Genuine High-Order Einstein-Podolsky-Rosen Steering

1 MB·English

by  Che-Ming Li

Genuine High-Order Einstein-Podolsky-Rosen Steering
Che-Ming Li1,∗ Kai Chen2,3,† Yueh-Nan Chen4, Qiang Zhang2,3, Yu-Ao Chen2,3, and Jian-Wei Pan2,3‡
1Department of Engineering Science, National Cheng Kung University, Tainan 701, Taiwan
2Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
3Shanghai Branch, CAS Center for Excellence and Synergetic
Innovation Center in Quantum Information and Quantum Physics,
University of Science and Technology of China, Shanghai 201315, China and
4Department of Physics and National Center for Theoretical Sciences,
National Cheng-Kung University, Tainan 701, Taiwan
(Dated: July 2, 2015)
Einstein-Podolsky-Rosen (EPR)steering demonstrates that two partiesshare entanglement even
5
if the measurement devices of one party are untrusted. Here, going beyond this bipartite concept,
1
0 we develop a novel formalism to explore a large class of EPR steering from generic multipartite
2 quantumsystemsofarbitrarilyhighdimensionalityanddegreesoffreedom,suchasgraphstatesand
hyperentangled systems. All of these quantum characteristics of genuine high-order EPR steering
l
u canbeefficientlycertifiedwithfewmeasurementsettingsinexperiments. Wefaithfullydemonstrate
J for the first time such generality by experimentally showing genuinefour-partite EPR steering and
applications to universal one-way quantum computing. Our formalism provides a new insight into
1
theintermediatetypeofgenuinemultipartiteBellnonlocalityandpotentialapplicationstoquantum

information tasks and experiments in thepresence of untrustedmeasurement devices.
]
h
p PACSnumbers: 03.65.Ud,03.67.Lx,42.50.Dv
-
t
n
a Einstein-Podolsky-Rosen (EPR) steering was origi- party. Each time, after receiving particles, they measure
u nally introduced by Schr¨odinger in 1935 [1] to describe theirrespectivepartsandcommunicateclassically. Since
q
the EPR paradox [2]. Recently, it has been formulated B doesnottrustA ,A ’staskistoconvinceB thatthe
[ s s s s
by Wiseman et al. [3] to show a strict hierarchy among state shared between them is entangled. As will succeed
3 Bell nonlocality, steering, and entanglement [4, 5] and inthistaskifandonlyifA canpreparedifferentensem-
s
v stimulated new applications to quantum communication blesofquantumstatesforB bysteeringB ’sstate. Here
s s
2
[6]. Several experimental demonstrations of EPR steer- we say an N-particle state generated from the source to
5
ing have been reported [5, 7, 8]. The steering effect re- possess genuine N-partite EPR steerability if by which
4
1 veals that different ensembles of quantum states can be As succeed in the task for all possible bipartitions As
0 remotely prepared by measuring one particle of an en- and B of the N-particle system. This interpretation is
s
1. tangled pair. We go a step further and consider the fol- consistent with the definition recently introduced by He
0 lowing question: how to experimentally observe genuine and Reid [11]. In a wider scope, Schr¨odinger’s original
5 multipartiteEPRsteering? Forinstance,givenanexper- concept can even be applied to quantum systems with
1
imental output state ρ which is created according a many degrees of freedom (DOFs), e.g., hyperentangled
: expt
v target four-qubit cluster state of the form [9, 10] systems, and, as will be shown presently, extended as
i genuine multi-DOF EPR steering. In this Letter we call
X
1
G = (+ 0 + 0 + 1 + 1 these two sorts of quantum characteristics genuine high-
ar | 4i 2 | i1| i2| i3| i4 |−i1| i2| i3| i4 order EPR steering.
+ + 0 1 + 1 0 ),
| i1| i2|−i3| i4 |−i1| i2|−i3| i4 Theconceptofverifyinggenuinehigh-orderEPRsteer-
where = (0 1 )/√2, how do we describe the ing leads us naturally to consider quantum scenarios
|±ik | ik ±| ik based on genuine high-order entanglement [12–17] in
effect of genuine multipartite EPRsteering and then de-
which B ’s measurement apparatus are trusted, while
tect such steerability of ρ in the laboratory? s
expt
A ’s are not; see Fig. 1. Demonstrations of genuine
Inspired by the task-oriented formulation of bipartite s
high-orderEPRsteerabilityguaranteefaithfulimplemen-
steering[3],genuinemultipartiteEPRsteeringcanbede-
tations of the quantum scenarios in the presence of un-
fined from an operational interpretation as the distribu-
characterizedparties. So far, while verifying genuine tri-
tionofmultipartite entanglementby uncharacterized (or
partite steering becomes possible [11], the fundamental
untrusted)parties. Letusconsiderasystemcomposedof
problemsuchastheverificationconsideredaboveandthe
N parties and a source creating N particles. Eachparty
cases for arbitrary large N remains open.
ofthesystemcanreceiveaparticlefromthesourcewhen-
everanN-particlestateiscreated. Wedividethesystem Here we develop new quantum witnesses to observe a
intotwogroups,sayA andB ,andassumethatA isre- large class of genuine high-order EPR steering indepen-
s s s
sponsibleforsendingparticlesfromsuchasourcetoevery dent of the particle or DOF number. Let us start with
2
(a)! (b)! (c)!
Measurement devices
Qudit graph state

… sMetetinagsu srwemitcehnets
d result H
H
N

} indicators
Inα

β
.
…

… }
Z
H
Z
H
Z
…

… (d)! (e)!
trusted untrusted
vertex
edge
2

device
device
Measurement
Readout
Quantum channel
.Classical channel
(cid:1)
(cid:1)
B(β)
4!
3
(β)
4

3
N-qudit
ρexpt
WN = …
BB(
3α
)
entasnogulrecme
ent
qudit
1

. ……
……

2

2
1
B(
α2
)
1!
FIG.1. (coloronline). GenuinemultipartiteEPRsteerabilityandapplications. Genericentanglement-basedquantumprotocols
relyonbothcharacterizedmeasurementdevicesandgenuinemultipartiteentanglement,suchasone-wayquantumcomputation
[12,13]andmultipartyquantumcommunications[14]. GenuinemultipartiteEPRsteerabilityenablesthemtoperformquantum
applications in the presence of untrusted measurement apparatus (a). Here, we develop quantum steering witnesses to ensure
that an experimental state ρ of an N-partite quantum d-dimensional system (N qudits) has such ability, for example, an
expt
experimental graph state [18–21] used for one-way computing. To implement gate operations such as circuits, composed of
one-qubit (d = 2) gate Hˆ and two-qubit controlled-Z (CZ) gates (b) and (c) for input states (cid:12)(cid:12)Inαβ(cid:11), one needs to prepare
chain-type (d), i.e., G , and box-type(e) cluster states, respectively. By performing measurements B(α) and B(β) on qubits
| 4i 2 3
2 and 3, respectively, the rest of the qubits 1,4 together with the outcomes of measurements on qubits 2, 3 would provide
a readout of the gate operations. See the Supplemental Material for detailed discussions [21]. In addition to releasing the
assumptionsaboutthemeasurementdevices,genuinemultipartitesteerabilitypromiseshigh-qualityone-waycomputation(see
Fig. 3).
a demonstration of the first experimental genuine four- A , the maximum value of W is
s 4
partite EPR steerability for states close to the cluster
1
state G . In our scenario, we assume that two pos- W max W =1+ 1.7071,
sible m| e4aisurements can be performed on each particle 4C ≡As,{va}C,{vb}QM 4 √2 ∼
(mk =1,2 for the kth particle) and that eachlocalmea- whereAs denotestheindexsetofAs forallpossiblepar-
surement has two possible outcomes, vk(mk) ∈{0,1}. We titionsand{va}C indicatesthattheoutcomesetforAsis
take the measurements for each party who implements derived from such a preexisting-state scenario. The out-
quantum measurements to observables with the nonde- come set vb QM of Bs is obtained by performing quan-
{ }
generate eigenvectors 0 = 0 , 1 = 1 for tum measurements on the preexisting quantum states.
{| ik,1 | ik | ik,1 | ik}
mk =1 and {|0ik,2 =|+ik,|1ik,2 =|−ik} for mk =2. Hence we posit that if anexperimental state ρexpt shows
The genuine four-partite EPR steerability of the ideal that
cluster state G4 is revealed by the following relations 1
of measureme|nt riesults: v1(1)+v2(2)+v3(1) =. 0 and v3(1)+ W4(ρexpt)>1+ √2, (1)
v(2) =. 0,andtherelation: v(1)+v(2)+v(1) =. 0andv(2)+
4 2 3 4 1
v(1) =. 0, where =. denotes equality modulo 2. When A then ρexpt can exhibit genuine four-partite EPR steer-
2 s ability close to G . This certification rules out all the
andBs shareastate|G4i,As cansteerthestatesofBs’s possibilities of r|es4uilts mimicked by tripartite steerabil-
particles by measuring the particlesheld as described by
ity, including all possible mixtures of them. States certi-
the above relations whatever the partition A and B is
s s fiedby this witness willenable the quantumprotocolsto
considered. Thus we construct the kernel of quantum
beimplementedevenwhenuncharacterizedmeasurement
steering witness in the form
apparatusareunavoidablyused(Fig.1). Itis alsoworth
W P(v(1)+v(2)+v(1) =. 0,v(1)+v(2) =. 0) noting that the steering witness (1) is efficient. Only the
4 ≡ 1 2 3 3 4 minimum two local measurement settings are sufficient
+P(v(1)+v(2)+v(1) =. 0,v(2)+v(1) =. 0), to show the steerability.
2 3 4 1 2
To experimentally observe the genuine-four partite
where P() denotes the joint probability of obtaining ex- EPR steerability, we utilize the technique developed in
·
perimental outcomes satisfying the designed conditions. the previousexperiment[22]togenerateasourceoftwo-
If B has a preexisting state known to A , rather than photon four-qubit cluster states entangled both in po-
s s
part of a genuine multipartite entanglement shared with larization and spatial modes. As illustrated in Fig. 2,
3
UV laser
The measured witness kernel W (ρ ) can also re-
PBS
4 expt
vealtheinformationaboutthestatefidelity, F (ρ )=
S expt
LB
Tr[G4 G4 ρexpt]. To derive such connection, let us
| ih |
L first represent the witness (1) in the operator form,
A
BBO
Wˆ W Iˆ ˆ , where Iˆ denotes the identity op-
HWP
RB
IF
eraGt4or≡and4Cˆ4 c−orWre4spondsto W4 suchthatW4(ρexpt)=
RA
Tr[ ˆ ρ W]. If Wˆ = Tr[Wˆ ρ ] < 0, then ρ
W4 expt h G4i G4 expt expt
QWP
is genuinely four-partite steerable. One can use WˆG4 to
derive a steering witness operator of the form Wˆ′
G4 ≡
W /2Iˆ G G [21], which provides a steering wit-
4C 4 4
−| ih |
FIG. 2. (color online). Experimental setup. A pulse (5 ps) ness FS(ρexpt) > W4C/2 ( 0.8536). They satisfy the
∼
of UV light with a central wavelength of 355 nm and an av- relation Wˆ′ γWˆ 0, where γ is some positive
erage power of 200 mW at repetition rate of 80 MHz dou- constant, wGh4ic−h meGan4s≥that when a state is detected
blepassesatwo-crystalstructuredBBOtoproducepolariza- by Wˆ′ , it is certified by Wˆ as well. Such a re-
tion entangled photon pairs either in the forward direction G4 G4
or in the backward direction. To create desired entangled lation can be manipulated to give G4 G4 ˆ4/2,
| ih | ≤ W
pairsinmodeRA,RB andinLA,LB,twoquarterwaveplates i.e., to provide the upper bound of the state fidelity,
(QWPs) are properly tilted along their optic axis. Half wave F (ρ ) W (ρ )/2. Similarily, we use Wˆ′ to
plates (HWPs), polarizing beam splitters (PBSs), and eight coSnsterxupctt a≤noth4er esxtpetering witness operator comGpo4sed
single-photondetectorsareusedaspolarization analyzersfor of ˆ [21] and derive the lower bound of F (ρ ):
4 S expt
theoutputstates. Here3-nmbandpassfilters(IFs)withcen- W
ˆ Iˆ G G . Hence, F (ρ ) is estimated as
tral wavelength 710 nm are placed in front of them. W4− ≤| 4ih 4| S expt
0.8829 0.0049 F (ρ ) 0.9415 0.0025. (4)
S expt
an ultraviolet (UV) pulse passes twice through two con- ± ≤ ≤ ±
tiguous type-I-β barium borate (BBO) to produce po-
Such characteristic of genuine four-partite steering
larization entangled photon pairs in the forward (spa-
serves as a source for implementing faithful one-way
tial modes R ) and the backward (L ) directions.
A,B A,B
quantum computing. We have realized the quantum
Through temporal overlaps of modes R and L and of
A A
gates illustrated in Fig. 1 [21] and objectively evaluated
modes R and L , one can create a four-qubit state
B B
their performance and connection with steerability. We
G′ = 1 (H H + V V ) R R use W4′(ρexpt) (3) to estimate three different fidelities
| 4i 2 | iA| iB | iA| iB | iA| iB [24]: their averagecomputation fidelity (F ) [25], the
(cid:2) comp
+(H H V V ) L L ,(2) quantum process fidelity (F ) [26], and the average
| iA| iB −| iA| iB | iA| iB process
(cid:3) state fidelity (Fav) [26]; see Fig. 3. To connect steer-
entangled both in spatial modes and horizontal(H) and
ability with the gate operation shown in Fig 1(b), we
vertical (V) polarizations. By encoding logical qubits
first construct a steering witness operator of the form
as |H(V)iA(B) ≡ |0(1)i1(2) and |R(L)iA(B) ≡ |0(1)i3(4), WˆCZ W4CIˆ ˆCZ [21],wherethe operator ˆCZ spec-
G′ is equivalentto the cluster state G up to atrans- ≡ −W W
|for4miation Hˆ Hˆ where Hˆ 0(1) | =4i+( ) . The ifies the relation between ideal input and output states
1 ⊗ 4 k| ik | − ik of the quantum circuit [27]. This means the steerability
witness kernel for this target state has a correspond-
can be verified by performing one-way computation to
ivn(g2)c+havn(1g)e=.in0m,ve(a1)su+revm(1e)n=t. s0e)tt+inPgs(vb(y1)W+4′v(≡2)P+(vv(1(22)) +=. uchseedcktohWeˆsCtiZmia>teWth4Ce.loFwoelrloawnidngupthpeersabmouenmdsetohfoFda(ρsthat)
2 3 3 4 2 3 4 S expt
0,v(1)+v(1) =. 0). [21], we get W (ρ ) 1 ˆ W (ρ )/2+1.
1 2 4 expt − ≤ hWCZi ≤ 4 expt
In the experiment, we obtain a high generation rate Hence, with the relation F = ˆ /2 [27] and the
comp CZ
of cluster state about 1.2 104 per sec with 200 mW estimation of ˆ , we arrive athWthe isteering witness
UVpump. We measurethe×twojointprobabilitiesinthe F >W /h2WaCnZdithe fidelity estimation
comp 4C
witness kernel with the designed measurement settings
[23] and have P(v(2)+v(2)+v(1) =. 0,v(1)+v(1) =. 0)= 1 1
1 2 3 3 4
0.9490 0.0022andP(v(1)+v(2)+v(2) =. 0,v(1)+v(1) =. W4(ρexpt)−1≤Fcomp ≤ 4W4(ρexpt)+ 2. (5)
± 2 3 4 1 2
0) = 0.9339 0.0027. Then we observe genuine four-
±
partite steerability verified by F and F are further determined by F [24].
process av comp
W′(ρ )=1.8829 0.0049. (3) The concept and method of the criterion (1) can be
4 expt ± directly applied to quantum states with complex struc-
This result is clearly larger than the maximum value tures. One of the extensions is to certify steerability of
the preexisting-statescenariocanachieve, W , by 36 a general d-dimensional N-partite and q-colorablegraph
4C
∼
standard deviations. state G [18–20] [Fig. 1(a)]. The kernel of the steering
| i
4
(a)! (b)! (c)!
1
1
1.00
0.8
0.8
0.98

Fidelity 00..64

Fidelity00..64

elity 000...999246
0.2
0.2
d
Fi 0.90
0
0
0.88
++ 0− −1+ ++ −−
+− 0− +1+ +− +− 0.86
−+ 0+ −1− −+ −+ 0.84
−− 0+ +1− −− ++ Fcomp
Fprocess
Fav
FIG.3. (coloronline). Steerabilityandperformanceofquantumgates. Twokindsofgateoperationsareexperimentallyrealized
forthetargetquantumgates[Figs.1(b),1(c)]. Wemeasurethefidelitiesoftheoutputstatesbyinputtingfourorthogonalstates
into the experimental gates, mn for m,n= +, [24]: (a) a mean fidelity 0.935 0.004 for the gate operation, Fig. 1(b),
| i − ∼ ±
and (b) an average fidelity 0.960 0.004 for the target gate, Fig. 1(c). (c) From the estimate of the average computation
∼ ±
fidelityF (5),weobtainthelowerboundsofthequantumprocessfidelityF ( 0.9415 0.0025) andtheaveragestate
comp process
∼ ±
fidelity F ( 0.9532 0.0020), which indicate good qualities of our experimental gates, regardless of the input states [24].
av
∼ ±
Since both created gates are based on thesame source, their estimations of the threedifferent fidelities are identical [21]. The
gate performance reveals genuinefour-partite steerability bythesteering witness Fcomp >W4C/2 0.8536. Thedouble-arrow
∼
dashed line shows theregion where thesource states are truly four-partite steerable.
witness is of the form applicationsandexperiments. Onecanprobemoresorts
of steerability, for example, in resonating valence-bond
q
W (q,d) P(v(2)+ v(1) =. 0 j Y ), states, which would allow an analogue quantum simula-
N ≡mX=1 j i∈Xǫ(j) i |∀ ∈ m tor[15]torunwithoutfullycharacterizedquantummea-
surements. Similarly, other quantum strategies basedon
for v(1),v(2) 0,1,...,d 1 measured in two comple- both characterizedmeasurements and genuine multipar-
i j ∈ { −. }
mentary bases [21], where = denotes equality modulo d titeentanglement,likequantummetrology[16],canben-
and ǫ(j) represents the set of vertices that form edges efit from it as well. Moreover, since genuine multipar-
with the vertex j in the color set Y . The quantum titeEPRsteerabilitycannotbemimickedbypartsofthe
m
steering witness is described by whole system, such ability together with steering wit-
ness could facilitate multipartite secret sharing [14] in a
1 2(q2 2q+2)+γ generic one-sided device-independent mode [6, 11]. It is
q
W (q,dρ )> (q+ − ), (6)
N | expt 2 r d interesting to compare our method with the recently de-
veloped single-system steering for quantum information
where γ = 0 and γ = [2(q 3)+1]+γ for q 3
2 q − q−1 ≥ processing [32] and to investigate further from the all-
[21]. If ρ is detected by Eq. (6), then ρ possesses
expt expt versus-nothing (AVN) point of view. Subtle AVN proof
genuine N-partite EPR steerability close to G . This
| i for steering and their experimental demonstration are
witness has the same features as the witness (1) [28] and
given for special classes of two qubits [33, 34].
possesses high robustness against noise [21]. With only
q localmeasurementsettings, W (q,d)canbe efficiently
N
realized regardless of the number of qudits. We remark We thank Q.-Y. He, J.-L. Chen, and Y.-C. Liang for
that, for states that do not belong to the above state discussions. This work has been supported by the Chi-
types,forexample,theW states[29],usefulsteeringwit- nese AcademyofScience, the NationalFundamentalRe-
nessesstillcouldbederivedfromone’sknowledgetothis searchProgram,andtheNationalNaturalScienceFoun-
target state [21]. For more extensions, how to observe dation of China. C.-M. L. acknowledges partial support
EPRsteeringinall DOFs [30,31]under considerationis from the Ministry of Science and Technology, Taiwan,
shown in the Supplemental Material. The concrete ex- under Grant No. MOST 101-2112-M-006-016-MY3. Y.-
perimental illustrations andapplications of such genuine N. C. acknowledges partial support from the National
high-order EPR steering are also detailed therein [21]. Center for Theoretical Sciences and the Ministry of Sci-
In conclusion, we have developed a novel formalism ence and Technology, Taiwan, under Grant No. MOST
to explore genuine high-order EPR steering and experi- 103-2112-M-006-017-MY4.
mentally demonstrated such generality and applications
with photonic cluster states. Being capable of reveal-
ing genuine high-order EPR steering pushes beyond the Note added.— Recently, we became aware of two ex-
capability of bipartite steering and promotes potential perimental demonstrations of tripartite steering [35, 36].
5
α-phase shifter, measurement B(α) can be realized for
k
one-way quantum computing [Figs. 1(d),(e) and [21]].
Whereas for the measurement mk = 1, they are left
∗ [email protected] empty. If a PBS is placed at the mode intersection, one
† [email protected] can realize the measurements m1,2 = 1 and m3,4 = 1
‡ [email protected] simultaneously.
[1] E. Schr¨odinger, Naturwissenschaften 23, 807 (1935); 23 [24] Here, F is considered by using two complementary
comp
823 (1935); 23 844 (1935). sets of input states [25]. F determines the similar-
process
[2] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, ity between the ideal and experimental process of quan-
777 (1935). tumgates[26].F isthequantumstatefidelitybetween
av
[3] H. M. Wiseman, S. J. Jones, and A. C. Doherty, Phys. the ideal and experimental gate outputsuniformly aver-
Rev.Lett. 98, 140402 (2007). aged over all input states on a state space [26]. There
[4] E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. exist useful relations between them. F can be effi-
process
D.Reid, Phys.Rev. A 80, 032112 (2009). cientlyestimatedbyF [25]: F 2F 1.In
comp process comp
[5] D.J.Saunders,Jones,H.M.Wiseman,andG.J.Pryde, addition, F can be represented in term≥s of F − by
av process
Nat.Phys. 6, 845 (2010). F =(MF +1)/(M +1) [26], where M is the di-
av process
[6] C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. mension of a quantumgate. Henceone can useF to
comp
Scarani, and H. M. Wiseman, Phys. Rev. A 85, estimateboththefidelitiesF andF bytheabove
process av
010301(R) (2012). connections [21].
[7] D.-H.Smith,G.Gillett,M.P.deAlmeida,C.Branciard, [25] H. F. Hofmann, Phys. Rev.Lett. 94, 160504 (2005).
A.Fedrizzi,T.J.Weinhold,A.Lita,B.Calkins,T.Ger- [26] A. Gilchrist, N. K. Langford, and M. A. Nielsen, Phys.
rits,H.M.Wiseman,S.W.Nam,andA.G.White,Nat. Rev. A 71, 062310 (2005).
Commun. 3, 625 (2012). [27] The kernel of the witness operator Wˆ is designed as
CZ
[8] fBo.rdW,iNtt.mBanrunn,nSe.rR,aHm.elMow.,WF.isSemteainnl,ecRhn.eUr,rNsin.,Ka.nLdanAg-. WˆCZ = P(α,β)∈{0,π},{±π2}|α+i22hα+| ⊗ |β+i33hβ+| ⊗
Zeilinger, New J. Phys. 14, 053030 (2012). UˆCZ(cid:12)(cid:12)Inαβ(cid:11)(cid:10)Inαβ(cid:12)(cid:12)UˆC†Z, where UˆCZ = (Hˆ ⊗ Hˆ)UCZ
[9] P.Walther,K.J.Resch,T.Rudolph,E.Schenck,H.We- and UCZ is the two-qubit controlled-Z gate. Here,
itnufruert(eLro,nVd.oVn)ed4r3a4l,,M16.9A(s2p0e0l5m);eyRe.rPanredveAd.eZl,ePil.inWgearl,thNear-, |inαp(βu)t+sitkate=(cid:12)(cid:12)In(αβ|0(cid:11)ik=+|−eiαα+(βi)2|1⊗ik|)−/√β+2i3defotrermthienetsartgheet
F. Tiefenbacher, P. B¨ohi, R. Kaltenbaek, T. Jennewein, gate and shows the one-to-one correspondence be-
and A. Zeilinger, ibid 445, 65 (2007). tween (cid:12)(cid:12)Inαβ(cid:11) and the output state UˆCZ(cid:12)(cid:12)Inαβ(cid:11). For the
[10] G.Vallone,E.Pomarico,P.Mataloni,F.DeMartini,and present quantum circuit, Fcomp is defined by Fcomp
≡
Vlo.neB,eEra.rdPio,mPahryicso.,RFev..DLeeMtta.r9ti8n,i,18a0n5d02P.(2M0a0t7a)l;oGni.,Vibaild- w1/h8ePre(α,β)∈{d0e,πn}o,{te±sπ2}th(cid:10)eInαβex(cid:12)(cid:12)pUˆeC†rZimECeZn(t(cid:12)(cid:12)aIlnαβg(cid:11)at(cid:10)eInoαβp(cid:12)(cid:12))eUrˆaCtZio(cid:12)(cid:12)nIns.αβ(cid:11),
CZ
100, 160502 (2008). If the Eideal probability P(α ,β ) = 1/4 is assigned
+ +
[11] Q.Y. Heand M. D.Reid, Phys.Rev. Lett.111, 250403 to all the settings of (α,β), we have the relation
(2013). F = ˆ /2. See Supplemental Material [21] for
[12] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, decotmaipled dhisWcuCsZsiions.
5188 (2001). [28] Fourfeaturesandimplicationsofthesteeringwitness(1)
[13] R. Raussendorf, D. E. Browne, and H. J. Briegel, Phys.
are summarized as follows. First, the states detected by
Rev.A 68, 022312 (2003).
this witness enable the quantum protocols to be im-
[14] M. Hillery, V. Buˇzek, and A. Berthiaume, Phys. Rev. A
plemented even when untrusted measurement appara-
59, 1829 (1999). tus are unavoidably used (Fig. 1). Second, the state fi-
[15] XPh.-ySs..M7,a3,9B9.(2D0a1k1i)c., W. Naylor,and A. Zeilinger, Nat. adneldityF,SF(ρSe(xρpetx)pat)l,socaanctsbeasesatniminadteicdatforormshWow4i(nρgexgpetn)u(i4n)e,
[16] V. Giovannetti, S. Lloyd, and L. Maccone, Nature Pho-
multipartite steerability. Third, W (ρ ) can be used
ton. 5, 222 (2011). 4 expt
to estimate the computation fidelity of one-way compu-
[17] R. Raussendorf, J. Harrington, and K. Goyal, New J. tation (5), and the steerability can be verified by per-
Phys.9, 199 (2007). forming one-way computation as well. Finally, only the
[18] H. J. Briegel and R. Raussendorf, Phys. Rev. Lett. 86, minimum two local measurement settings are sufficient
910 (2001). to efficiently show the steerability.
[19] D. L. Zhou, B. Zeng, Z. Xu, and C. P. Sun, Phys. Rev. [29] W. Du¨r, G. Vidal, and J. I. Cirac, Phys. Rev. A 62,
A 68, 062303 (2003).
062314 (2000).
[20] M. Hein, J. Eisert, and H. J. Briegel, Phys. Rev. A 69, [30] P. G. Kwiat, J. Mod. Opt.44, 2173 (1997).
062311 (2004). [31] J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G.
[21] See Supplemental Material for complete discussions, de- Kwiat, Phys. Rev.Lett. 95, 260501 (2005).
tailed derivations, and concrete illustrations..
[32] C.-M. Li, Y.-N. Chen, N. Lambert, C.-Y. Chiu and F.
[22] K. Chen, C.-M. Li, Q. Zhang, Y-A. Chen, A. Goebel,
Nori, arXiv:1411.3040 (2014).
S. Chen, A. Mair, and J.-W. Pan, Phys. Rev. Lett. 99,
[33] J.-L.Chen,X.-J.Ye,C.Wu,H.-Y.Su,A.Cabello, L.C.
120503 (2007). Kwek, and C. H.Oh, Sci. Rep.3, 2143 (2013).
[23] Weusethreedifferentsetupstoimplement spatialmode [34] K. Sun, J.-S. Xu, X.-J. Ye, Y.-C. Wu, J.-L. Chen, C.-F.
analysis. Whenperforming themeasurement mk =2for Li, and G.-C. Guo, Phys.Rev. Lett.113, 140402.
k = 3 (4), a beam splitter is set at the intersection of
[35] S. Armstrong, M. Wang, R. Y. Teh, Q. Gong, Q. He,
modes R and L . Furthermore, combined with
A(B) A(B)
6
J. Janousek, H.-A. Bachor, M. D. Reid, P. K. Lam, [36] D.Cavalcanti,P.Skrzypczyk,G.H.Aguilar,R.V.Nery,
arXiv:1412.7212. P.H.SoutoRibeiro,andS.P.Walborn,arXiv:1412.7730.
Genuine High-Order Einstein-Podolsky-Rosen Steering: Supplementary Material
Che-Ming Li1,∗ Kai Chen2,3,† Yueh-Nan Chen4, Qiang Zhang2,3, Yu-Ao Chen2,3, and Jian-Wei Pan2,3‡
1Department of Engineering Science, National Cheng Kung University, Tainan 701, Taiwan
2Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
3Shanghai Branch, CAS Center for Excellence and Synergetic
Innovation Center in Quantum Information and Quantum Physics,
University of Science and Technology of China, Shanghai 201315, China and
4Department of Physics and National Center for Theoretical Sciences,
National Cheng-Kung University, Tainan 701, Taiwan
(Dated: July 2, 2015)
In this supplementary information, we provide the required material to support our theoretical
5
andexperimentalresultspresentedinthemaintext. Novelapplicationsofourtheoreticalschemeto
1
0 theexisting experimentsare supplementedaswell. Firstly, wedescribe indetail thebasicprinciple
2 ofourexperimentalquantumgates. Then,weprovideacompleteproof oftheintroducedquantum
steeringwitnessforgenuinemultipartiteEinstein-Podolsky-Rosen(EPR)steerabilityofstatesclose
l
u toad-dimensional N-partite andq-colorable graph state. The robustnessof thesteering witness is
J discussed. Aconcrete illustration ofgenuineN-degree-of-freedom EPRsteering andcertification of
suchgenuinehigh-orderEPRsteeringareintroducedindetail. Inaddition,weprovethatthestate
1
fidelity of a generated state with respect to the target state FS(ρexpt) can be estimated from the
] measuredsteeringwitnesskernel,andfurthermore,thatFS(ρexpt)canserveasanindicatorshowing
h genuinemultipartitesteerability. Wealso showthat thesteerability can bedetectedbyperforming
p one-way quantum computing and that how we use the measured witness kernel to estimate the
- averagecomputation fidelity(F ),thequantumprocessfidelity(F ),andtheaveragestate
t comp process
n fidelity (F ). Finally, a method of constructing quantum steering witnesses based on full state
av
a knowledge is introduced in theend.
u
q PACSnumbers: 03.65.Ud,03.67.Lx,42.50.Dv
[

3
v
2
5
4
1
0
.
1
0
5
1
:
v
i
X
r
a
2
Experimentalquantumgates.—Foragivenclusterstate,aquantumcomputingisdefinedbyconsecutivesingle-qubit
measurements in basis B(α) = α , α and their results of measurements, where α =(0 eiα 1 )/√2
k {| +ik | −ik} | ±ik | ik± | ik
foranyrealα[1,2]. Ameasurementoutcomeof α denotess =0while α signifiess =1. Thismeasurement
| +ik k | −ik k
basis determines a gate operation R(α) = exp( iαZ/2) where Z 0 0 1 1 [3], followed by a transformation
z
Hˆ. A chain-type graph (horseshoe cluster) stat−e G [Fig. 1(d) in≡th|eimh a|i−n t|eixht]|can be used to realize a two-qubit
4
| i
controlled-Z gate(U ) [Fig.1(b)]: U j k =( 1)jk j k , for j,k =0,1. When measuringalongbasesB(α) and
CZ CZ| i| i − | i| i 2
B(β), the state of the qubits 1 and 4 would be
3
(Xs2 Xs3)(Hˆ Hˆ)U R(−α) R(−β) + + .
⊗ ⊗ CZ z ⊗ z | i| i
Let us focus on measurements with the outcome s = 0 and s = 0. It is easy to find that, for such a case, the
2 3
post-measurement state of the 2nd and the 3rd qubits, α β , determines the input state and the connection
| +i2⊗| +i3
between input and output states of the quantum gate (Hˆ Hˆ)U by
CZ
⊗
Outα =(Hˆ Hˆ)U Inα , (S1)
β ⊗ CZ β
(cid:12) (cid:11) (cid:12) (cid:11)
where (cid:12) (cid:12)
Inα =R(−α) R(−β) + + = α β . (S2)
β z ⊗ z | i| i |− +i|− +i
(cid:12) (cid:11)
Then,onecaninputdifferentstates(cid:12)Inα intothegatetoseetheeffectofthegateoperation(Hˆ Hˆ)U byanalyzing
β ⊗ CZ
the output states Outα . For exam(cid:12)ple(cid:11)s, one can set the angles (α,β) as (0,0), (0,π), (π,0) and (π,π) to prepare
β (cid:12)
an orthonormal se(cid:12)t of th(cid:11)e input states + + , + , + , and , respectively. Here, for (α,β) 0,π ,
(cid:12) | i| i | i|−i |−i| i |−i|−i ∈ { }
the output states are entangled [Fig. 3(a)]. One can also design another orthonormal set of input states which is
complementary to the above set of input states [4]. As (α,β) are chosen as ( π/2, π/2), ( π/2,π), (π/2, π/2)
− − − −
and (π/2,π/2), we have +i +i , +i i , i +i , and i i , respectively, where i =(0 i 1 )/√2.
One can analyze the ga|teio|periat|ioniU|− i(Hˆ|− iHˆ|)Ui rea|l−izeid|−inithe one-way mode [Fi|g±s.i1(c),|(ei)]±by|foillowing the
CZ CZ
samemethodasthatusedforthetargetgate(H⊗ˆ Hˆ)U . FortheboxclusterstateshowninFig.1(e),measurements
CZ
⊗
on the qubits 2,3 along the basis B(α) and B(β), respectively, will give an output state of the qubits 1,4 with
2 3
(Z X)s3(X Z)s2U (Hˆ Hˆ)U R(−α) R(−β) + + .
⊗ ⊗ CZ ⊗ CZ z ⊗ z | i| i
For the cases where s = 0 and s = 0, the post-measurement state of the 2nd and the 3rd qubits, α β ,
2 3 | +i2⊗| +i3
shows the connection between input and output states of the target quantum gate by
Outα =U (Hˆ Hˆ)U Inα ,
β CZ ⊗ CZ β
(cid:12) (cid:11) (cid:12) (cid:11)
where Inα = α β . One can de(cid:12)sign two orthonormal sets(cid:12)of input states which are complementary each
β |− +i|− +i
other, f(cid:12)or e(cid:11)xamples, the states correspond to (α,β) 0,π and (α,β) π/2, π/2 illustrated above to analyze
(cid:12) ∈ { } ∈ { − }
the gate operation.
To demonstrate the quantum gate (Hˆ Hˆ)U in the one-way realization, we have created states close to G′
[Eq.(2)inthe maintext],whichis equival⊗entto tChZe clusterstate G upto atransformationHˆ Hˆ . Furtherm|or4ei,
4 1 4
| i ⊗
with the wave plate sets together with BS and PBS (Fig. 2), we perform the required measurements B(α) and B(β)
2 3
to prepare different input states of the target gate operation as describe above. See Ref. [23] in the main text.
Similarly, our created states can be directly used for implementing a quantum circuit composed of two controlled-Z
gates [Fig. 1(c)], since the box-cluster state [Fig. 1(e)] needed for realizing such a quantum circuit distinguishes only
the state G′ from a transformation Hˆ on every qubit and swap between qubits 2 and 3.
| 4i
Proof of quantumsteeringwitnesses.—Aspresentedinthemaintext,thekernelofthesteeringwitnessforageneral
d-dimensional N-partite and q-colorable graph state G is of the form
| i
q
W (q,d) P(v(2)+ v(1) =. 0 j Y ), (S3)
N ≡ j i |∀ ∈ m
mX=1 i∈Xǫ(j)
for v(1),v(2) v 0,1,...,d 1 , where =. denotes equality modulo d and ǫ(j) represents the set of vertices that
i j ∈ ≡ { − }
form edges with the vertex j in the color set Y . For examples, since the state G corresponds to a chain-type
m 4
| i
3
two-color graph (q = 2 and d = 2), the corresponding witness kernel W [see Eq. (1) in the main text] then is a
4
concrete illustration of the above representation. For the measurement setting, we design the first and second kind
measurements for each party who implements quantum measurements to observables with the eigenvectors
v(1) = v(1) , and v(2) =Fˆ† v(2) , (S4)
{(cid:12) k Ek,1 (cid:12) k Ek} {(cid:12) k Ek,2 k(cid:12) k Ek}
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
respectively. These two bases are complementary to each other, i.e., v(1) v(2) = 1/√d for all v(1),v(2) v.
|k,1h k | k ik,2| k k ∈
Here, Fˆk is the quantum Fourier transformation defined by Fˆk(cid:12)vk(m)Ek =1/√d dv−=01ωvk(m)v|vik [3].
ThewitnesskernelW (q,d)isdesignedaccordingtothestate(cid:12)vectorofthetaPrgetgraphstate G . AtrulyN-qudit
N (cid:12) | i
graphstate[5–7]canberepresentedbyafully-connectedgraphG(V,E)[Fig.1(a)]. ThegraphGconsistsofthesetV
of vertices with cardinality V =N, representing the qudits, and the set E of edges each of which joins two vertices,
| |
representing interacting pairs of qudits of the graph state. A graph is called a q-colorable graph if the vertices of the
graphG can be divided into q sets,say Y for m=1,2,...,q, andthe vertices ofeachset are givena color suchthat
m
adjacentverticeshavedifferentcolors. Anedge,(i,j) E,correspondstoanunitarytwo-qudittransformationamong
∈
the two qudits (vertices) i and j by U = d−1 v v (Z )v, where v is an orthonormal basis for the ith
(i,j) v=0| iiih |⊗ j {| ii}
qudit and the operator Z for the jth qudit iPs defined by Z = d−1ωk k k , ω =exp(i2π/d). Note that Z =Z
j j k=0 | ijjh | j
for d = 2. An explicit presentation of the state vector of the gPraph G can be generated by applying the operators
U based on G to an initial state f = N Fˆ 0 : G = U f . It is worth noting that all the
(i,j) | 0i k=1 k| ik | i (i,j)∈E (i,j)| 0i
information about the probability amplitudeNs of G has been consQidered in the steering witness kernel WN(q,d).
| i
For the general preexisting-state scenario, the untrusted measurement devices of A can declare random variables
s
instead of results obtained from quantum measurements on parts of the all qudits of the genuinely multipartite
entangledstate. Thenthe extreme valuesofthe witnesskernelW (q,d) isdeterminedbythe followingmaximization
N
for such a situation:
W (q,d) max W (q,d), (S5)
NC N
≡As,{va}C,{vb}QM
where A denotes the index set for the subsystem A for all possible bipartitions of the N-partite system, v
s s a C
represents the set of declared random variables from A , and v is the set of outcomes derived from{ B}’s
s b QM s
{ }
quantum measurements on preexisting quantum states. Considering the concrete steering witness kernel W (q,d)
N
(S3) for generic q-colorable N-qudit graph states and performing the above maximization, we have
W (q,d)=maxλ[Fˆ† 0 0 Fˆ +(q 1) 0 0]
NC λ k| ikkh | k − | ikkh |
1 2(q2 2q+2)+γ
q
= (q+ − ), (S6)
2 r d
where λ[U] denotes the eigenvalues of the operator U, γ = 0 and γ = [2(q 3)+ 1]+γ for q 3. If the
2 q q−1
− ≥
experimental state ρ provides a measured witness kernel W (q,dρ ) satisfying
expt N expt
|
W (q,dρ )>W (q,d),
N expt NC
|
then ρ possesses genuine N-partite Einstein-Podolsky-Rosen (EPR) steerability close to G . Hence, we get the
expt
| i
quantum steering witness (6) shown in the main text.
Robustness of quantum steering witnesses.—To investigate the robustness of our steering witness, we consider that
a pure state G is mixed with white noise to be
| i
p
ρ(p )= noiseIˆ+(1 p ) G G , (S7)
noise dN − noise | ih |
where Iˆdenotes the identity operator and p is the probability of uncolored noise. The mixed state ρ(p ) is
noise noise
identified to possess truly high-order EPR steerability by the steering witness (6) if W (q,dρ(p )) >W (q,d).
N noise NC
|
FigureS1 depicts the thresholdprobabilityofuncolorednoisep whichgivesW (q,dρ(p))=W (q,d). We consider
N NC
|
p asanindicatorofrobustnessofthe steeringwitness. For largedimensionalityd, the criterionisveryrobustandthe
noise tolerance is up to p <1/2,independent of the number of qudits and the types of graphstates. In addition,
noise
4
(a)
GHZ states
(b)
cluster states
40

d10

d10
els
9
els
9
35
ev
8
ev
8
dit l
7
dit l
7
30
qu
6
qu
6
of
5
of
5
25
er
4
er
4
mb
3
mb
3
20
Nu
2
Nu
2

3 4 5 6 7 8 9 10

3 4 5 6 7 8 9 10
p
%15

Number of qudits N
Number of qudits N
FIG. S1. (color online). Robustness of steering witness (6). If the probability of white nose p < p [see Eq. (S7)], then
noise
the genuine multipartite steering can be certified by the steering condition (6). Here the threshold of noise intensity p is an
indicatorshowingthenoisetoleranceorrobustnessofthesteeringcriterion. Concreteexamplesofrobustnessofsteeringwitness
for two-color (q =2) star (a) and chain graph states (b) are illustrated. For large d, the condition (6) is robust against noise
up to p=50% for both cases.
as illustrated by detecting states close to two-color (q = 2) star [Greenberger-Horne-Zeilinger (GHZ) state] (a) and
chain (cluster state) graph states (b), our criterion is also robust for the cases of finite d.
Experimental state fidelities F (ρ ).—The steering witness (S3) can be represented in the operator form:
S expt
Wˆ W (q,d)Iˆ ˆ (q,d), (S8)
G NC N
≡ −W
where ˆ (q,d) is the operator presentation of the witness kernel W (q,d) (S3) such that W (q,dρ ) =
N N N expt
W |
Tr[ ˆ (q,d)ρ ]. If Wˆ (ρ ) = Tr[Wˆ ρ ] < 0, then ρ is truly multipartite steerable and close to
N expt G expt G expt expt
W D E
the target graph state G . We use Wˆ to construct a second witness operator for detecting steerability. Let us
G
assumethatthiswitness|opieratorisoftheform, Wˆ′ δ Iˆ G G. Theunknownparameterδ canbedetermined
G ≡ G −| ih | G
by showing
Wˆ′ γWˆ 0, (S9)
G− G ≥
where γ is some positive constant. This relationmeans that when a state is detected by Wˆ′ , it is certified by Wˆ as
G G
well. When setting γ as 1/2, the above relation holds to give δ = W (q,d)/2. Hence, from the steering witness
G NC
operator Wˆ′, we get the criterion of genuinely N-partite steering on the state fidelity F (ρ )=Tr[G G ρ ]
G S expt | ih | expt
1 2(q2 2q+2)+γ
q
F (ρ )> (q+ − ). (S10)
S expt 4 r d
For q =2, such as GHZ (star graph) and cluster states, the criterion reads F (ρ )>1/2(1+1/√d). Furthermore,
S expt
as d=2 is considered for the experimental state, it provides the steering witness
1 1 1
F (ρ )> (1+ )= W 0.8536,
S expt 4C
2 √2 2 ∼
as introduced in the main text for the target state G .
4
| i
The steering witness in terms of F (S10) is especially useful when one already has the state fidelity information
S
obtained from experiments. One is allowed to evaluate the existing experimental results for steerability, while they
didnotmeasurethesteeringwitnesskernelbefore. Forinstances,theN-qubitentangledionsforN =2,3,..,6created
in the experiment [8] can be identified as genuinely N-partite steerable when considering their fidelities. Similarly,
genuine tripartite steering can be confirmed in a superconducting circuit as well [9].
With the condition (S9), Wˆ′ Wˆ /2 0, we also have
G− G ≥
1
G G ˆ (q,d),
N
| ih |≤ 2W
whichimpliestheupperboundofthestatefidelity. Toderivethelowerboundoftheexperimentalstatefidelityfromthe
measuredwitness kernelW (q,dρ ), weuse the same approachasthatshownaboveto constructanothersteering
N expt
|