Optimal Pointers for Joint Measurement of sigma-x and sigma-z via Homodyne Detection

0.28 MB·English

by  Bas Janssens

σ σ
Optimal Pointers for Joint Measurement of and via
x z
Homodyne Detection
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0 Bas Janssens∗ and Luc Bouten†
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2

n ∗Department of Mathematics, Radboud University Nijmegen,
a Toernooiveld 1, 6525 ED, Nijmegen, The Netherlands
J

1
3 †Physical Measurement and Control 266-33,California Institute of Technology,
1200 E. California Blvd., Pasadena,CA 91125,USA
3
v
6
8
0
0
1
5 Abstract
0
/
h We study a model of a qubit in interaction with the electromagnetic field. By means of
p homodynedetection,thefield-quadratureAt+A∗t isobservedcontinuouslyintime. Duetothe
-
t interaction, information about the initial state of the qubit is transferred into the field, thus
n influencingthehomodynemeasurementresults. Weconstructrandomvariables(pointers)on
a
the probability space of homodyne measurement outcomes having distributions close to the
u
q initial distributions of σx and σz. Using variational calculus, we find the pointers that are
optimal. These optimal pointersare veryclose tohittingtheboundimposed byHeisenberg’s
:
v uncertainty relation on joint measurement of two non-commuting observables. We close the
i
X paperby giving theprobability densities of thepointers.
r
a
1 Introduction
Theimplementationofquantumfilteringandcontrol[5]inrecentexperiments[2],[13]hasbrought
newinteresttothefieldofcontinuoustimemeasurementofquantumsystems[11],[12],[5],[7],[10],
[25],[9]. Inparticular,homodynedetectionhasplayedaconsiderableroleinthisdevelopment[10].
In this paper, we aim to gain insight into the transfer of information about the initial state of a
qubitfromthisqubit,atwo-levelatom,tothehomodynephotocurrent,whichisobservedinactual
experiments. Our goal is to perform a joint measurement of two non-commuting observables in
the initial system. In order to achieve this, we construct randomvariables (pointers) on the space
ofpossible homodyne measurementresults,having distributions close(in a sense to be defined) to
the distributions of these observables in the initial state.
1
The problem of joint measurement of non-commuting observables has been studied by several
authors before, see [22], [12], [16] and the references therein. As a measure for the quality of
an unbiased measurement, we use the difference between the variance of the pointer in the final
state and the variance of the observable in the initial state, evaluated in the worst case initial
state [18]. In other words, the quality of measurement is given in terms of the worst case added
variance. Theseworstcaseaddedvariancesfortwopointers,correspondingtotwonon-commuting
observablesoftheinitialsystem,satisfyaHeisenberg-likerelationthatboundshowwelltheirjoint
measurement can be performed [18].
The paper concentrates on the example of a qubit coupled to the quantized electromagnetic field.
We study this system in the weak coupling limit [15], i.e. the interaction between qubit and
field is governed by a quantum stochastic differential equation in the sense of Hudson and Par-
thasarathy [17]. In the electromagnetic field we perform a homodyne detection experiment. Its
integratedphotocurrentisthemeasurementresultformeasurementofthefield-quadratureA +A
t ∗t
continuouslyintime. UsingthecharacteristicfunctionsintroducedbyBarchielliandLupieri[4],we
findthe probabilitydensityforthesemeasurementresults. Inthisdensitythex- andz-component
oftheBlochvectoroftheinitialstateappear,indicatingthathomodynedetectionisinfactajoint
measurement of σ and σ in the initial state.
x z
The goal of the paper is to construct random variables (pointers) on the probability space of
homodyne measurement results having distributions as close as possible to those of the observa-
bles σ and σ in the initial state of the qubit. ‘As close as possible’ is taken to mean that the
x z
pointermustgiveanunbiasedestimateoftheobservable,withitsworstcaseaddedvarianceaslow
as possible. Using anargumentdue to Wiseman[24], we firstshowthat optimalrandomvariables
will only depend on the endpoint of a weighted path of the integrated photocurrent. Allowed
to restrict our attention to this smaller class of pointers, we are able to use standard variational
calculus to obtain the optimal random variables. They do not achieve the bound imposed by the
Heisenberg-like relation for the worst case added variances [18], but will be off by less than 5.6%.
Theremainderofthepaperisorganizedasfollows. InSection2weintroducethemodelofthequbit
coupledtothefieldintheweakcouplinglimit. Section3introducesthequalityofameasurementin
terms of the worst case added variance. This section also contains the Heisenberg-like relation for
joint measurement. In Section 4 we calculate the characteristic function of Barchielli and Lupieri
for the homodyne detection experiment. Section 5 deals with the variational calculus to find the
optimal pointers. In Section 6 we calculate the densities of the optimal pointers and then capture
our main results graphically. In the last section we discuss our results.
2 The model
We considera two-levelatom, i.e. a qubit, in interactionwith the quantized electromagneticfield.
ThequbitisdescribedbyC2 andtheelectromagneticfieldbythesymmetricFock space overthe
Hilbert space of quadratically integrable functions L2(R) (space of one-photon wave fFunctions),
i.e.
∞
:=C L2(R)⊗sk.
F ⊕
k=1
M
2
With the Fock space we can describe superpositions of field-states with different numbers of
photons. The joint sysFtem of qubit and field together is described by the Hilbert space C2 .
⊗F
The interaction between the qubit and the electromagnetic field is studied in the weak coupling
limit [14], [15], [1]. This means that in the interaction picture the unitary dynamics of the qubit
and the field together is given by a quantum stochastic differential equation (QSDE) in the sense
of Hudson and Parthasarathy[17]
0 0 0 1
dU = σ dA σ dA 1σ σ dt U , with σ = , σ = , U =I. (1)
t − ∗t − + t− 2 + − t − 1 0 + 0 0 0
n o (cid:18) (cid:19) (cid:18) (cid:19)
The operators σ and σ are the annihilator and creator on the two-level system. The field
+
−
annihilation and creation processes are denoted A and A , respectively. Keep in mind that the
t ∗t
evolutionU acts nontriviallyonthe combinedsystemC2 ,whereasσ andA areunderstood
t t
to designate the single system-operatorsσ I and I A⊗.FThroughout±the paper we will remain
t
±⊗ ⊗
in the interaction picture. Equation (1) should be understood as a shorthand for the integral
equation
t t t
U =I + σ U dA σ U dA 1 σ σ U dτ,
t Z0 − τ ∗τ −Z0 + τ τ − 2Z0 + − τ
where the integrals on the right-hand side are stochastic integrals in the sense of Hudson and
Parthasarathy [17]. The value of these integrals does not lie in their actual definition (on which
we will not comment further), but in the Itoˆ rule satisfied by them, allowing for easy calculations.
Theorem 2.1: (Quantum Itˆo rule [17], [20])LetX andY bestochasticintegralsoftheform
t t
dXt =CtdAt+DtdA∗t +Etdt
dY =F dA +G dA +H dt
t t t t ∗t t
forsome stochasticallyintegrableprocessesC ,D ,E ,F ,G andH (see [17], [20]for definitions).
t t t t t t
Then the process X Y satisfies the relation
t t
d(X Y )=X dY +(dX )Y +dX dY ,
t t t t t t t t
where dX dY should be evaluated according to the quantum Itoˆ table:
t t
dA dA∗ dt
t t
dA 0 dt 0
t
dA∗ 0 0 0
t
dt 0 0 0
i.e. dX dY =C G dt.
t t t t
Asacorollarywehavethat,foranyf C2(R),theprocessf(X )satisfiesd(f(X ))=f (X )dX +
t t ′ t t
∈
1f (X )(dX )2, where (dX )2 should be evaluated according to the quantum Itoˆ table.
2 ′′ t t t
3
First a matter of notation. The quantum Itoˆ rule will be used for calculating differentials of
products of stochastic integrals. Let Z be stochastic integrals. Then we write
i i=1,...,p
{ }
d(Z Z ...Z )= [ν]
1 2 p
ν⊂{X1,...,p}
ν=
6 ∅
wherethesumrunsoverallnon-emptysubsetsof 1,...,p . Foranyν = i ,...,i ,the term[ν]
1 k
{ } { }
isthecontributiontod(Z Z ...Z )comingfromdifferentiatingonlythe termswithindicesinthe
1 2 p
set i ,...,i andpreservingthe orderofthe factorsin the product. The differentiald(Z Z Z ),
1 k 1 2 3
{ }
forexample,containsterms oftype [1],[2],[3], [12],[13],[23]and[123]. We have[2]=Z (dZ )Z ,
1 2 3
[13]=(dZ )Z (dZ ), [123]=(dZ )(dZ )(dZ ), etc.
1 2 3 1 2 3
Let us return to equation (1). In order to illustrate how the quantum Itoˆ rule will be used, we
calculatethetimeevolutiononthequbitexplicitly. Thealgebraofqubit-observablesisthealgebra
of 2 2-matrices, denoted M (C). The algebra of observables in the field is given by B( ), the
2
boun×ded operators on . If id : M (C) M (C) is the identity map and φ : B( ) CFis the
2 2
F → F →
expectation with respect to the vacuum state Φ := 1 0 0 ... (i.e. φ(Y) := Φ,YΦ ),
then time evolution on the qubit T : M (C) M (C)⊕is g⊕iven⊕by T∈(XF):=id φ(U Xh IUi).
Onthe combined system,the full titme ev2olutio→nj 2: M (C) B( ) t M (C) ⊗B( )t∗is g⊗ivenbty
t 2 2
⊗ F → ⊗ F
j (W):=U WU . In a diagram this reads
t t∗ t
M Tt M
2 2
−−−−→
id I id φ (2)
⊗ ⊗
M B( ) jt M xB( ).
2⊗y F −−−−→ 2⊗ F
In the Schr¨odingerpicture the arrowswould be reversed. A qubit-state ρ would be extended with
the vacuum to ρ φ, then time evolved with U , and in the last step the partial trace over the
t
⊗
field would be taken, resulting in the state ρ T .
t
◦
Using the Itoˆ rule we can derive a (matrix-valued) differential equation for T (X), i.e.
t
dT (X)=id φ d(U X IU )
t ⊗ t∗ ⊗ t
=id⊗φ(cid:0)(dUt∗)X ⊗IUt(cid:1)+Ut∗X ⊗I(dUt)+(dUt∗)X ⊗I(dUt) (3)
=id⊗φ(cid:0)Ut∗L(X)⊗IUt dt (cid:1)
=Tt L(X(cid:0)) dt, (cid:1)
where L is the Lindblad ge(cid:0)nerato(cid:1)r
L(X):= 1(σ σ X +Xσ σ )+σ Xσ .
−2 + − + − + −
In the derivation (3) we used the QSDE for U which easily follows from (1)
t∗
dU =U σ dA σ dA 1σ σ dt , U =I.
t∗ t∗ + t− − ∗t − 2 + − 0∗
n o
Furthermore, we used that stochastic integrals with respect to dA and dA vanish with respect
t ∗t
to the vacuum expectation, leaving us only with the dt terms. The differential equation (3) with
initial condition T (X)=X is solvedby T (X)=exp(tL)(X), which is exactly the time evolution
0 t
of a two-level system spontaneously decaying to the ground state, as it should be. Although the
arguments above are completely standard (cf. [17]), they do illustrate nicely and briefly some of
the techniques used also in following sections.
4
3 Quality of information transfer
Now suppose we do a homodyne detection experiment, enabling us to measure the observables
A +A in the field continuously in time [3]. If initially the qubit is in state ρ, then at time t the
t ∗t
qubit and field together are in a state ρt on M (C) B( ) given by ρt(W) := ρ φ(U WU ) =
2 ⊗ F ⊗ t∗ t
ρ id φ(U WU ) . Since
⊗ t∗ t
(cid:0) d id φ(U(cid:1) I (A +A )U ) =id φ d(U I (A +A )U )
⊗ t∗ ⊗ t ∗t t ⊗ t∗ ⊗ t ∗t t
(cid:16) (cid:17) (cid:16) (cid:17)
=id φ [1]+[2]+[3]+[12]+[13]+[23]+[123]
⊗
(cid:16) (cid:17)
=id φ U (σ +σ ) IU dt
⊗ t∗ − + ⊗ t
=exp(tL(cid:16))(σ +σ+)dt=e−2t(cid:17)σxdt,
−
we have that regardless the initial state ρ of the qubit, the expectation of (A +A ) in the final
t ∗t
state ρt will equal the expectation of (2 2e−2t)σx in the initial state ρ.
−
3.1 Defining the quality of information transfer
The process at hand is thus a transfer of information about σ to a ‘pointer’ A +A , which can
x t ∗t
be read off by means of homodyne detection. This motivates the following definition.
Definition 3.1: (Unbiased Measurement [18]) Let X be an observable of the qubit, i.e. a
self-adjoint element of M (C), and let Y be an observable of the field, i.e. a self-adjoint operator
2
in (or affiliated to) B( ). An unbiased measurement M of X with pointer Y is by definition a
completely positive maFp M : B( ) M (C) such that M(Y)=X.
2
F →
Needlesstosay,foreachfixedtthemapM : B( ) M (C)givenbyM(B):=id φ(U I BU )
is a measurement of σx with pointer Y = (2−F2e→−2t)−21(At +A∗t). This means⊗that,t∗af⊗ter thte
measurement procedure of coupling to the field in the vacuum state and allowing for interaction
with the qubit for t time units, the distribution of the measurement results of the pointer Y has
inherited the expectation of σ , regardless of the initial state ρ. However, we are more ambitious
x
and would like its distribution as a whole to resemble that of σ . This motivates the following
x
definition.
Definition3.2: (Quality [18])LetM : B( ) M (C)beanunbiasedmeasurementofX with
2
F →
pointer Y. Then its quality σ is defined by
σ2 :=sup Var (Y) Var (X) ρ (M ) ,
ρ M ρ 2
◦ − ∈S
where (M ) denotes the state spnace of M (C) (i.e. all(cid:12)positive noromalized linear functionals
2 2 (cid:12)
on M (SC)).
2
This means that σ2 is the variance added to the initial distribution of X by the measurement
5
procedure M for the worst case initial state ρ. A small calculation shows that
Var (Y) Var (X)=ρ M(Y2) M(Y)2 ,
ρ M ρ
◦ − −
whichimpliesthatσ2 = M(Y2) M(Y)2 ,whereX(cid:0) X denotesth(cid:1)eoperatornormon M (C).
2
k − k 7→k k
Inparticularthis showsthatσ2 is positive, asone mightexpect. Itfollowsfrom[18] thatσ equals
zero if and only if the measurement procedure M exactly carries over the distribution of X to Y.
In short, σ is a suitable measure for how well M transfers information about X to the pointer Y.
3.2 Calculating the quality of information transfer
Let us return to the example at hand, i.e. M(B) = id φ(U I BU ), with field-observable
Yeva=lua(2tin−g2Me−(2tY)2−)1=(A(t2+−A2∗te)−a2ts)−a2Mpoin(Atetr+foAr∗tσ)x2..LTeottuhsis⊗caalicmu,latw∗tee⊗witisllqfiurtasltitiyn,trwohdiuchceasmomouenitdseatos
which will be of use to us in later calculations as well.
(cid:0) (cid:1)
Definition 3.3: Let f and h be real valued functions, h twice differentiable. Let Y be given by
t
dY =f(t)(dA +dA ), Y =0. For X M (C) we define
t t ∗t 0 ∈ 2
Fh(X,t):=id⊗φ Ut∗X ⊗h(Yt)Ut .
When no confusion can arise we shall shorten F (cid:0)(X,t) to F (X).(cid:1)
h h
The homodyne detection experimenthas givenus a measurementresult(the integratedphotocur-
rent) which is just the path of measurement results for A +A continuously in time. Given this
t ∗t
result,wepost-processitbyweightingtheincrementsofthepathwiththefunctionf(t)andletting
h(y) act on the result. The following lemma will considerably shorten calculations.
Lemma 3.4:
dF (X)
hdt =Fh L(X) +f(t)Fh′(σ+X +Xσ−)+ 21f(t)2Fh′′(X)
(cid:0) (cid:1)
Proof. Using the notationbelow Theorem2.1 with Z =U , Z =I h(Y ) and Z =U , we find
1 t∗ 2 ⊗ t 3 t
dF (X)=id φ [1]+[2]+[3]+[12]+[13]+[23]+[123] .
h
⊗
Again we will use that the vacuum(cid:0)expectation kills all dA and dA terms.(cid:1)Using Theorem 2.1
t ∗t
we see that after the vacuum expectation the terms [1], [3] and [13] make up F L(X) dt. Since
h
third powers of increments are 0 we again have [123]=0. From
(cid:0) (cid:1)
dh(Yt)=h′(Yt)f(t)(dAt+dA∗t)+ 12h′′(Yt)f(t)2dt,
we find that, after taking vacuum expectations, the terms [12] and [23] make up the second term
f(t)Fh′(σ+X +Xσ−)dt and [2] provides the last term 21f(t)2Fh′′(X)dt.
6
We are now well-equipped to calculate M (A +A )2 . Choose f(t) = 1 and h(x) = x2. (The
t ∗t
maps x7→xn will be denoted Xn hereafter.(cid:0)) Then M ((cid:1)At+A∗t)2 =FX2(I) and by Lemma 3.4
dFXd2t(I) =2FX(σ−+σ+)+F1(I)=2FX((cid:0)σ−+σ+)+(cid:1)I, FX2(I,0)=0. (4)
Applying Lemma 3.4 to FX(σ +σ+), we obtain
−
dFX(σd−t+σ+) =−21FX(σ−+σ+)+2F1(σ+σ−), FX(σ−+σ+,0)=0. (5)
Finally, F (σ σ ) satisfies
1 +
−
dF (σ σ )
1 dt+ − =−F1(σ+σ−), F1(σ+σ−,0)=σ+σ−. (6)
Solving (6), (5) and (4) successively leads first to F1(σ+σ )= e−tσ+σ , then to FX(σ +σ+)=
4(e−2t e−t)σ+σ and finally to FX2(I) = 8(e−2t 1)2σ−+σ +tI. C−onsequently, the−quality of
the me−asurement−M of σx with pointer Y =(2−2e−−2t)−1(A−t+A∗t) is given by
σ2 ==k(M2 (Y2te2−)−2t)M2 +(Y1).2k=(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)8(e−2t(2−−1)22eσ−+2tσ)−2 +tI −I(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)=(cid:13)(cid:13)(cid:13)(cid:13)2σ+σ−+(cid:18)(2−2te−2t)2 −1(cid:19)I(cid:13)(cid:13)(cid:13)(cid:13)
−
This expression takes its minimal value 2.228 at t=2.513, leading to a quality σ =1.493.
The calculation above has an interesting side product. The observable M (A +A )2 depends
t ∗t
linearly on σ , indicating that in addition to information on σ , also information on σ in the
z x z
(cid:0) (cid:1)
initial qubit-state ends up in the measurement outcome. Indeed, if we use as a pointer
(A +A )2 tI
Y˜ := t ∗t − I, (7)
4(e−2t 1)2 −
−
then we have M(Y˜)=σ , so that M is also a measurement of σ with pointer Y˜.
z z
Note that the pointers Y and Y˜ commute, i.e. measuring A +A via the homodyne detection
t ∗t
scheme is an indirect joint measurement of σ and σ . If we would also like to gain some infor-
x z
mationabout σ , we couldfor example sweepthe measuredquadraturethrough[0,2π) intime by
y
measuringeiωtA +e iωtA instead. Inthispaperhowever,wewillrestrictourselvestocontinuous
t − ∗t
time measurementofA +A ,asadditionalinformationonσ woulddeterioratethe qualityofσ -
t ∗t y x
and/or σ -measurement. The following theorem is a Heisenberg-like relation that gives a bound
z
on how well joint measurements can be performed.
Theorem 3.5: (Joint Measurement [18]) Let M : B( ) M (C) be an unbiased measure-
2
ment ofself-adjointobservablesX M (C) andX˜ M (CF) w→ith self-adjointcommuting pointers
2 2
Y and Y˜ in (or affiliated to) B( )∈, respectively. T∈hen for their corresponding qualities σ and σ˜
F
the following relation holds
2σσ˜ [X,X˜] .
≥k k
7
Denote by σ˜ the quality oftheσ measurementwith the pointerY˜ defined in(7). Since [σ ,σ ]=
z x z
2iσ , the qualities σ and σ˜ (corresponding to the pointers Y and Y˜, respectively) satisfy the
y
−
inequality
σσ˜ 1. (8)
≥
Using similar techniques as before, that is recursively calculating FX4(I) via Lemma 3.4, we find
σ˜2 = t2 + 2t−4(e−2t −1)2.
8(e−2t 1)4 (e−2t 1)2
− −
This expression takes its minimal value 8.836 at t = 2.513. This leads to a quality σ˜ = 2.973,
which means that σσ˜ = 4.437, i.e. we are far removed from hitting the bound 1 in (8). However,
there is still some room for manoeuvring by post-processing of the homodyne measurement data.
4 The weighted path
Let us presently return to our homodyne detection experiment. We observeA +A continuously
τ ∗τ
intime,i.e.theresultofourmeasurementisapathω ofmeasurementresultsω (thephotocurrent
τ
integrateduptotimeτ)forA +A . ThismeansthatwehaveaspaceΩofallpossiblemeasurement
τ ∗τ
paths and that we can identify an operator A + A with the map from Ω to R mapping a
τ ∗τ
measurementpathω Ωtothemeasurementresultω attimeτ. Thatis,wehavesimultaneously
τ
∈
diagonalized the family of commuting operators A +A τ 0 and viewed them as random
{ τ ∗τ| ≥ }
variables on the spectrum Ω. The spectral projectors of the operators A +A 0 τ t
{ τ ∗τ| ≤ ≤ }
endow Ω with a filtration of σ-algebras Σ . Furthermore, the states ρτ, defined by ρτ(W) :=
t
ρ φ(U WU )provideafamilyofconsistentmeasuresP on(Ω,Σ ),turningitintotheprobability
sp⊗ace (Ωτ∗,Σ ,τP). (See e.g. [8].) τ τ
t
We aim to find random variables on (Ω,Σ ,P) having distributions resembling those of σ and σ
t x z
in the initial state ρ. In the previous section we used the random variables
ω ω2 τ
Y(ω)= τ and Y˜(ω)= τ − 1, τ =2.513 (9)
2 2e−τ2 4(e−τ2 1)2 −
− −
for σ and σ , respectively. Our next goal is to find the optimal random variables, in the sense of
x z
the previously defined quality.
4.1 Restricting the class of pointers
In ourspecific example,M is givenby M(B)=id φ(U I BU ). Note that stochastic integrals
⊗ τ∗ ⊗ τ
with respect to the annihilator A acting on the vacuum vector Φ are zero. Therefore, we can
τ
modify U to Z , given by
τ τ
dZ = σ (dA +dA ) 1σ σ dτ Z , Z =I,
τ − ∗τ τ − 2 + − τ 0
n o
8
without affecting M [6]. Therefore, for all B B( ), we have M(B) = id φ(U I BU ) =
∈ F ⊗ τ∗ ⊗ τ
id φ(Z I BZ ). The solution Z can readily be found, it is given by
⊗ τ∗ ⊗ τ t
Z = e−12t 0 .
t (cid:18) 0te−12τ(dAτ +dA∗τ) 1(cid:19)
Note that Z , as a matrix valued functiRon of the measurement path, is an element of M (C) ,
t 2 t
⊗C
where isthecommutativevonNeumannalgebrageneratedbyA +A , 0 τ t. Moreoverwe
Ct τ ∗τ ≤ ≤
see that Z is not a function of all the (A +A )’s separately,it is only a function of the endpoint
t τ ∗τ
coofmthmeuwtaetiigvhetevdonpNatehumYtan=n a0ltgee−br21aτ(gdeAnτer+ateddAb∗τ)y[Y24,].theTnhewreefeovreenifhawveedZefineMSt(C⊂)Ct to. be the
t t 2 t
R ∈ ⊗S
Denote byC E[C ]the unique classicalconditionalexpectationfrom onto thatleavesφ
t t t
invariant, i.e.7→φ(E[C|S ])= φ(C) for all C . We can extend E[ ] bCy tensoSring it with the
t t t
identitymaponthe2|S 2matricestoobtain∈aCmapid E[ ]from·M|S (C) ontoM (C) .
t 2 t 2 t
Fromthepositivityof×E[ ]asamapbetweencomm⊗utat·iv|Sealgebras,itfol⊗lowCsthatid E[⊗S ]
t t
is completely positive. Sin·c|Se E[ ] satisfiesE[CS ]=E[C ]S for allC andS⊗ ·,|Swe
t t t t t
find that id E[ ] satisfies t·h|eSmodule property,|iS.e. |S ∈C ∈S
t
⊗ · |S
id E[ ](A BA )=A id E[ ](B) A ,
t 1 2 1 t 2
⊗ · |S ⊗ · |S
(cid:16) (cid:17)
forallA ,A M (C) andB M (C) . Moreover,ifρisastateonM (C),thenitfollows
1 2 2 t 2 t 2
from the inva∈riance of⊗φSunder E[∈ ] th⊗atCid E[ ] leaves ρ φ invariant. We conclude
t t
that, given ρ on M (C), the map id· |SE[ ] f⊗rom M· |S(C) ont⊗o M (C) is the unique
2 t 2 t 2 t
⊗ · |S ⊗C ⊗S
conditional expectation in the noncommutative sense of [21] that leaves ρ φ invariant. We will
use the shorthand E for id E[ ] in the following. ⊗
St ⊗ · |St
Lemma 4.1: Let C be a pointer with quality σ such that M(C)=X. Then C˜ :=E[C ]
t C t
is also a pointer with∈MC(C˜)=X, and with quality σ σ . |S
C˜ ≤ C
Proof. Note that for all states ρ on M (C) we have
2
ρ M(C˜) =ρ⊗φ Zt∗I⊗C˜Zt =ρ⊗φ Zt∗ESt(I ⊗C)Zt =ρ⊗φ ESt Zt∗I⊗CZt
(cid:0) (cid:1)=ρ φ(cid:0)(Z I CZ )(cid:1)=ρ M(C(cid:16)) =ρ(X), (cid:17) (cid:16) (cid:0) (cid:1)(cid:17)
⊗ t∗ ⊗ t
where we used the module property and(cid:0)the fa(cid:1)ct that Z is an element of M in the third
t 2 t
step and the invariance of ρ φ in the fourth step. Since this holds for all states⊗ρSon M (C), we
2
conclude that M(C˜)=X. ⊗
As for the variance, we note first that the conditional expectation E is a completely positive
identity preserving map. Therefore, for all self-adjoint C , we haveSt
t
∈C
2
E (I C2) E (I C)
St ⊗ ≥ St ⊗
(cid:16) (cid:17)
by the Cauchy-Schwarz inequality for completely positive maps [23].
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We can now apply the same strategy as before. For all states ρ on M (C) we have
2
ρ M(C2) = ρ φ(Z I C2Z )
⊗ t∗ ⊗ t
(cid:0) (cid:1) = ρ φ E Z I C2Z
⊗ St t∗ ⊗ t
= ρ φ(cid:16)Z E(cid:0) (I C2)Z(cid:1)(cid:17)
⊗ t∗ St ⊗ t
(cid:16) 2(cid:17)
ρ φ Z E (I C) Z
≥ ⊗ t∗ St ⊗ t
= ρ M((cid:16)C˜2)(cid:16). (cid:17) (cid:17)
ThusM(C2) M(C˜2),andinparticularσ2 (cid:0)= M(C(cid:1)2) M(C)2 M(C˜2) M(C˜)2 =σ2.
≥ C k − k≥k − k C˜
This has a very useful consequence: if we are looking for pointers that record, say σ or σ in
x z
an optimal fashion, then it suffices to examine only pointers in . Instead of sifting through the
t
S
collection of all random variables on the measurement outcomes, we are thus allowed to confine
the scope of our search to the rather transparent collection of measurable functions of Y . In the
t
following, we will look at such pointers h (Y ). We will usually drop the subscript t on h to make
t t
the notation lighter.
4.2 Distribution of Y
t
At this point we are interested in the probability distribution of the random variable Y . Its
t
characteristic function [4] is given by
E(k):=Eρt exp(−ikYt) =ρ⊗φ Ut∗I ⊗exp(−ikYt)Ut =ρ Fexp(−ikX)(I) ,
(cid:2) (cid:3) (cid:16) (cid:17) (cid:0) (cid:1)
so that we need only calculate Fexp( ikX)(I). For notational convenience we will replace the
subscript exp( ikX) by k in the follo−wing. Using Lemma 3.4, we find the following system of
−
matrix valued differential equations:
dF (I) k2e t
dkt = −ike−2tFk(σ−+σ+)− 2− Fk(I),
dF (σ +σ ) k2e t
k −dt + = −21Fk(σ−+σ+)−2ike−2tFk(σ+σ−)− 2− Fk(σ−+σ+),
dF (σ σ ) k2e t
k dt+ − = −Fk(σ+σ−)− 2− Fk(σ+σ−),
with initially F (I,0)=I, F (σ +σ ,0)=σ +σ , F (σ σ ,0)=σ σ .
k k + + k + +
− − − −
Solving this system leads to
Fk(I)=e−k2(1−2e−t) I ik 1 e−t σ +σ+ k2 1 e−t 2σ+σ .
− − − − − −!
(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)
We define the Fourier transform to be (f)(x) := 1 ∞ f(k)eikxdk. Then the probability
F √2π
−∞
densityofYtwithrespecttotheLebesguemeasureisgivenbRy √12πF(E)(x)= √12πρ F Fk(I) (x) .
(cid:16) (cid:0) (cid:1) (cid:17)
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