Joint measurements on qubits and cloning of observables

0.1 MB·English

by  Alessandro Ferraro

Jointmeasurements onqubits and cloning ofobservables
Alessandro Ferraro1,∗ and Matteo G. A. Paris2,†
1ICFO - Institut de Cie`ncies Foto`niques, E-08860Castelldefels (Barcelona), Spain
2Dipartimento di Fisica dell’Universita` di Milano, Italia
Cloningofobservables,unlikestandardcloningofstates,aimsatcopyingtheinformationencoded
inthestatisticsofaclassofobservablesratherthenonquantumstatesthemselves.Insuchaprocess
the emphasis is on the quantum operation (evolution plus measurement) necessary to retrieve the
originalinformation. We analyze, for qubitsystems, the cloningof a class generatedby two non-
commutingobservables,elucidatingtherelationshipbetweensuchaprocessandjointmeasurements.
7
Thishelpsinestablishinganoptimalitycriterionforcloningofobservables. Weseethat,evenifthe
0
cloningmachineisdesignedtoactonthewholeclassgeneratedbytwononcommutingobservables,
0
2 thesameoptimalperformancesofajointmeasurementcanbeattained. Finally,theconnectionwith
statedependentcloningisenlightened.
n
a
J

4
1
I. INTRODUCTION

1
v Itiswellknown that, in general, the act ofmeasuring aquantum system necessarily disturb the system
9
itself. A clear exemplification of this fact is given by joint (or simultaneous) measurements, i.e. measure-
8
0 mentsinwhichonetriestosimultaneously acquireinformationabouttwononcommutingobservables from
1
a single system. In this case, the possibility to sharply measure (i.e., to obtain measurement results not
0
affected by more noise than the one intrinsically present into the system) one observable is forbidden if
7
0 a sharp measure of the other one is performed. Nevertheless, information about both observables can be
/
h retrieved, paying the price of an extra noise in the statistics of the measurement results. An optimal joint
p measurementistheoneintroducingtheminimumextranoisecompatiblewiththelawsofquantummechan-
-
t ics. The first investigations in this issue can be traced back to the seminal paper by Arthurs and Kelly [1].
n
There, position andmomentum observables havebeen considered andtheminimumextranoisewasfound
a
u tobefourtimeslargerthantheintrinsicnoiserelatedtotheHeisenberguncertaintyrelation. Noticethattwo
q
points are crucial in their analysis: i) the equality between the jointly inferred mean values and the mean
:
v values obtained if the measurements were performed non simultaneously is imposed; ii) the dependence
i
X on an a priori knowledge of the initial ensemble is needed, in order to adjust the relative sharpness of the
r two measurements. The usual implementation of an Arthurs and Kelly measurement concerns the field of
a
quantum optics, in which two incompatible quadratures of the field are considered. Typically, an hetero-
dyne detection scheme (or equivalently adouble homodyne detection) implements the joint measurement,
howeveralsoadifferentstrategycanbedevised,inthecaseoffieldsdescribedbyacoherentstate. Onecan
infactfirstgeneratetwo(imperfect)copiesoftheoriginalstate,viaanoptimalcloningprocess[2],andthen
measurethetwoquadraturesoneforeachclones. Boththestrategiesperformanoptimaljointmeasurement.
Thisfact, together withthequite general relationship between cloning andmeasurement [3],mightlead to
think that optimal cloning can be associated to optimal joint measurements. However, this is not the case
ingeneral. Asan example, consider aqubit system and twoPauli observables, for which again an optimal
joint measurement involves an extra noise four times larger then the intrinsic one [4, 5]. An experimental
implementation ofsuch ajoint measurement isreported inRef.[6],using anentangled two-photon state as
information carrier from thevery beginning. Then, itiseasy todemonstrate that optimal universal cloning
for qubits [7] does not allows to perform an optimal joint measurements of the two Pauli observables (see
∗Electronicaddress:[email protected]
†Electronicaddress:[email protected]
2
below). This is due to the fact that cloning machines are in general designed such to optimize the fidelity
between input and output states, not such to optimize the joint measurement of a specific couple of non-
commuting observables. It is the latter viewpoint that we will instead adopt in this work. More precisely,
we will actually consider a further step, which can be understood in the spirit of state dependent cloning
[8]. Recall that the latter cloning strategy, unlike the universal case, is specifically designed to act only on
aparticular class out of the whole Hilbert space of the system (e.g., equatorial qubits). Following an anal-
ogous strategy, we will design our process (from now on called cloning of observables) in order to clone
and possibly jointly measure neither allpossible observables nor twoobservables only, but rather the class
generated by any linear combination of two observables. More in details, a cloning machine for the given
setofobservablesisadeviceinwhichasignalqubitinteractswithaprobequbitviaagivenunitarywiththe
aim of reproducing the statistics of each observable on both the qubits at the output, independently of the
signal qubit. Theoptimality ofourcloner willbeassessed byinvestigating theoptimality ofthejoint mea-
surement(ofthetwoobservables generating theclass), buttheadditional requirement ofcloning thewhole
class of observables poses a clear distinction between a joint measurement and a cloning of observables.
Furthermore, unlike a joint measurement, our protocol is not focused only on the measurement process,
which can actually not be performed. Indeed, our approach is motivated, besides by the exploitation of a
newaspectofthefundamentalissueofquantumcloning,alsoinviewofquantumcommunicationpurposes,
inarepeater-likeconfiguration. Morespecifically,onecanaddressthetransmissionofinformationencoded,
rather than in a set of states, in the statistics of a set of observables, independently of the quantum state at
the input. In the following we will consider set of observables generated by two Pauli operators, whereas
formoregeneral classesofobservables werefertoRef.[9].
The paper is organized as follows. In Sec. II we describe the general concept of cloning machine for
observables, introducing some basic properties. We point out in Sec. III the impossibility of designing a
perfect cloning machines for a class of two noncommuting observables, analogously to the impossibility
ofperfectly cloning twononorthogonal states. Then, anapproximate cloning machine forthiscase willbe
introduced. InSec.IVweassesstheoptimalityofsuchacloningmachineandcompareittoajointmeasure-
mentprocess. Wewillseethat, evenifthecloning machineisdesigned toactonthewholeclassgenerated
by two noncommuting observables, the same performances of a joint measurement can be attained, when
one restricts the attention to the only two observables generating the class. Sec. V closes the paper with
someremarks.
II. CLONINGOFOBSERVABLES
We consider a device in which a signal qubit (say, qubit ”1”) is prepared in the (unknown) state ̺ and
then interacts, via a given unitary U, with a probe qubit (”2”) prepared in the known state ̺ . For a given
p
class of qubit observables X X(j) where is a subset of the real axis and X(j) [C2], we
≡ { }j∈J J ∈ L
introducetheconceptofcloningasfollows[9]. AcloningmachinefortheclassofobservablesXisatriple
(U,̺ ,X)suchthat
p
X = X X = X ̺ X X, (1)
1 2
∀ ∀ ∈
whereX Tr [̺X]isthemeanvalueoftheobservable X attheinputand
1
≡
X Tr [RX I] X Tr [RI X] (2)
1 12 2 12
≡ ⊗ ≡ ⊗
are the mean values of the same observable on the two output qubits (see Fig. 1). The density matrix
R = U ̺ ̺ U† describes the(generallyentangled) stateofthetwoqubitsaftertheinteraction, whereasI
p
⊗
denotestheidentity operator.
Theabove definition identifies the cloning of an observable with the cloning of its mean value. This is
justified by the fact that for any single-qubit observable X the cloning of the mean value is equivalent to
3
X X
%
U
%p
X
FIG.1: Schematicdiagramofacloningmachineforobservables(U,̺ ,X): asignalqubitpreparedintheunknown
p
state̺interacts,viaagivenunitaryU, withaprobequbitpreparedintheknownstate̺ . Theclassofobservables
p
XisclonedifameasurementofanyX Xoneitherthetwoqubitsattheoutputgivesthesamestatisticsasitwas
∈
measuredontheinputsignalqubit,independentlyontheinitialqubitpreparation̺.
the cloning of the whole statistics. In fact, any X [C2] has at most two distinct eigenvalues λ ,λ ,
0 1
∈ L { }
occurring with probability p ,p . For a degenerate eigenvalue the statement is trivial. For two distinct
0 1
eigenvalueswehaveX= λ p +λ p which,togetherwiththenormalizationcondition1 = p +p ,proves
1 1 0 0 0 1
thestatement. Inother words, wesay thatthe class ofobservables X hasbeen cloned ifameasurement of
any X X on either the two qubits at the output gives the same statistics as it was measured on the input
∈
signalqubit,independently ontheinitialqubitpreparation.
Aremark about this choice isin order. In fact, in view of the duality among states and operators on an
Hilbertspace,onemayarguethataproperfigureofmerittoassessacloningmachineforobservableswould
be a fidelity–like one. This is certainly true for the d-dimensional case, d > 2, while for qubit systems a
properassessmentcanbealsomadeintermofmean–valueduplication, whichsubsumesalltheinformation
carried bythesignal. Furthermore, inview ofthe possibility torealize ajoint measurement viaacloner of
observables, werecall that theEq. Eq.(1)isastandard requirement insuchascenario, asalready recalled
inSec.I
Beforebeginningouranalysisletusillustrateabasicpropertyofcloningmachines,whichfollowsfrom
the definition, and which will be used throughout the paper. Given a cloning machine (U,̺ ,X), then
p
(V,̺ ,Y) is a cloning machine too, where V = (W† W†) U (W I) and the class Y = W†XW
p
⊗ ⊗
is formed by the observables Y(j) = W†X(j)W, j . The transformation W may be a generic
∈ J
unitary. We will refer to this property to as unitary covariance of cloning machine. The proof proceeds
as follows. By definition Y(j) = Tr [̺ W†X(j)W] = Tr [W ̺W†X(j)]. Then, since (U,̺ ,X) is a
1 1 p
cloningmachine, wehave
Y(j) = Tr [U (W̺W† ̺ )U†(X(j) I)]
1,2 p
⊗ ⊗
= Tr [U (W I)(̺ ̺ )(W† I)U† (W W)(Y(j) I)(W† W†)]
1,2 p
⊗ ⊗ ⊗ ⊗ ⊗ ⊗
= Tr [V (̺ ̺ )V†(Y(j) I)] = Y (j) . (3)
1,2 p 1
⊗ ⊗
ThesameargumentholdsforY (j)[11]
2
Anotherresultwhichwillbeusedinthefollowingistheparameterizationofatwo-qubittransformation,
which corresponds to a SU(4) matrix, obtained by separating its local and entangling parts. A generic
two-qubitgateSU(4)matrixmaybefactorized asfollows[10]:
3
i
U = L U L = L exp θ σ σ L (4)
2 E 1 2 h2X j j ⊗ ji 1
j=1
where θ R and the σ ’s are the Pauli’s matrices. The local transformations L and L belongs to
j j 1 2
∈
the SU(2) SU(2) group, whereas U accounts for the entangling part of the transformation U. In our
E
⊗
4
context, decomposition (4), together with unitary covariance of cloning machines, allows to ignore the
local transformations L , which corresponds to a different state preparation of signal and probe qubits at
1
the input. On the other hand, as we will see in the following, the degree of freedom offered by the local
transformations L willbeexploited todesignsuitable cloningmachinesfornoncommuting observables.
2
III. NONCOMMUTINGOBSERVABLES
In order to introduce cloning machines for a class of noncommuting observables let us consider the
specific class Xnc = {x1σ1 + x2σ2}x1,x2∈R. If a cloning machine (U,̺p,Xnc) existed, then the mean
values as well as the statistics of any observable belonging to X would be cloned at its output. As a
nc
consequence, one would jointly measure any two non-commuting observables belonging to X (e.g., σ
nc 1
ontheoutputsignalandσ ontheoutputprobe)withoutanyaddednoise,thusviolatingtheboundsimposed
2
byquantum mechanics [1,4,5]. Generalizing thisargument toanytwo–parameter class ofnoncommuting
observables (i.e., toany class Xgnc = cC+dD c,d∈R, with C, D generic non-commuting observables),
{ }
wethen conclude that acloning machine for ageneric two–parameter class of noncommuting observables
doesnotexist[9]. Thisresultsresemblestheno–cloningtheoremforstates,anditcanactuallybeseenasits
counterpart for cloning of observables. Furthermore, it permits acomparison among cloning machines for
observablesandforstates. Letuswritethegenericinputsignalas̺= 1(σ +s σ),whereσ = (σ ,σ ,σ ),
2 0 · 1 2 3
s = (s ,s ,s ) is the Bloch vector and σ is the identity operator. The state–cloning counterpart of the
1 2 3 0
statement above can then be obtained by considering the class of observables X : If a cloning machine
nc
(U,̺ ,X ) existed, then the components s and s of Bloch vector s would be cloned for any input
p nc 1 2
signal. The same situation occurs in the case of a two–parameter class generated by any pair of Pauli
operators. In other words, it is not possible to simultaneously copy a pair of components of the Bloch
vector of a generic state, even completely disregarding the third one [12, 13]. Let us stress the fact that,
evenifaperfectcloningofobservablescouldbeperformed(forexample,inthecaseofaclassgeneratedby
commutingobservables[9]),correlationsbetweenthemeasurementswouldexist,otherwisestateestimation
limitswouldbeovercame.
Now, a question arises on whether, analogously to state–cloning, we may introduce the concept of
approximate cloning machines, i.e. cloning ofobservables involving added noise. Indeed this can be done
andoptimalapproximate cloningmachinescorresponding tominimumaddednoisemaybefoundaswell.
An approximate cloning machine for the class of observables X is defined as the triple (U,̺ ,X)
p apx
suchthatX = X/g andX = X/g i.e.
1 1 2 2
Tr [̺X] = g Tr [RX I] = g Tr [RI X] , (5)
1 1 12 2 12
⊗ ⊗
for any X X. The quantities g , j = 1,2 are independent on the input state and are referred to as the
j
∈
noisesaddedbythecloning process.
Letusbeginbyagainconsidering theclassXnc = {x1σ1+x2σ2}x1,x2∈R. Byusingthedecomposition
ofageneric SU(4)matrix inEq.(4)onemayattempt tofindan approximate cloning machine considering
onlytheactionoftheentanglingkernelU . Unfortunately,itcanbeshownthatnoU ,g andg existwhich
E E 1 2
realize approximate cloning for ̺ = 0 0. A further single-qubit transformation should be introduced
p
| ih |
afterU . Inparticular, theunitaryF = i/√2(σ +σ )flipsthePaulimatricesσ andσ (i.e.,F†σ F =
E 1 2 1 2 1,2
σ )andpermitstherealization ofanapproximate cloning machine. Indeed,theunitary
2,1
T = (I F)Unc Unc = eiθ2(σ1⊗σ1−σ2⊗σ2),
⊗
realizestheapproximate cloning machine(T, 0 0,X ) withaddednoises
nc apx
| ih |
1 1
g = g = . (6)
1 2
cosθ sinθ
5
Inordertoprovethecloningproperties ofT letusstartfromtheunitary (I F)U ,whereU isageneric
E E
⊗
entangling unitary oftheformgiveninEq. (4). Then, byimposing approximate cloning foranyX X ,
nc
∈
oneobtainsthefollowingsystemofEquations:
g Tr [(I ̺ )U† (σ I)U ] = σ (7a)
1 2 ⊗ p E 1⊗ E 1
g Tr [(I ̺ )U† (σ I)U ] = σ (7b)
1 2 ⊗ p E 2⊗ E 2
g Tr [(I ̺ )U† (I σ )U ] = σ (7c)
2 2 ⊗ p E ⊗ 2 E 1
g Tr [(I ̺ )U† (I σ )U ] = σ . (7d)
2 2 ⊗ p E ⊗ 1 E 2
System (7) admits the solution θ = θ = θ/2, θ = 0—i.e., U U with θ free parameter—with
1 2 3 E nc
− ≡
g = 1/cosθandg = 1/sinθ. Noticethatothersolutionsfortheθ ’sparametersmaybefound,which
1 2 1,2,3
howevergivethesameaddednoiseastheoneconsidered above.
Remarkably,similarcloningmachinesmaybeobtainedforanyclassofobservables generatedbyapair
of operators unitarily equivalent to σ and σ . In fact, given the two-parameter classes of noncommuting
1 2
observables defined as XV = cC +dD c,d∈R, with C = V†σ1V, D = V†σ2V and V generic unitary
{ }
onehasthatanapproximate cloningmachineisgivenbythetriple(U , 0 0,X ) ,withU = (V†
V V apx V
| ih | ⊗
V†)(I F)U (V I),withadded noises g = 1/cosθ andg = 1/sinθ. Thestatement easily follows
nc 1 2
⊗ ⊗
fromthefactthat(T, 0 0,X ) isacloningmachineandfromunitarycovariance. Similarresultshold
nc apx
| ih |
foranyclassofobservables unitarily generated byanypairof(noncommuting) Paulioperators.
IV. OPTIMALITYANDJOINTMEASUREMENTS
Letusnowconsideriftheapproximatecloningmachineforobservablesintroducedaboveisoptimal. In
ordertoassessthequalityandtodefineoptimalityofatriple(U,̺ ,X) weconsideritasatooltoperform
p apx
ajoint measurements of noncommuting qubit observables [6]. Forexample, consider the cloning machine
(T, 0 0,X ) andsupposetomeasureσ andσ onthetwoqubitsattheoutput. Weemphasizeagain
nc apx 1 2
| ih |
that the cloning machine (T, 0 0,X ) clones every observable belonging to X , while we are now
nc apx nc
| ih |
consideringonlytheobservablesσ andσ which,inasense,generatetheclass. Thedifferencewithajoint
1 2
measurement specifically designed tomeasureonlyacouple ofobservables isevident. Indeed aprioriitis
not clear at all if a cloning for observables, being in this sense far more general than a joint measurement,
canreachtheminimumdisturbance necessary toperform anoptimaljointmeasurement. Nevertheless, this
turns out to be the case, as we will see in the following. We have that the measured expectation values of
σ andσ attheoutput aregivenby σ = g σ (withh = 1,2), wherethe σ ’saretheinput mean
1 2 h m h h h
h i h i h i
values. Itfollowsthatthemeasureduncertainties (∆O = O2 O 2)attheoutputaregivenby
h i−h i
∆ σ = g2∆σ
m h h i h
where∆σ denote theintrinsic uncertainties forthetwoquantities attheinput. SinceforanyPauliopera-
i h
torswehaveσ2 = Ionemayrewrites
h
∆ σ = tan2θ+∆σ (8)
m 1 i 1
∆ σ = cot2θ+∆σ . (9)
m 2 i 2
Asaconsequence, themeasureduncertainty product isgivenby:
∆ σ ∆ σ =∆σ ∆σ +cot2θ∆σ +tan2θ∆σ +1.
m 1 m 2 i 1 i 2 i 1 i 2
Sincethearithmeticmeanisbounded frombelowbythegeometric meanwehave
cot2θ∆σ +tan2θ∆σ 2 ∆σ ∆σ ,
i 1 i 2 ≥ p i 1 i 2
6
withtheequalsigniff∆σ = tan4θ∆σ ,thenitfollowsthat
i 1 i 2
2
∆ σ ∆ σ ∆σ ∆σ +1 .
m 1 m 2 ≥(cid:16)p i 1 i 2 (cid:17)
Iftheinitialsignalisaminimumuncertainty state—i.e.,∆σ ∆σ =1—onefinallyhasthatthemeasured
i 1 i 2
uncertaintyproductisboundedby∆ σ ∆ σ 4. Noticethatanoptimaljointmeasurementcorresponds
m 1 m 2
≥
to have ∆ σ ∆ σ = 4. In our case this is realized when θ is chosen such that tan4θ = ∆σ /∆σ .
m 1 m 2 i 1 i 2
As already recalled in Sec. I, such an a priori knowledge of the initial state is a standard requirement of
jointmeasurements. Therefore,since(T, 0 0,X ) addstheminimumamountofnoiseinajointmea-
nc apx
| ih |
surement performed on minimum uncertainty states weconclude that itis anoptimal approximate cloning
machine for the class under investigation. An optimal approximate cloning machine for the more general
classX maybealsodefined,usingtheconceptofjointmeasurement fornoncanonical observables [6].
gnc
Let us now consider the comparison with a joint measurement of σ and σ performed with the aid of
1 2
an optimal universal cloning machine for states [7]. It is easy to show that the best result in this case is
given by ∆ σ ∆ σ = 9, indicating that cloning of observables is more effective than cloning of states
m 1 m 2 2
to perform joint measurements (for the case of three observables see Refs. [14, 15]). In fact, a symmetric
cloning machine for states shrinks the whole Bloch vector s by a factor 2, whereas a cloning machine for
3
observables shrinks the components s and s of s only by a factor 1/√2 (considering equal noise g =
1 2 1
g = √2). Notice that such a behavior is different from what happens in the case of continuous variables,
2
forwhichtheoptimal covariant cloning ofcoherent states alsoprovides theoptimal joint measurements of
two conjugated quadratures [2]. This is due to the fact that coherent states are fully characterized by their
complex amplitude, that is by the expectation values of two operators only, whereas the state of a qubit
requires theknowledge ofthethreecomponents oftheBlochvector.
Asafinalremarkwenoticethatiftherequirementofuniversalityisdropped, thencloningmachinesfor
states can befound that realize optimal approximate cloning of observables. Forexample, an approximate
cloningmachineforthetwo–parameterclassX canbeobtainedbyconsideringaphase–covariantcloning
nc
forstates[8]. Inordertoclarifythispoint,letusrecallthataphase–covariant cloningmachineforstatesof
theformusesaprobeinthe 0 stateandperformsthefollowingtransformation:
| i
0 0 0 0
| i| i → | i| i
1 0 cosθ 1 0 +sinθ 0 1 ), (10)
| i| i → | i| i | i| i
where, in general, θ [0,2π]. If we now consider the X class it is straightforward to show that Eqs.
nc
∈
(5) are satisfied for any X X using the machine (10), with the optimal added noises given by Eq.
nc
∈
(6). This can be intuitively understood by considering that a phase covariant cloning machine extracts the
optimal information about states lying on the equatorial plane of the Bloch sphere, which in turn include
the eigenstates of σ and σ . Nevertheless, notice that there exist optimal approximate cloning machines
1 2
forobservables whichdonotcoincide withphase–covariant cloning forstates. Asanexample,considering
a cloning machine with added noise g = 1/cosθ and g = 1/sinθ one would still obtain optimal
1 2
− −
cloning ofobservables, evenifsuch amachine could not perform aphase covariant cloning ofstates. This
remark shows that, in general, one cannot simply transfer the results for state cloning into the cloning of
observables case.
V. CONCLUSIONS
Wehaveanalyzedindetailsthecloningforclassesofobservableswithfocusonclassesgeneratedbytwo
noncommuting observables. Wehave elucidated the relationship between cloning of observables and joint
measurements and shown that even if the cloning machine is designed to act on the whole class generated
bytwononcommutingobservables, thesameoptimalperformancesofajointmeasurementcanbeattained.
7
Acknowledgments
Thisworkhasbeensupported byMIURthrough theprojectPRIN-2005024254-002. AFacknowledges
thefinancialsupportfromUniversita` diMilanounderGrant“Borsediperfezionamento all’estero”.
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[11] Noticethatacloningmachineoftheform(V,̺ ,Y)withV = (W† A†)U(W I),whereAisanarbitrary
p
⊗ ⊗
unitarymatrixalsoassuresthatY(j) = Y1(j).Ontheotherhand,inordertohaveY(j) = Y2(j)atthesame
time,therelationA=W shouldhold.
[12] P.Kienzler,Int.J.Th.Phys.37,257(1998).
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