The canonical Naimark extension for the Pauli quantum roulette wheel

0.35 MB·English

by  Carlo Sparaciari

ThecanonicalNaimarkextensionforgeneralizedmeasurementsinvolvingsetsofPauliquantum
observableschosenatrandom
Carlo Sparaciari
Dipartimento di Fisica, Universita` degli Studi di Milano, I-20133 Milano, Italy
Matteo G. A. Paris∗
Dipartimento di Fisica, Universita` degli Studi di Milano, I-20133 Milano, Italy and
CNISM, UdR Milano, I-20133 Milano, Italy
(Dated:January21,2013)
WeaddressmeasurementschemeswherecertainobservablesX arechosenatrandomwithinasetofnon-
k
degenerate isospectral observables and then measured on repeated preparations of a physical system. Each
(cid:80)
observablehasaprobabilityz tobemeasured, with z = 1, andthestatisticsofthisgeneralizedmea-
k k k
surementisdescribedbyapositiveoperator-valuedmeasure(POVM).Thiskindofschemesarereferredtoas
3
quantum roulettes since each observable X is chosen at random, e.g. according to the fluctuating value of
1 k
an external parameter. Here we focus on quantum roulettes for qubits involving the measurements of Pauli
0
matricesandweexplicitlyevaluatetheircanonicalNaimarkextensions, i.e. theirimplementationasindirect
2
measurementsinvolvinganinteractionschemewithaprobesystem. Wethusprovideaconcretemodeltoreal-

n izetheroulettewithoutdestroyingthesignalstate,whichcanbemeasuredagainafterthemeasurement,orcan
a betransmitted.Finally,weapplyourresultstothedescriptionofStern-Gerlach-likeexperimentsonatwo-level
J
system.

8
1 PACSnumbers:03.65.Ta,03.67.-a

]
h
I. INTRODUCTION involving an independent preparation of an ancillary (probe)
p
system [6], an interaction of the probe with the system un-
-
t der investigation, and a final step where only the probe is
n In this paper we deal with a specific class of general-
subjected to a (projective) measurement [7, 8]. A question
a ized quantum measurements usually referred to as Quan-
u thus arises on the canonical implementation of the quantum
tum Roulettes. These quantum measurements are achieved
q roulette’ POVM, and on the resources needed to realize the
through the following procedure. Consider K projective
[ corresponding interaction scheme. This is the main point of
measurements, described by the set {X } of non-
k k=1,...,K this paper. In particular, we focus on quantum roulettes in-
3 degenerateisospectralobservablesinaHilbertspaceH. The
v system is sent to a detector which, at random, performs the volvingthemeasurementsofPaulimatricesonaqubitsystem
8 andexplicitlyevaluatetheircanonicalNaimarkextensions.
measurement of the observable X . Each observable has a
4 probabilityz tobemeasured,withk(cid:80) z =1. Thisscheme WeremindthathavingtheNaimarkextensionofageneral-
0 k k k
isreferredtoasquantumroulettesincethemeasuredobserv- izedmeasurementis,ingeneral,highlydesirable,sinceitpro-
0
1. able Xk is chosen at random, e.g. according to the fluctu- videsaconcretemodeltorealizeanapparatuswhichperforms
atingvalueofaphysicalparameter,asithappensfortheout- the measurement without destroying the state of the system
1
comeofaroulettewheel.Thegeneralizedobservableactually underinvestigation. Thereby,thestateafterthemeasurement
2
1 measuredbythedetectorisdescribedbyapositiveoperator- canbemeasuredagain,orcanbetransmitted,andthetradeoff
: valuedmeasure(POVM),whichprovidestheprobabilitydis- betweeninformationgainandmeasurementdisturbancemay
v
tributionoftheoutcomesandthepost-measurementstates[1– be evaluated [9–12]. Alternatively, the scheme may serve to
i
X 3]. performindirectquantumcontrol,[13].
r As a matter of fact, any POVM on a given Hilbert space It should be emphasized that the concept of quantum
a
maybeimplementedasaprojectivemeasurementinalarger roulette provides a natural framework to describe measure-
one,e.g. see[4]forsingle-photonqudits. Thismeasurement ment scheme where the measured observable depends on an
scheme is usually referred to as a Naimark extension of the externalparameter,whichcannotbefullycontrolledandfluc-
POVM. Indeed, it is quite straightforward to find a Naimark tuatesaccordingtoagivenprobabilitydistribution. Apromi-
extensionforthePOVMofanyquantumrouletteintermsofa nentexampleisgivenbytheStern-Gerlachapparatus,which
jointmeasurementperformedon thesystemunderinvestiga- allows one to measure a spin component of a particle in the
tionandanancillaryone. direction individuated by an inhomogeneous magnetic field
Ontheotherhand,foranyPOVMtheNaimarktheorem[5] [14,15]. Indeed,wheneverthefieldisfluctuating,ortheun-
ensures the existence of a canonical Naimark extension, i.e. certainty in the splitting force is taken into account [16], the
theimplementationofthePOVMasanindirectmeasurement measurementschemeisdescribedbyaquantumroulette.Also
in this case, if we have the canonical Naimark extension for
the roulette, then we have a concrete way to realize a mea-
surement without destroying the state [17]. We remind that
∗Electronicaddress:matteo.paris@fisica.unimi.it in the continuous variable regime quantum roulettes involv-
2
ing homodyne detection with randomized phase of the local whichstatesthatageneralizedmeasurementinaHilbertspace
oscillatorhasbeenalreadystudiedtheoretically[18]andreal- H maybealwaysseenasanindirectmeasurementinalarger
A
izedexperimentally[19]. Hilbert space given by the tensor product H ⊗ H . This
A B
The paper is structured as follows. In the next Section indirect measure is known as canonical Naimark extension
we introduce notation, briefly review the Naimark theorem, for the generalized measurement. Conversely, when we per-
and gather all the necessary tools, e.g. the Cartan decompo- form a projective measure on the subsystem H of a com-
B
sition of SU(4) transformations, which allows us to greatly positesystemH ⊗H ,thedegreesoffreedomofH may
A B B
reduce the number of parameters involved in the problem of betracedoutandweobtainthesameprobabilitydistribution
finding the canonical Naimark extension. In Section III we oftheoutcomesoftheprojectivemeasurementandthesame
introduce the concept of quantum roulette, derive the corre- post-measurement states performing a generalized measure-
spondingPOVM,andillustrateanexampleofnon-canonical mentonthesubsystemH .
A
Naimark extension. In Section IV we derive the canonical
extensionforone-parameterPauliquantumroulettesanddis-
cvuostesddetotaailnsaolyfztheeSirteimrnp-Gleemrleanctha-tliiokne,wexhpeerreiamseSnetcstaiosnqVuainstud|ρmeA-ᐳ Πx |ρAxᐳΠx |ρAᐳ |ρAᐳU |ρAxᐳ |ρAxᐳ
rnoeutilcetfiteesl,di.ies.rtaankdinogmilnytofluacctcuoautinntgt.heFipnoaslslyib,iSlietcytitohnatVthIecmloasges-(a) |ρAᐳ(a) ||ρωABxᐳᐳᐸ|ρωBᐳB| (|bω)Bᐳᐸ|ρωBᐳB| UPxPx PxP
x
(b)
thepaperwithsomeconcludingremarks.
FIG.1: ThetwomeasurementschemeslinkedbytheNaimarkthe-
orem. (a): a generalized measurement described by the POVM
Π = M†M ; (b): its canonical Naimark extension, defined by
II. NOTATIONANDTOOLS x x x
thetriple{ρ ,U,{P }}, describingtheprobestateρ , theevolu-
B x B
tionoperatorU andtheprojectivemeasurement{P }ontheprobe
x
A. POVMsandtheNaimarktheorem system,respectively.
Whenwemeasureanobservableonaquantumsystem,we The Naimark theorem gives a practical recipe to evaluate
can not predict which outcome we will obtain in each run. the canonical extension for a generalized measurement in a
Whatweknowisthespectrumofpossibleoutcomesandtheir HilbertspaceH :
A
probability distribution. Given a system described by a state
ρintheHilbertspaceH,toobtaintheprobabilitydistribution Π =Tr (cid:2)I⊗ρ U† I⊗P U(cid:3) (1)
x B B x
oftheoutcomesxweusetheBornRule:
where ρ ∈ L(H ) describes the state of the probe system
B B
px =Tr[ρΠx] (or ancilla), the operators {Px} ∈ L(HB) are a set of pro-
jectorswhichdescribethemeasurementontheancillaandthe
Inordertosatisfythepropertiesofthedistributionp ,theop-
x unitaryoperatorU ∈L(H ⊗H )worksonboththesystem
A B
erators Π do not need to be projectors. The operators Π
x x andtheancilla. AcanonicalNaimarkextensionforthegener-
have to be positive, Π ≥ 0, since the probability distribu-
x alizedmeasurementgivenbytheoperators{Π }∈L(H )is
x A
tionp hastobepositiveforevery|ϕ(cid:105)∈H,andnormalized,
x thusindividuatedbythetriple{ρ ,U,{P }}.
(cid:80) Π =I,sincep isnormalized.Adecompositionofiden- B x
x x x EvaluatingthecanonicalNaimarkextensionforageneral-
tity by positive operators Π will be referred to as a positive
x izedmeasurementisdesirablesinceitgivesaconcretemodel
operator-valued measure (POVM), and the operators Π are
x torealizeanapparatuswhichperformsthemeasurementwith-
theelementsofthePOVM.
outdestroyingthestate.Then,thepost-measurementstatecan
We use Π to get information about the probability dis-
x betransmitted,ormeasuredagain.
tribution p , but if we are interested in post-measurement
x
states we have to introduce the set of operators M , the de-
x
tectionoperators.Theseoperatorsshouldgivethesameprob-
B. TheCartandecompositionofSU(4)transformations
ability distribution given by Π , thus they are obtained from
x
p = Tr(cid:2)M ρM†(cid:3) = Tr[ρΠ ]. Therefore, detection op-
x x x x √
erators that satisfy Π = M M† are M = U Π , with
x x x x x x
U a unitary operator such that U U† = I, and this leaves a
x x x
residual freedom on the post-measurement states. The post- S R
1 1
measurementstatesarethengivenby:
V
1
ρ = M ρM†
x p x x S R
x 0 0
A measurement described by the operators Π is referred to
x
asgeneralizedmeasurement.
In order to link general measurements with physical FIG. 2: The Cartan decomposition of the operator X ∈ SU(4),
givenby(R ⊗R )V(S ⊗S ).
schemes of measurement, we have the Naimark theorem, 1 0 1 0
3
Inthefollowingwearegoingtodealwithtwo-qubitinter- Intermsoftheαparameterstheboundedregioncorrespond-
actions, i.e. unitary operators (with unit determinant) of the ingtothecanonicalclassvectorsisgivenby
group SU(4), which are individuated by 15 parameters. In
−π ≤α ≤0
ordertoreducethenumberoftheseparameterswewillmake 1
use of the Cartan decomposition, which allows us to factor 0≤α2 ≤−α1
a general operator in SU(4) into local operators working on α +α ≤2α ≤0
1 2 3
single qubits and a single two-qubit operator V individuated
ifα =0thenα −α ≥−π
3 1 2
by3parameters[20–22],see(Fig. 2).
According to the Cartan decomposition any X ∈ SU(4) As we will see in the next section, the Cartan decomposi-
can be rewritten as X = (R ⊗ R )V(S ⊗ S ), where tionsimplifiestheproblemoffindingthecanonicalNaimark
1 0 1 0
R ,R ,S ,S ∈ SU(2) and V = exp{i(cid:80)3 k σ ⊗σ }, extension for the quantum roulette since we will be able to
1 0 1 0 j=1 j j j neglectsingle-qubitoperationsandthusreducingthethe15-
with k ≡ (k ,k ,k ) ∈ R3. The operators σ are the Pauli
1 2 3 i parameters operator U to the 3-parameters operator V. Fur-
matrices. If we introduce the following equivalence relation
thermore,wewillrestricttheintervaloftheparametersofV
inSU(4)
totheboundregiondefinedabove.
A∼B if A=(R ⊗R )B(S ⊗S ) (2)
1 0 1 0
III. THEPAULIQUANTUMROULETTEWHEEL
we can split the whole space of unitary operators with unit
determinantintoequivalenceclasses. Then,sinceanoperator
LetusconsiderKobservables{X }inaHilbertspaceH
X ∈ SU(4) is represented (by the equivalent relation given k A
withdimensiond = dim(H ). Alltheobservablesarenon-
here) by the matrix V, it is possible to establish a link be- A
degenerate and isospectral. Since the observables are non-
tweenoperatorsinSU(4)andrealvectors: X ∼ k,wherek
degenerateandtheHilbertspaceisfinite-dimensional,eachof
istheclassvectorofX. Thefollowingoperationsareclass-
them has d eigenvalues. We use a detector which chooses at
preserving:
randomoneoftheseobservablesandperformsameasurement
Shift: kcanbeshiftedby±π alongoneofitscomponents. ofthatobservable.Eachobservablehasaprobabilityzkofbe-
2 ingselectedbythedetectorand(cid:80) z =1. Thisscheme,de-
k k
Reverse: thesignoftwocomponentsofkcanbereversed. noted by the K-tuple {{X1,z1},{X2,z2},...,{XK,zK}},
isreferredtoasQuantumRoulette.
Swap: twocomponentsofkcanbeswapped. If we have a system represented by the state ρ ∈ L(HA)
and we send it to the detector, the probability distribution of
Bytheuseoftheseoperationsitisalwayspossibletoreduce theoutcomesisgivenby
anykintoaboundedregionK givenby: (cid:88) (cid:88) (cid:104) (cid:105)
p = z p(k) = z Tr ρP(k) (4)
x k x k x
1. π2 >k1 ≥k2 ≥k3 ≥0 k(cid:34) k (cid:35)
(cid:88)
2. k1+k2 ≤ π2 =Tr ρ zkPx(k) =Tr[ρΠx]
k
3. If k =0, thenk ≤ π
3 1 4 where p(k) is the probability distribution of the outcome x
x
Thek∈K arereferredtoascanonicalclassvectors. for the observable Xk and Px(k) = |x(cid:105)(k)(k)(cid:104)x| is the 1-
TheexpressionoftheoperatorVmaybesimplifiedusinga dimensional projector on the eigenspace of the eigenvalue x
differentsetofparameters,e.g. for the observable Xk. In the last equality of the Eq. (4) we
have introduced the POVM of the roulette, whose elements
k =−α1−α2 aregivenby
1 4 (cid:88)
k =−α1+α2 Πx = zkPx(k) (5)
2 4 k
α
k =− 3 The Πx’s are positive operators, indeed, given any |ϕ(cid:105) ∈
3 2 H , we have (cid:104)ϕ|Π |ϕ(cid:105) = (cid:80) z |(cid:104)ϕ|x(cid:105)(k)|2 ≥ 0, and they
A x k k
representadecompositionofidentity,since
Besides, we introduce the operators Σ = 1σ ⊗σ , which
i 2 i i (cid:88) (cid:88)(cid:88) (cid:88) (cid:88) (cid:88)
are normalized in the space of 4×4 operators, with the in- Π = z P(k) = z P(k) = z I=I
x k x k x k
ner product (cid:104)A,B(cid:105) = Tr[B†A]. Eventually, we obtain the x x k k x k
followingmatrixV:
On the other hand, Π are not orthogonal projectors since
x
Π Π (cid:54)=δ Π . Indeed:
(cid:26) (cid:20)1 1 (cid:21)(cid:27) x x(cid:48) xx(cid:48) x
V =exp −i 2(α1−α2)Σ1+ 2(α1+α2)Σ2+α3Σ3 Π Π =(cid:88)z z P(k)P(k(cid:48)) (cid:54)=0
x x(cid:48) k k(cid:48) x x(cid:48)
(3)
k,k(cid:48)
4
In fact, while (k)(cid:104)x|x(cid:48)(cid:105)(k) has to be equal to δ for a fixed overall state of the composite system. The operator U ∈
xx(cid:48)
value of k, the quantity (k)(cid:104)x|x(cid:48)(cid:105)(k(cid:48)) (with k (cid:54)= k(cid:48)) could be SU(4),thenitisdefinedby15parameters. Therefore,theto-
differentfromzeroalsowhenx(cid:54)=x(cid:48). tal number of parameters that defines the Naimark extension
is 19 (4 parameters from |ω (cid:105) and |x(cid:105) plus 15 from U). As
B
wewillseethisnumbercanbegreatlyreducedbyemploying
A. Thenon-canonicalNaimarkextension theCartandecomposition.
The quantum roulette has a Naimark extension that is not
A. ApplicationoftheCartandecompositiontotheNaimark
the canonical one (i.e. the indirect measurement scheme de-
extension
scribedbytheNaimarktheorem)thatcanbeobtainedasfol-
low: consideranadditionalHilbertspaceH ,describingthe
B
ancilla,withdimensionequaltothenumberKofobservables The Naimark theorem provides a practical connection be-
X ∈ L(H ),andabasis{|θ (cid:105)}inH . Thenweintroduce tween the generalized measurement given by the quantum
k A k B
projectors Q = (cid:80) P(k) ⊗ |θ (cid:105)(cid:104)θ | in the larger Hilbert rouletteandtheindirectmeasurementdescribedbytheexten-
x k x k k sion. Indeedboththeprobabilitydistributionoftheoutcomes
spaceH ⊗H andwepreparetheinitialstateoftheancilla
in|ω (cid:105)=A (cid:80) B√z |θ (cid:105),obtainingtheprobabilitydistribution px andthepost-measurementstatesρAx havetobeequalfor
B k k k thesetwoschemes. Thatis:
p =Tr [ρ⊗|ω (cid:105)(cid:104)ω |Q ]
x AB B B x p =Tr [Uρ ⊗ρ U†I⊗P ]=Tr [ρ Π ] (6)
x AB A B x A A x
thatgivesusthePOVM’selements
Πx =TrB[I⊗|ωB(cid:105)(cid:104)ωB|Qx]=(cid:88)zkPx(k) ρAx = p1 TrB[UρA⊗ρBU†I⊗Px]= p1 MxρAMx† (7)
x x
k
wherethedistributionp andthestateρ inthefirstequality
Moreover,thepost-measurementstateswillbegivenby x Ax
belong to the projective measurement while those in the last
1 equalitybelongtothegeneralizedmeasurement.
ρ = Tr [Q ρ⊗|ω (cid:105)(cid:104)ω |Q ]
x p B x B B x WefocusnowontheBornruleEq.(6)inordertoevaluate
x
1 (cid:88) theoperatorsΠx;afterstraightforwardcalculation,weobtain
= p zk Px(k)ρPx(k) theelementsofthePOVM:
x
k
Π =S† Tr [(I⊗S ρ S†)V†(I⊗R†P R )V]S
This measurement scheme does not involve an evolution op- x 1 B 0 B 0 0 x 0 1
eratorU ∈ L(HA ⊗HB)andtheprojectivemeasureisper- ConsidernowS ρ S† andR†P R ;theoperatorsR ,S
0 B 0 0 x 0 0 0
formed on both system and ancilla, unlike the canonical ex- ∈ L(H )representarotationinthequbitHilbertspaceH .
B B
tensionthatinvolvesaprojectivemeasurementonthesolean- Sincebothρ andP arenotyetdefinedanddependonsome
B x
cilla.
parameters, wecancombinetherotationtothemandweare
leftwithatransformationfromL(H )toL(H ):
B B
IV. THECANONICALNAIMARKEXTENSIONOFTHE ρ →ρ(cid:48) =S ρ S†
PAULIQUANTUMROULETTEWHEEL B B 0 B 0
We focus on quantum roulettes which work on qubit sys- Px →Px(cid:48) =R0†PxR0
tems with Hilbert space H ≡ C2 and address quantum
A
i.e. we can neglect this transformation by a suitable
roulettes involving the measurement of Pauli operators. In
reparametrization of ρ(cid:48) and P(cid:48). Furthermore, we assume
order to obtain the canonical Naimark extension for these B x
the operator S to be the identity (S = I). We make this
rouletteswehavetoaddaprobesystem(theancilla). Weas- 1 1
assumptioninordertosimplifytheresearchofthecanonical
sumeatwo-dimensionalancillaandshowthatthisisenough
extension. Thisansatzwillbejustifiedaposteriori: oncewe
to realize the canonical extension using Eq. (1), where the
find the Naimark extension, if the probability distribution p
elementsofthePOVMaregivenbyEq. (5). x
obtainedfromtheextensionisequaltotheoneobtainedfrom
SinceH isaHilbertspacewithdimensiontwo,wechoose
B thePOVM,thentheextensioniscorrectandS =I.
thefollowingrepresentationforthestateρ =|ω (cid:105)(cid:104)ω |and 1
B B B
for the projector P = |x(cid:105)(cid:104)x|, where |ω (cid:105) = cosθ|0(cid:105) + NowwehavePOVM’selementsobtainedbythecanonical
x B 2 extension:
eiϕsinθ|1(cid:105) and |x(cid:105) = cosα|0(cid:105)+eiβsinα|1(cid:105), with the pa-
2 2 2
rametersα,θ ∈ [0;π]andβ,ϕ ∈ [0;2π). Noticethat|0(cid:105),|1(cid:105) Π =Tr [(I⊗ρ )V†(I⊗P )V] (8)
x B B x
areabasisinH ;weassumethat|0(cid:105)istheeigenvectorofσ
B 3
relatedtotheeigenvalue1,while|1(cid:105)istheeigenvectorrelated andtheseelementshavetobeequaltothoseevaluatedforthe
to-1. quantum roulette in exam. Using the Cartan decomposition
The last tool to individuate the canonical extension is the on the canonical Naimark extension reduces the number of
evolution operator U ∈ L(H ⊗ H ) which works on the parameterto7(4fromρ andP and3fromV).
A B B x
5
B. DetectionoperatorsfortheQuantumRoulette operatorV into
V →(R ⊗R )V(cid:48)(S ⊗S )
TheoperatorR isnotinvolvedinthedefinitionoftheel- 1 0 1 0
1
ementsofthePOVM,butitisnecessaryfortheevaluationof
and the operators R ,S ∈ SU(2) modify both the initial
thepost-measurementstateρ . SincetheoperatorsR and 0 0
Ax 0
S wereabsorbed into, respectively, P and ρ and S = I, stateofHB andtheorthogonalprojector
0 x B 1
thenthedecompositionofU isU = (R ⊗I)V andtheleft
partofEq.(7)becomes: 1 ρB(θ(cid:48),ϕ(cid:48))=S0ρB(θ,ϕ)S0†
P (α(cid:48),β(cid:48))=R† P (α,β)R
1 x 0 x 0
ρ = Tr [(R ⊗I)Vρ ⊗ρ V†(R†⊗P )]
Ax px B 1 A B 1 x Hence, to transform the αi and keep the correct extension is
= 1 R Tr [Vρ ⊗ρ V†(I⊗P )]R† necessarytomodifyρB andPx.
p 1 B A B x 1
x
Therefore, the operator R describes a residual degree of
1 E. Thecanonicalextension
freedom in the design of possible post-measurement states.
This freedom was expected, since when we define a POVM
WenowfocusonquantumroulettesgivenbyPauliopera-
Π , the post-measurement states can be evaluated using the
x tors σ and on their canonical Naimark extension. The most
detectionoperatorsM .TheseoperatorsaredefinedasM = i
√ x x general quantum roulette of this kind is {σ ,z } and
U Π ,whereU isaunitaryoperator.TheoperatorU pro- i i i=1,2,3
x x x x itscanonicalextensiondependsontwoundefinedparameters
videsthesamefreedomgivenbyR tothepost-measurement
1 (e.g.z andz ). Findingtheextensionforthegeneralroulette
states. 1 2
is analytically challenging and thus we focus to roulettes in-
volvingtwoPaulioperators.
Let us consider the roulette {{σ ,z},{σ ,1−z}}, where
1 3
C. Thegeneralsolution
z getsvaluesfromtheinterval(0;1). ThePOVM’selements
aregiveby
The problem of finding the canonical Naimark extension
for a given quantum roulette is now basically reduced to the 1(cid:18)2−z z(cid:19) 1(cid:18) z −z (cid:19)
Π = Π = (9)
solution of four equations dependent on seven parameters. 1 2 z z −1 2 −z 2−z
Indeed, the considered roulettes are always in qubit spaces;
√
hence, the elements of the POVM Πx are self-adjoint 2×2 and the detection operators by M = U Π , x = ±1.
operatorsonthefieldCandtherelationgivenbyEq.(8)pro- Upon expanding them on the Paulixbasis, ix.e. Mx = a I+
x 0
videsfourequations: onefromtheelementΠx11 thatisreal, a σ + a σ + a σ , the coefficients a are evaluated us-
1 1 2 2 3 3 i
twofromtheelementΠx12(realpartandimaginarypart)and ing the inner product (cid:104)X,Y(cid:105) = Tr[XY†]. For the roulette
onefromtheelementΠx22. {{σ ,z},{σ ,1−z}},weobtaina =a (z)(fori=0,1,3)
1 3 i i
and a = 0. In other words, the detection operators of
2
a roulette involving the Pauli operators σ and σ have no
1 3
D. Exchangeoftheparameters component on the missing one, i.e. σ . This result also
2
holds for the other roulettes depending on two σ’s , e.g. for
One may wonder if it is possible to look for the canonical {{σ2,z},{σ3,1−z}}, the detection operators Mx have no
extensionwhentheparametersαi getvaluesfromallR. The componentbyσ1.
Cartandecompositiondoesnotimposerestrictionontherange Thesolutionforthecanonicalextensioncorrespondstothe
ofthecomponentsoftheclassvectors(thatis,theparameters parameters
α ), butweknowthateachoperatorinSU(4)isrelated(via
i (cid:115)
the equivalence relation Eq. (2)) to a canonical class vector, 1
whose components lie on the bounded region K ∈ R3. On α1 =−π; α2 =0; α3 =arcsin(− 1− z −1)
2
the other hand if we find an extension with α outside of K,
i π
it is possible to use the 3 class-preserving operations (shift, α=arccos(z−1); β =π; θ = ; ∀ϕ,
2
reverseandswap)tobringbacktheparameterstoK.
May the parameters be brought back to K after we have where α ,α ,θ,ϕ and β are in the correct range and we
1 2
found the canonical extension? This is not possible, unless are left to check whether also α and α lies in the correct
3
wealsomodifytheotherobjectsoftheextension. Indeed, if range. First of all, cosα = z − 1, i.e. cosα ∈ (−1;0);
we have found the extension, then we have defined both α (cid:113)
i then α ∈ (π;π). Finally, sinα = − 1 −1, that is
andθ,ϕ,α,β. Butiftheα aremodifiedbyoneofthethree 2 3 1−z
i 2
class-preservingoperations,thenalsotheotherparametersare sinα3 ∈ (−1;0); then α3 ∈ (−π2;0). The parameter α3
modified and the state |ωB(cid:105) and the operator Px change. In has to be in [α1+2α2;0], and since α1 = −π and α2 = 0, its
fact,theoperationsareclass-preserving,sotheytransformthe greatestrangeis[−π;0].
2
6
Theingredientsofthecanonicalextensionarethusthestate hand,wemayassumethatb (cid:28) B sothatwecanneglectthe
|ωB(cid:105)= √1 |0(cid:105)+ e√iϕ|1(cid:105),theprojectors othercomponentsofB. TheinteractionHamiltonianisgiven
2 2 byH =(B−bz)σ (neglectingthevacuumpermittivity)and
3
1(cid:18) 2−z (cid:112)z(2−z)(cid:19) the corresponding evolution operator by U = e−iτ(B−bz)σ3,
P1 = 2 (cid:112)z(2−z) z (10) whereτ isaneffectiveinteractiontime. Startingfromanini-
tialstatewhichisfactorizedintoaspinandaspatialpart,i.e.
P−1 =I−P1 |Ψ(cid:105)(cid:105)=(c0|0(cid:105)+c1|1(cid:105))⊗|ψ(q)(cid:105),theevolvedstateisgivenby
andtheunitary U|Ψ(cid:105)(cid:105)=c |0(cid:105)⊗|ψ (q)(cid:105)+c |1(cid:105)⊗|ψ (q)(cid:105)
0 − 1 +
f(z) 0 0 0  where |ψ (q)(cid:105) = e±iτ(B−bz)|ψ(q)(cid:105). The evolution is thus
±
0 0 if∗(z) 0 couplingthespinandthespatialpart. Movingtothemomen-
V =  (11)
 0 if∗(z) 0 0  tumrepresentation
0 0 0 f(z)
(cid:90)
(cid:114)(cid:113) |ψ(cid:101)±(p)(cid:105)= d3qe−iq·p|ψ±(q)(cid:105)
withf(z)= 2−2z + √i .
2−z z2−1 =|ψ(cid:101)(p±τbe3)(cid:105)
InordertoobtainthecanonicalNaimarkextensionforthe
roulettes {{σ ,z},{σ ,1−z}} and {{σ ,z},{σ ,1−z}}, and tracing out the spin part after the interaction, we have
1 2 2 3
wehavetoremoveourpreviousassumptionS = I. Infact, that the motional degree of freedom after the interaction is
1
to rotate a Pauli operator σ by an angle θ we have to use a describedbythedensityoperator
i
rotationoperatorW =e−i(n·σ)θ,wherenistheversorofthe
direction around which the rotation is made. Then, to move (cid:37)p =|c1|2|ψ(cid:101)+(p)(cid:105)(cid:104)ψ(cid:101)+(p)|+|c0|2|ψ(cid:101)−(p)(cid:105)(cid:104)ψ(cid:101)−(p)|
fromatwoPaulioperatorsroulettetoanother,weneedtoap-
plythecorrectrotationinordertomodifytheσ .Forexample, As a consequence the beam is divided in two parts, and it
i
tomovefrom{{σ ,z},{σ ,1−z}}to{{σ ,z},{σ ,1−z}} is possible to perform measurements on a screen placed at a
1 3 2 3
we have to apply the operator W = e−iπ4σ3, which changes givendistancefromthemagneticfield,wherewecanseethe
σ intoσ andleavesσ unchanged. beamsastwodifferentspots.
1 2 3
Therefore,theextensionsfortheotherroulettesdepending If for some reason the direction of the magnetic field is
ontwoPaulioperatorsaredefinedbythesameparametersof tilted we have B = (B − bt)eα, where t is a cohordinate
theextensionfor{{σ1,z},{σ3,1−z}},buttheelementsΠx along the new direction and eα = cosαe3 +sinαe⊥, e⊥
arerotatedbytheoperatorW,thatis: denoting any direction perperdicular to the z-axis, say e1.
The above analysis is still valid if we perform the substitu-
Π →WΠ W† tionH → (B −bt)σ whereσ isaPaulimatrixdescribing
x x θ θ
a sping component along a tilted axis. Assuming a rotation
Thismeansthat,whileforthefirstfoundextensiontheoper- along the x-axis we have that σ corresponds to the rotated
θ
ator S1 can be assumed equal to I, for the extensions of the operator
other roulettes the operator S has to be equal to the conju-
1
gate transpose of the rotation operator W. We find that, for σ =U σ U†
θ θ 3 θ
theroulettegivenbyσ2andσ3,S1 =eiπ4σ3 whilefortheone
givenbyσ1andσ2,S1 =e−iπ4σ1. whereUθ =e−iσ1θ andθ =α/2.
V. THESTERN-GERLACHAPPARATUSASAQUANTUM A. Thecontinuousquantumroulette
ROULETTE
Usually,themagneticfieldoftheapparatusisassumedtoa
The so called Stern-Gerlach apparatus allows one to mea- beastableclassicalquantity. However, inanypracticalsitu-
sureacomponent(e.g.thecomponentalongthez-axisS )of ationthemagneticfieldunavoidablyfluctuates. Inparticular,
z
thequantumobservablespin,i.e.theintrinsicangularmomen- we focus on Stern-Gerlach apparatuses where the magnetic
tum of a particle. The measurement is usually performed on fieldfluctuatesinonedimensionaroundapre-establisheddi-
a collimated beam of particles (e.g. neutral atoms) sent with rectionandprovideamoredetailedanalysisofnon-idealse-
thermalspeedintoaregionofinhomogeneousmagneticfield. tups [23, 24]. As mentioned above, a measurement of spin
Here the particles are deflected by the field in some beams madewithatiltedmagneticfieldcorrespondstomeasurethe
which, after propagating into the vacuum, are collected by a operatorσ . Ifthemagneticfieldisfluctuating,thenwemay
θ
screen. The magnetic field is usually assumed of the form describe this situation using a continuous quantum roulette
B=(B−bz)e ,wherezisthecohordinatealongthez-axis, where θ is randomly fluctuating around the z-axis according
3
Bisthefieldintheorigin,andbisaconstant.Actually,thisis toagivenprobabilitydistribution.
anartificialmodel,sinceafieldlikethisonedoesnotrespect Inprinciple,themagneticfieldmayfluctuateinanydirec-
the Maxwell equations, as ∇·B = −b (cid:54)= 0. On the other tiononthezy-plane,i.e. theangleθtakesvaluesbetween−π
2
7
and π. On the other hand, in a realistic situation, the mag- Let consider the parameters α and α ; the codomain of
2 1 2
neticfieldmovesawayfromthepre-establisheddirection(the thefunctionf(∆)is(0;1]. Therefore,ifcosα = −2f(∆),
2 1
z-axis,inthiscase)justbyasmallangle. Wethusintroducea thencosα ∈(0;−1]andα ∈(−π;−π]. Instead,cosα =
1 1 2 2
Gaussianprobabilitydistributionz(θ)forthefluctuatingval- 2f(∆),i.e.cosα ∈ (0;1]andα ∈ (π;0]. Bothα andα
2 2 2 1 2
uesofθ dependonthefunctionf,sowhenwechooseavalueforf the
twoparametersarefixed. Forexamplewhenf(∆)→0,then
z(θ)= 1 exp{− θ2 } α1 →−π2 andα2 → π2;ontheotherhand,iff(∆)= 12 then
A 2∆2 α = −π and α = 0. The ranges of these two parameters
1 2
arecorrectandwehaveα ≤−α ∀f(∆). Finally,sincewe
wherethenormalizationAis: 2 1
havefixedα =0,wehavetocheckwhetherα −α ≥−π,
3 1 2
(cid:90) π2 θ2 √ π and this is case: as it can be easily checked α1 −α2 = −π
A= exp{− }dθ = 2π∆Erf( √ ) for all ∆ ∈ [0;+∞). The canonical extension is thus given
−π2 2∆2 2 2∆ bythestate|ωB(cid:105) = |0(cid:105), theobservableσ3 (measuredonthe
ancilla),andtheunitaryV ∈L(H ⊗H )
In order to evaluate the elements of the POVM which de- A B
scribesthiscontinuousquantumroulette,weneedtheprojec-
torsontheeigenspacesofσθ,i.e. 1(cid:88)3
V = v σ ⊗σ (15)
(cid:18) cos2θ icosθsinθ(cid:19) 4 k k k
P (θ)= (12) k=0
1 −icosθsinθ sin2θ (cid:114) (cid:114)
1 1
P−1(θ)=I−P1(θ) v0/3 = 2 +f(∆)± 2 −f(∆) (16)
∆→1
Therefore,theelementsofthePOVMaregivenby (cid:39) 1±∆
v =iv (17)
(cid:90) π 1/2 0/3
2
Π = z(θ)P (θ)dθ (13)
x x
−π For a particle with spin up, represented by the pure state
2
|0(cid:105), the probability distribution of the outcomes is given by
that is the equation equivalent to Eq. (5) in the continuous
p = 1 +f(∆),p =1−p andthus,whensuchaparticle
case. We can evaluate the elements of the POVM using the 1 2 −1 1
is measured, there is always a probability that the apparatus
distribution z(θ) and the projectors P (θ) given above, and
x measuresthespindown|1(cid:105).
weobtain:
(cid:18)1 +f(∆) 0 (cid:19)
Π = 2 (14)
1 0 1 −f(∆)
2
Π =I−Π VI. CONCLUSIONS
−1 1
wherethefunctionf(∆) : [0;+∞)→[1;0)isgivenby
2 WehaveaddressedPauliquantumroulettesandfoundtheir
canonical Naimark extensions. The extensions are minimal,
Erf(π−√i4∆2)+Erf(π+√i4∆2) i.etheyinvolveasingleancillaqubit,andprovideaconcrete
f(∆)= 2 2∆ 2 2∆ .
4e2∆2Erf( √π ) model to realize the roulettes without destroying the signal
2 2∆ state,whichcanbemeasuredagainafterthemeasurement,or
We have f(∆) (cid:39) 1/2 − ∆2 for vanishing ∆ and f(∆) (cid:39) can be transmitted. Our results provide a natural framework
1/8∆2for∆→∞. todescribemeasurementschemewherethemeasuredobserv-
abledependsonanexternalparameter,whichcannotbefully
controlledandmayfluctuateaccordingtoagivenprobability
distribution. As an illustrative example we have applied our
B. Thecanonicalextensionforthecontinuousroulette
resultstothedescriptionofStern-Gerlach-likeexperimentson
atwo-levelsystem,takingintoaccountpossibleuncertainties
Welookforthecanonicalextensionforthisrouletteinor-
inthesplittingforce.
dertoobtainapracticablemeasurementschemewiththesame
behavior(sameprobabilitydistributionandpost-measurement
states)oftheStern-Gerlachexperimentwithfluctuatingmag-
neticfield. Acanonicalextensionmaybefound,correspond-
ingtotheparameters Acknowledgments
α =arccos(−2f(∆)); α =arccos(2f(∆)); α =0
1 2 3 ThisworkhasbeensupportedbyMIURthroughtheproject
α=π; β =0; θ =0; ϕ=0
FIRB-RBFR10YQ3H-LiCHIS.
8
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