩZ × ΩX , with margins M1 , M2 on the value spaces
ΩZ and ΩX , respectively. Then for ε1 , ε2 > 0 with ε1 + ε2 ≤ 1,
max Od z, Wε1 (M1 , Z)
z∈ΩZ
· max Od x, Wε2 (M2 , X)
x∈ΩX
≥ 2(1 − ε1 − ε2 )2 . (14.53)
340
14 Qubits
If one assumes
max Od z, Wε1 (M1 , Z)
z∈ΩZ
= max Od x, Wε2 (M2 , X)
x∈ΩX
= 1,
(14.54)
√
it follows again that ε1 + ε2 ≥ 1 − 1/ 2. Let ε1 , ε2 be such that
1 − ε1 = min tr
z M1
{z} ,
z
1 − ε2 = min tr
x M2
{x} ,
x
z∈ΩZ
x∈ΩX
= 21 (I + zˆz · σ),
= 21 (I + x xˆ · σ).
(Here z , x are the eigenstates of Z, X.) Then, since tr z M1 {z}
tr x M2 {x} ≥ 1 − ε2 , Eq. (14.54) must hold, and therefore
min tr
z∈ΩZ
z M1
{z}
+ min tr
x∈ΩX
x M2
{x}
≥ 1 − ε1 and
1
≤1+ √ .
2
Finally, we can replace the eigenstate notation by the spectral projections of Z, X to
obtain
min tr Z {z} M1 {z}
z∈ΩZ
+ min tr X {x} M2 {x}
x∈ΩX
1
≤1+ √ .
2
(14.55)
This is a bound on the proximity of M1 to Z and M2 to X, measured in terms of
the minimal “overlaps"of the associated effects, which are not allowed to become
too large due to the competing properties of the noncommutativity of Z, X and the
compatibility of M1 , M2 . Note that in this example we see once more the close
connection between preparation and measurement uncertainty, here expressed in
terms of overall width and error bar width.
14.6 Appendix
Proof of Propositions 14.3 and 14.4
(a) If (D1 , D2 ) is an admissible point, then also (D2 , D1 ) is an admissible point.
Proof If C with C+ = 21 (c0 I + c · σ) and D with D+ = 21 (d0 I + d · σ) realise the
distances D1 and D2 , respectively, then choose c0 , c and d0 , d as follows: c0 = d0 ,
c has the length of d and its angle relative to a is equal to the angle between d and
b; similarly, d0 = c0 , d has the length of c and its angle relative to b is the same as
the angle between c and a. This ensures that (D1 , D2 ) = (D2 , D1 ).
(b) Assume that (D1 , D2 ) = D∞ C, A , D∞ D, B is an admissible point. As
shown in Proposition 14.2, the joint measurability of C and D implies that of C(1)