Testing Bell inequalities with circuit QEDs by joint spectral measurements

0.23 MB·English

by  Hao Yuan

Testing Bell inequalities with circuitQEDs by jointspectral measurements
Hao Yuan,1,2 L. F. Wei,1,3,∗ J. S. Huang,1 X. H. Wang,1 and Vlatko Vedral2,4,5,†
1Quantum Optoelectronics Laboratory, School of Physics and Technology,
Southwest Jiaotong University, Chengdu 610031, China
2Clarendon Laboratory, UniversityofOxford, ParksRoad, OxfordOX13PU,UnitedKingdom
3State Key Laboratory of Optoelectronic Materials and Technologies,
School of Physics and Engineering, Sun Yat-Sen University, Guangzhou 510275, China
4CentreforQuantumTechnologies,NationalUniversityofSingapore, 3ScienceDrive2,Singapore117543,Singapore
5DepartmentofPhysics,NationalUniversityofSingapore, 2ScienceDrive3,Singapore117542, Singapore
(Dated:January14,2011)
WeproposeafeasibleapproachtotestBell’sinequalitywiththeexperimentally-demonstratedcircuitQED
1 system,consistingoftwowell-separatedsuperconductingchargequbits(SCQs)dispersivelycoupledtoacom-
1 monone-dimensionaltransmissionlineresonator(TLR).Ourproposalisbasedonthejointspectralmeasure-
0 mentsofthetwoSCQs,i.e.,theirquantumstatesinthecomputationalbasis{|kli, k,l=0,1}canbemeasured
2
bydetectingthetransmissionspectraofthedrivenTLR:eachpeakmarksoneofthecomputationalbasisandits

n relativeheightcorrespondstotheprobabilitysuperposed.Withthesejointspectralmeasurements,thegenerated
a BellstatesofthetwoSCQscanberobustlyconfirmedwithoutthestandardtomographictechnique. Further-
J more,thestatisticalnonlocal-correlationsbetweenthesetwodistantqubitscanbedirectlyreadoutbythejoint

3 spectralmeasurements,andconsequentlytheBell’sinequalitycanbetestedbysequentiallymeasuringtherel-
1 evant correlations relatedtothesuitably-selected setsof theclassicallocal variables {θj,θj′,j = 1,2}. The
experimentalchallengesofourproposalarealsoanalyzed.

]
h PACSnumber(s):03.65.Ud,42.50.Dv,85.25.Cp
p
-
t
n I. INTRODUCTION Recently, a novel superconductingelectrical circuit archi-
a tecture (called circuit QED) analogous to cavity QED has
u
beenfirstsuggestedbyBlaisetal.[9],andthenrealizedinthe
q
experimentbyWallraffetal.[10]. IncircuitQED,asupercon-
[ Historically,thewell-knownEinstein,Podolsky,andRosen
(EPR) paradox [1] concerningthe completeness of quantum ducting qubit is strongly coupled to one-dimensional trans-
1 mechanicswasproposedbasedonagedankenexperimentin missionlineresonator(TLR)whichactsasquantizedcavity.
v
1935. EPR claimed that quantum mechanics is incomplete Now much attention has been focused on this field because
0
5 andso-calledlocalhiddenvariables(LHV)shouldexist. Sub- it opens the possibility of studying quantum optics phenom-
ena in solid-state system and quantum information process-
5 sequently, it has provokedmuch debate on the completeness
2 ofquantummechanicsandtheexistenceofLHVtheories. In ing.Experimentally,remarkableadvanceshavebeenachieved
. 1964,BellactuallyquantifiedtheEPRargumentandderived such as: observation of the vacuum Rabi splittings [10] and
1
0 astrictinequality[2]withrespecttothecorrelationstrengths ACStackshifts[11],observingBerry’sphase[12],measuring
the Lamb shifts [13], generations of microwave single pho-
1 possibletoachievebyalltheLHVmodels. Ifthisinequality
1 isviolated,thentherearenoLHVandthequantummechan- tons[14]andFockstates[15],couplingtwoqubitsviacavity
v: icalpredictionof the existenceof quantumnonlocalcorrela- asquantumbus[7,16], two-qubitGroverandDeutsch-Jozsa
i tion (i.e., entanglement) is sustained. However, Bell’s sem- algorithms [17], etc. In this system the qubit-state readouts
X were achieved by detecting the state-dependentshifts of the
inal inequality did not allow for the practically-existing im-
r perfections and thus was not accessible to the experimental frequencyof the dispersively-coupledresonator[17, 19–22].
a
Indeed, in the dispersive regime[18], the transition frequen-
tests. Soonlater,Clauser,Horne,ShimonyandHolt(CHSH)
addressedthisissueandderivedanexperimentallytestablein- cies of the superconducting qubits are far detuned from the
frequencyofthecoupledresonator.Asaconsequence,asuffi-
equality[3].Duringthepastdecades,anumberofexperimen-
tal tests of CHSH-type Bell’s inequality have been reported cientlylargestate-dependentfrequencyshiftsoftheresonator
can be induced [9, 10], which could be tested by measuring
using, e.g., entangled photon pairs [4], trapped ions [5], an
atomandaphoton[6],twosuperconductingphasequbits[7], thetransmissionspectraofthedrivenresonator.
andeventwoentangleddegreesoffreedom(comprisingspa- Veryrecently,anefficientapproachtoimplementthejoint
tial and spin components) of single neutrons [8], and so on. spectral measurements of the multi-qubits has been pro-
These experimental evidences strongly convince that Bell’s posed by detecting the transmission spectra of the common
inequality could be violated and thus agree well with quan- resonator [23]. By considering all the quantum correla-
tummechanicalpredictions,rulingouttheLHVtheories. tion of qubits and resonator (i.e., beyond the usual coarse-
grained/mean-fieldapproximation),itisshownthateachpeak
in the measured transmission spectra of the drivenresonator
marks one of the logic states and the relative height of such
∗Electronicaddress:[email protected] apeakisrelatedtoitscorrespondingsuperposedprobability.
†Electronicaddress:[email protected] Withthesejointspectralmeasurements,Huangetal.,[24]pre-
2
sentedahighefficiencytomographicreconstructionsofmore
thanone-qubitquantumstates. Asanotherapplication,inthis
paper we discuss how the proposed joint spectral measure-
mentscanbefurtherutilizedtotestBell’s inequalitywithan
experimentally-demonstratedtwo-qubit circuit QED system.
In Out
ComparedwiththepreviousschemesfortestingBellinequal-
ities, the advantageof the presentproposalis that, the desir-
ablecoincidedmeasurementsonthetwoqubitscanbedirectly
implementedbythesuitably-designedjointspectralmeasure-
ments. This is because the states of the two qubits jointly FIG. 1: (Color online) Circuit QED architecture with two su-
determinethe frequencyshiftsand consequentlythe spectral perconducting charge qubits (SCQs) dispersively coupling to one-
distributionsof thedrivenresonator. Therefore,thenonlocal dimensional transmission line resonator (TLR). The joint spectral
correlationsbetweenthewell-separatedqubitscanbedirectly measurementsofthetwoSCQscanberealizedbydetectingthestate-
dependentfrequencyshiftsinthetransmissionspectraofthedriven
read out by analyzing the measured transmission spectra of
resonator.
thedrivenresonator. We noteinpassingthatnoneofthecir-
cuit QED proposals so far can be used to close the locality
loophole. (RWA),thesystemoftwoSCQsplusresonatorisdescribedby
Thepaperisorganizedasfollows:InsectionII,wepropose theusualDickeHamiltonian(~=1)[25]
amethodtodeterministicallygenerateanidealBellstatewith
ω
the experimentally-demonstratedcircuitQED system, via an H =ω a†a+ [ jσ +g (a†σ +aσ )]. (2)
r zj j −j +j
2
iSWAP gate assisted by severalsingle-qubitgates[25]. Fur- jX=1,2
ther, a simple quantum interference method, instead of the
Here, ω is the single-modefrequencyof the TLR, a†(a) its
usualquantum-statetomographicreconstruction,isproposed r
creation (annihilation) operator; ω the transition frequency
toconfirmsuchapreparationbycoherentquantumoperations j
of jth SCQ thatcan beadjusted byexternalbiasflux and g
and projective measurements on two qubits in the computa- j
thecouplingstrengthofthejthSCQtotheresonator.Finally,
tionalbasis. InsectionIII,wediscusshowtoutilizethejoint
σ = 1 0, σ = 0 1 and σ = 0 0 1 1.
spectral measurements to implement this confirmation. Im- +j j −j j zj j j
| i h | | i h | | i h |−| i h |
Two SCQs in thiscircuitcan be controlledandmeasuredby
portantly, by applying two individual Hadamard-like opera-
driving the resonator. The Hamiltonian to describe such a
tions to encode the local variables θ ,θ into the gener-
1 2
{ } driveisgivenby
ated Bell state, we show that the statistical correlations be-
tween the distant qubits can be read out by these spectral H =ǫ(a†e−iωdt+aeiωdt), (3)
d
measurementsfor varioustypical sets of classical local vari-
CabHleSsH{-θtyj,pθej′,Bjel=l’s1i,n2e}q.uWaliitthytchaensebmeeteasstuerde.dcDoirsrceulsastiioonnsatnhde qwuheenrceyǫoifsthtiemeex-tienrdneaplleynadpepnltieredadlraivmep.litude and ωd the fre-
conclusionarepresentedinsectionIV. From the easily-preparedinitial state 00 12, an ideal Bell
| i
statecanbegenerateddeterministicallybythefollowingthree
steps,thecorrespondinggatesequenceisshowninFig.2.
II. GENERATIONANDCONFIRMATIONOFBELL
STATES 3
0 R ( ) R ( )
1 x 4 y 4
Inthissection,wewillshowhowtodeterministicallypre- iSWAP !
pareoneoftheidealBellstates 12
0 R ( )
2 x 4
1 1
Φ = (00 11 ), Ψ = (01 10 ), (1)
± ±
| i √2 | i±| i | i √2 | i±| i
FIG. 2: Gate sequence to generate the Bell state |Φ−i12 from the
withtheexperimentally-demonstratedcircuitQEDsystem.In initialstate|00i12.
fact,theBellstateinthissystemhavebeenpreparedbyseveral
methods, includingmeasuring-basedsynthesis [26] and gate Step 1: Perform the unitary operation R (π/4) =
x
sequence (see, eg. Refs. [7, 16, 19, 20, 25]). Here, we will exp(iσ π/4)oneachSCQtopreparethesuperpositionoftwo
x
presentanalternativegatesequencemethodtoimplementthe basisstatesofeachSCQ.Thatis,
desirable preparation based on an iSWAP gate (the previous
methodsusethecontrolledphasegate). 00 Rx1(π4)Rx2(π4) ψ = 1(0 +i1 ) (0 +i1 ) .(4)
12 12 1 2
| i → | i 2 | i | i | i | i
WeconsiderthecircuitQEDarchitecturesketchedinFig.1,
inwhichtwosuperconductingchargequbits(SCQs)arecou- The unitary operations R (π/4) and R (π/4) can be
x1 x2
pled to one-dimensional TLR. To obtain the maximal cou- achievedbyexternallyappliedmicrowavedriveswiththedu-
pling,twoSCQsareplacedclosetotheendsoftheresonator ration of t = π∆ /4ǫg and t = π∆ /4ǫg , respec-
1 r 1 2 r 2
− −
(voltageantimodesoftheresonator).WhentwoSCQsworkat tively[25]. Here,∆ = ω ω isthedetuningbetweenthe
r d r
−
theiroptimalpointandundertherotating-waveapproximation drivefrequencyandtheresonatorone.
3
Step 2: In the dispersive regime, i.e.,g /∆ 1 and thesameprobability(i.e.,P = P = 1/2). Notethatthis
1 1 00 11
| | ≪
g /∆ 1 (with ∆ = ω ω being the detuning be- isnotthesufficientconditionfortheconfirmationoftheBell
2 2 j j r
| | ≪ −
tween the jth SCQ and the TLR), and ∆ = ∆ = ∆ and state Φ ,sinceastatisticalmixtureofthelogicstates 00
1 2 − 12
| i | i
g = g = g (achieved by adjusting the external bias flux), and 11 withthesameprobability1/2mayalsoresultinthe
1 2
| i
theefficientHamiltonianofthissystemreads same outputs. Therefore, besides the above projective mea-
surementperformeddirectly,anotherprojectivemeasurement
H = g2 [4(a†a+ 1)(σ +σ ) in the computational basis is also required after the unitary
eff −2∆ 2 z1 z2 operationR (π/4) = R (π/4)R (3π/4)R (3π/4) on each
y z x z
(σ σ +σ σ )]. (5) SCQ. These operationsevolve the prepared Bell state to an-
x1 x2 y1 y2
−
otherBellstate,i.e.,
ThisHamiltoniangeneratesatwo-qubitgate(withtheevolu-
tiondurationts =3π∆/2g2) Φ Ry1(π4)Ry2(π4) Ψ = 1 (01 + 10 ) . (10)
− 12 + 12 12
| i → | i √2 | i | i
eiδ 0 0 0
 0 0 i 0  Thismeasurementimpliesthat,ifthepreparedstateisthede-
U(t )=exp( iH t )= , (6)
s − eff s 0 i 0 0 sirable Bell state Φ− 12, then the measured state should be
 0 0 0 e−iδ  Ψ+ 12 and thus |the ioutput is either 01 or 10 (with the
  s|amei probability 1/2). This is becau|seithe m|ixiture of the
where δ = 6π(n + 1/2) with n = a†a being the mean states 00 and 11 would not vanish by the quantum inter-
photonnumberintheresonator.Ifnishknowininadvance,the ference| afiterthe|deisignedquantumoperations.
globalphasefactorδcanbeeliminatedbyperformingsingle-
qubit phase rotations: 0 e−iδ/2 0 and 1 eiδ/2 1 ,
oneachqubit. Then,th|eiab→ovetwo-q|uibitgat|eire→ducesto|thie III. JOINTSPECTRALMEASUREMENTSOFTWO
QUBITSFORTESTINGBELL’SINEQUALITY
desirableiSWAPgate
1 0 0 0 A. Jointspectralmeasurementsofthequbitsforconfirming
0 0 i 0 thepreparationofBellstates
U = . (7)
iSWAP 0 i 0 0
 0 0 0 1 Customarily,completelycharacterizinganunknownquan-
 
tum state requires to tomographicallyreconstruct its density
Notethatnoexternalmicrowavedriveisappliedinthisstep.
matrix, by a series of quantum measurements performed on
With such an iSWAP gate, the entanglement between two
itsmanycopies(see, e.g.,[7,17,19,20,22]). Here,wewill
SCQsisgenerated,i.e.,
show that the Bell state generated above, e.g., Φ , can
− 12
| i
beconfirmedbyusingthejointspectralmeasurementsofthe
1
ψ 12 UiSWAP ψ′ 12 = (00 01 10 11 )12. (8) two qubits in a significantly simpler fashion. In the disper-
| i → | i 2 | i−| i−| i−| i
sive regime of two-qubit circuit QED system, the transition
Step 3: Apply the unitary operation R (3π/4) to the first frequencyof each SCQ is far detuned from the coupled res-
y
SCQ for generating a desirable Bell state Φ from the onator.Andinaframeworkrotatingatthedrivefrequencyω ,
− 12 d
state ψ′ preparedabove,i.e., | i theefficientHamiltonianofthewholesystemis
12
| i
|ψ′i12 Ry1→(34π)|Φ−i12 = √12(|00i−|11i)12. (9) H = (−∆r +Γ1σz1+Γ2σz2)a†a+ ω˜21σz1+ ω˜22σz2
+ǫ(a†+a), (11)
Here, the unitary operation R (3π/4) is implemented by
y
Ry(3π/4) = Rx(π/4)Rz(3π/4)Rx(3π/4). Rz(3π/4) and whereΓj = gj2/∆j and ω˜j = ωj +Γj, j = 1,2. From the
Rx(3π/4)canberealizedbyapplyingmicrowavedriveswith firstterminEq.(11),itcanbedirectlyseenthattheresonator
thedurationt3 =3π∆a/[2(∆a+g12/∆1)∆a+(2ǫg1/∆r)2] ispulledbydifferentlogicstatesoftwoqubits.Thefrequency
(∆Saim=ilωar1ly−,oωtdh)eranBdetll4s=tat−es3iπn∆Erq/.(41ǫ)g1c,arnesbpeeacltsivoeglye.nerated sΓhif+tsΓofcthoerrreesspoonnadtotorbthye−loΓg1ic−stΓat2e,s−oΓf1tw+oΓqu2,bΓits1−11Γ2,,1a0nd,
1 2
withtheabovemethod. 01 ,and 00 ,respectively. Thus,thefrequencysh|iftsio|fthie
Customarily, the quantum-state preparation is experimen- r|esoinator|canibeusedtomarkthelogicstatesoftwoqubits.
tally confirmed by quantum-state tomography (see, e.g., [7, Further, by considering all the statistical quantum corre-
17, 19, 20, 22]), i.e., reconstructing its density matrix via a lations between SCQs and resonator (i.e., beyond the usual
seriesofmeasurementsonmanycopiesofthepreparedstate. coarse-grained/mean-field approximation), the steady-state
However, based on the same logic proposedin Ref. [27] the transmissionspectraofthedrivenresonatorS = a†a /2ǫ
ss ss
Bell state generated above can be simply confirmed by the canbeanalyticallyderivedas[23,24] h i
followingtwosteps. First,weperformaprojectivemeasure-
ment on the prepared state in the computational basis. The 2(AC+BD)
S = , (12)
outputswillbeeitherthelogicstate 00 orthe 11 statewith ss − κ(A2+B2)
| i | i
4
with A = (Γ2 Γ2) + 2(κ2/4 ∆2)(Γ2 + Γ2) + Bell’sinequalitywithcircuitQEDviajointspectralmeasure-
(κ2/4 ∆2)2 1κ−2∆2,2B = 2κ∆−(Γ2r+ Γ21 + κ22/4 mentsofthedistantSCQs.
− r − r − r 1 2 −
∆2), C = κ σ (0)σ (0) Γ Γ κ∆ ( σ (0) Γ + First,withthesingle-qubitgates[25],aHadamard-likeop-
σr (0) Γ ) +h zκ1/2(3∆z22 i2κ21/42 −Γ2 rΓh2)z,1andi2D1 = eration[29]canbeconstructedas
h z2 i2 2 r − − 1 − 2
2 σ (0)σ (0) ∆ Γ Γ 2 σ (0) Γ (Γ2 Γ2 +
− h z1 z2 i2 r 1 2− j=1h zj ij j j− j′ R (θ ) = R (θ /2)R (π/4)R ( θ /2)
κΓ2/4=−g∆2/2r∆)+. ∆Hre(rΓe21, κ+dΓe22n+otPe3sκp2h/o4to−n∆le2ra)k,ajge6=rajt′e=of1t,h2e, j j 1z j 1 x ieiθj z − j
j j j = , (13)
resonator. Experimentally, the qubit decay rate (e.g., γ = √2(cid:18)ie−iθj 1 (cid:19)
2π 0.25MHz [28]) is negligible, since it is significantly
× whichplaysanimportantrolefortestingBell’sinequality.
smaller than the photon leakage rate of the resonator (e.g.,
Then, the classical variables θ ,θ can be encodedinto
κ = 2π 1.69MHz [22]). It has been numerically proven { 1 2}
× the generated Bell state Φ with Hadamard-like opera-
that, each peak in the above steady-state spectra marks one | −i12
tionsR (θ ). Thus,thestate Φ ischangedinto
ofthelogicstatesandtherelativeheightofsuchapeakcorre- j j | −i12
spondstothesuperposedprobabilityinthedetectedtwo-qubit Φ′ = R (θ )R (θ )Φ
state. | −i12 1 1 2 2 | −i12
1
= [(1+e−i(θ1+θ2))00 +i(eiθ2 e−iθ1)01
|11〉 |00〉 2√2 | i − | i
0.6
(a)
0.5 + i(eiθ1 e−iθ2)10 (1+ei(θ1+θ2))11 ] .(14)
12
− | i− | i
0.4
Thirdly, the correlation between two classical variables
0.2 θ ,θ is measuredby thedifferencein the probabilitiesof
1 2
{ }
twoqubitsfoundin thesameanddifferentlogicstates. This
mission −01 50 −100 −50 0 50 100 150 meansthatthecorrelationfunctionE(θ1,θ2)canbeobtained
Trans0.6 (b) |10〉 |01〉 as
0.5
E(θ ,θ ) = P (θ ,θ ) P (θ ,θ )
0.4 1 2 same 1 2 − diff 1 2
= P (θ ,θ )+P (θ ,θ ) P (θ ,θ )
00 1 2 11 1 2 01 1 2
0.2 −
P (θ ,θ ), (15)
10 1 2
−
0
−150 −100 −50 ω−ω0(MHZ) 50 100 150 whereP00(θ1,θ2) (P01(θ1,θ2), P10(θ1,θ2), P11(θ1,θ2))de-
d r
notes the probability of two qubits found in the logic state
FIG.3: Transmissionspectraofthedrivenresonatorversusthede- 00 12(01 12, 10 12, 11 12).
| i | i | i | i
tuning∆r =ωd−ωrforBellstates|Φ−i12(a)and|Ψ+i12(b).The Theoretically, the correlation of two classical variables
parametersarechosenas(Γ1,Γ2,κ)=2π×(13,4,1)MHz[20]. θ1,θ2 can be described by the operator T = 11 11 +
{ } | ih |
00 00 10 01 01 01 =σ σ andthusthecor-
z1 z2
| ih |−| ih |−| ih | ⊗
relationfunctioniscalculatedas
Therefore,ifthetwoSCQsarepreparedatthestate Φ ,
− 12
| i
then two peaks marking respectively 00 and 11 with the E(θ ,θ )= Φ′ T Φ′ =cos(θ +θ ). (16)
sameheightwouldbedetectedinthem| eaisured|speictra. Fur- 1 2 12 h −| | −i12 1 2
thermore, if the joint spectral measurements are performed Thus, for the set of classical local variables θ ,θ ,θ′,θ′ ,
{ 1 2 1 2}
after the evolution (10), then other two peaks marking re- theso-calledCHSHfunction[3]reads
spectively 10 and 01 should be detected in the spectra.
This is the|diriect evi|denice showing that the state generated f(|Φ′−i12) = |E(θ1,θ2)+E(θ1′,θ2)+E(θ1,θ2′)
above is just the desirable Bell state, i.e., the coherent su- E(θ′,θ′)
− 1 2 |
perpositionofthestates 00 and 11 , ratherthantheir mix- = cos(θ +θ )+cos(θ′ +θ )
ture. This argument cou|ldibe ver|ifieid by the numerical re- | 1 2 1 2
+cos(θ +θ′) cos(θ′ +θ′). (17)
sults shown in Fig. 3. For the typical experimental param- 1 2 − 1 2 |
eters (Γ ,Γ ,κ) = 2π (13,4,1)MHz [20], the Fig. 3(a)
1 2 Typically, for one set of classical local variables
×
shows the steady-state spectral distributions (versus the de- θ ,θ ,θ′,θ′ = π/4,3π/4,π/2,π , the CHSH func-
tuning∆ =ω ω )fortheBellstate Φ ,andFig.3(b) { 1 2 1 2} { }
r d r − 12 tion[3]iscalculatedas
− | i
foranotherBellstate Ψ .
+ 12
| i
f(Φ′ )=√2+1>2. (18)
| −i12
Obviously, the CHSH-type Bell’s inequality [3] f 2 is
B. TestingBell’sinequalityviajointspectralmeasurements ≤
violated. While, for another set of classical local variables
θ ,θ ,θ′,θ′ = π/4,0,7π/4,3π/2 , the CHSH func-
BeingabletomeasureBellstatesconstitutesagoodentan- {tio1n[32]is1 2} { }
glementwitness,butwecaninfactdobetterthanthat. With
the generated Bell state Φ , we now show how to test f(Φ′ )=2√2>2. (19)
| −i12 | −i12
5
Therefore, in this case the CHSH-type Bell’s inequality [3] 0.5 |11〉 |10〉 |01〉 |00〉 0.5 |11〉 |10〉 |01〉 |00〉
(a) (b)
f 2isviolatedmaximally. 0.4 0.426 0.4 0.426
≤
Theabovetheoreticalpredictionscanbenumericallytested
0.3 0.3
bytheexperimentaljointspectralmeasurements. Indeed,the
0.2 0.2
correlation function in Eq. (15) can be directly calculated
0.1 0.074 0.1 0.074
from the measured transmission spectra of the driven res- on 0 0
onator. For example, for the set of classical local variables missi −150−100 −50 0 50 100 150 −150−100 −50 0 50 100 150
ans0.5 0.5
|11〉 |10〉 |01〉 |00〉 |11〉 |10〉 |01〉 |00〉 Tr (c) 0.426 (d) 0.426
0.5(a) 0.5(b) 0.4 0.4
0.4 0.4 0.426 0.3 0.3
0.3 0.3 0.2 0.2
0.2 0.2 0.1 0.074 0.1 0.074
0.1 0.1 0.074 −1050−100 −50 0 50 100 150 −1050−100 −50 0 50 100 150
mission −1050−100 −50 0 50 100 150 −1050−100 −50 0 50 100 150 ωd−ωr(MHZ)
ans0.5 0.5 FIG. 5: Plot of the transmission spectra of the driven resonator
Tr0.4(c) 0.426 0.4(d) versus the detuning ∆r = ωd − ωr for the state |Φ′−i12
0.3 0.3 with another set of classical local variables {θ1,θ2,θ1′,θ2′} =
0.25 {π/4,0,7π/4,3π/2}. (a)-(d): thecorrelationfunctionscanbedi-
0.2 0.2 rectlycalculatedas(E(θ1,θ2),E(θ1′,θ2),E(θ1,θ2′),E(θ1′,θ2′)) =
0.1 0.074 0.1 (0.704,0.704,0.704,−0.704). Theparametersarethesameasthat
0 0 inFig.3.
−150−100 −50 0 50 100 150 −150−100 −50 0 50 100 150
ω−ω(MHZ)
d r
FIG.4: Transmissionspectraofthedrivenresonatorversusthede-
simplerandeasiertobeexperimentallyrealized. We discuss
tuning ∆r = ωd −ωr for the state |Φ′−i12 with one set of clas- itsadvantagesnext.
sical local variables {θ1,θ2,θ1′,θ2′} = {π/4,3π/4,π/2,π}. (a)-
(d): thecorrelationfunctionscanbedirectlycalculated,(E(θ1,θ2),
E(θ1′,θ2), E(θ1,θ2′), E(θ1′,θ2′)) = (−1,−0.704,−0.704,0), ac-
cordingtoEq.(15).TheparametersarearethesameasthatinFig.3.
IV. DISCUSSIONANDCONCLUSION
θ ,θ ,θ′,θ′ = π/4,3π/4,π/2,π ,Fig.4showstherele-
v{an1ttr2ans1mis2s}ions{pectra(correspond}ingtothestate Φ′ )
| −i12 We now briefly address the experimental feasibility of
versus the detuning of ∆ = ω ω . Here, the parame-
r d r our scheme. With the typical experimental parameters
−
ters are chosen as (Γ ,Γ ,κ) = 2π (13,4,1)MHz [20].
1 2 (ω ,ω ,ω ,g )=2π (6.442,4.5,4.85,0.133)GHz[19]and
× r 1 2 1(2)
WiththeFig.4(a)-(d),thecorrelationfunctionsbetweentwo ×
the amplitude of the drive chosen as ǫ=2π 1.2GHz, we can
classicallocalvariablescanbedirectlycalculated(E(θ ,θ ), ×
1 2 approximatelyestimate the time neededin our scheme. The
E(θ′,θ ),E(θ ,θ′),E(θ′,θ′))=( 1, 0.704, 0.704,0).
1 2 1 2 1 2 − − − required times for realizing the unitary operatons Rx(π/4),
ThustheCHSHfunction[3]iscalculatedas
R (3π/4)(settingω =2π 4.491GHztosatisfy∆ +Γ =
x d a j
×
0),R (3π/4)(settingω =2π 4GHz),aniSWAPgate(set-
f(|Φ′−i12)=2.408≈√2+1>2. (20) ting∆z =2π 1.18GHzd)aret1×(2)=1.5ns,t4=4.5ns,t3=1.5ns,
×
and t =50ns, respectively. Experimentally, the time interval
This indicates thatthe CHSH-type Bell’s inequality[3] f s
≤ forperformingajointspectralmeasurementintroducedabove
2 is violated. Similarly, for another set of classical lo-
isabout40ns[22]. Thus,thetimesforgeneratingaBellstate
cal variables θ ,θ ,θ′,θ′ = π/4,0,7π/4,3π/2 , with
{ 1 2 1 2} { } and confirming its existence are about 60.5ns and 93ns, re-
the same experimental parameters, we plot the transmission
spectively.Also,thetimeintervalfortestingBell’sinequality
spectra of the driven resonator versus the detuning ∆ =
r
ωd − ωr for the state |Φ′−i12 in Fig. 5. Again, from cscahnebmeeetsotipmearfteodrmasa∼tes1t6e0xnpse.rimFiennatlliys,atbhoeutto3ta1l3.t5imnse,iwnhoicuhr
the Fig. 5(a)-(d) the correlation functions are directly cal-
culated as (E(θ ,θ ), E(θ′,θ ), E(θ ,θ′), E(θ′,θ′)) = isstillshorterthanthequbit’srelaxationanddephasingtimes,
1 2 1 2 1 2 1 2 e.g., T =7.3µs and T =500ns[28]. It is clear, however, that
(0.704,0.704,0.704, 0.704). As a consequence, we easily 1 2
− this is not enough to close the locality loophole. Therefore,
obtaintheCHSHfunction[3]:
ourproposalshouldbe experimentallyrealized with the cur-
f(Φ′ )=2.816 2√2>2. (21) renttechniques.
| −i12 ≈ In conclusion, we have proposed a simple and feasible
Obviously,theCHSH-typeBell’sinequality[3]f 2issig- method to test Bell’s inequality with experimentally-
≤
nificantlyviolated. demonstrated circuit QED system by joint spectral mea-
Finally, we emphasize that compared with the previous surements. At first, an alternative method to generate an
schemes for testing Bell’s inequality (see, e.g., [7, 20]) with ideal Bell state is proposedwith an iSWAP gate and several
superconducting qubits, the present proposal seems much single-qubitgates[25]. Thenwe presentasimple methodto
6
confirmthegenerationoftheBellstatewithtwosingle-qubit out from the measured transmission spectra of the driven
unitary operations and two projective measurements on two resonator. As a result, the present proposal seems much
qubits. With the experimental joint spectral measurements, simpler and easier to be experimentally realized by joint
the confirmation can be easily achieved. Further, with two spectralmeasurements. Webelievethatourmethodcouldbe
Hadamard-like operations constructed with single-qubit generalizedtotestBell-likeinequalitiesformulti-qubitstates
gates [25], two classical variables θ ,θ can be encoded inastraightforwardway.
1 2
{ }
intothegeneratedBellstate. Forthetypicalsetsofclassical
local variables θ ,θ′,j = 1,2 , the correlation functions
{ j j }
in Bell’s inequalitycan be directly read outby jointspectral Acknowledgements: This work was supportedin part by
measurements because each detected peak marks one of the Natural Science Foundation of China under Grant Nos.
the logics states and its relative height corresponds to the 10874142and90921010,theFundamentalResearchProgram
superposed probability in the detected two-qubit states. In of China under Grant No. 2010CB923104, and the Fun-
this way, the Bell’s inequality could be efficiently tested. damental Research Funds for the Central Universities under
Compared with the previous schemes for testing Bell’s GrantNos. SWJTU09CX078and2010XS47. V.V.acknowl-
inequality,theadvantageofourproposalisthatthedesirable edges financial support from the Engineering and Physical
coincided measurements on the two qubits can be directly SciencesResearchCouncilinUnitedKingdomaswellasthe
realized by joint spectral measurements and the nonlocal NationalResearchFoundationandtheMinistryofEducation
correlations between the distant qubits can be directly read inSingapore.
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