Einstein-Podolsky-Rosen Steerability Criterion for Two-Qubit Density Matrices

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by  Jing-Ling Chen

Einstein-Podolsky-RosenSteerability CriterionforTwo-QubitDensity Matrices
Jing-LingChen,1,2,∗ Hong-Yi Su,1,2 Xiang-Jun Ye,1 ChunfengWu,2 and C. H. Oh2,3,†
1Theoretical Physics Division, Chern Institute of Mathematics,
Nankai University, Tianjin 300071, People’s Republic of China
2CentreforQuantumTechnologies, NationalUniversityofSingapore, 3ScienceDrive2,Singapore117543
3Department of Physics, NationalUniversityof Singapore, 2Science Drive3, Singapore 117542
(Dated:January31,2012)
WeproposeasufficientcriterionS =λ1+λ2−(λ1−λ2)2<0todetectEinstein-Podolsky-Rosensteering
forarbitrarytwo-qubitdensitymatrixρAB. Hereλ1,λ2 arerespectivelytheminimalandthesecondminimal
eigenvalues of ρTABB, which isthe partial transpose of ρAB. Byinvestigating several typical two-qubit states
suchastheisotropicstate, Bell-diagonal state, maximallyentangled mixedstate, etc., weshow thiscriterion
2 worksefficientlyandcanmakereasonablepredictionsforsteerability. Wealsopresentamixedstateofwhich
1
steerabilityalwaysexists,andcomparetheresultwiththeviolationofsteeringinequalities.
0
2
PACSnumbers:03.65.Ud,03.67.-a

n
a
J In 1935, Einstein-Podolsky-Rosen (EPR) questioned the tialtransposematrixρTB. Welistsomeexamplestoshowthe
AB
8 completeness of quantum mechanics based on locality and utilityofourcriterion.
2 realism [1]. Soon after, Schro¨dinger [2] published a semi- SteerabilityCriterion.—Let λ1,λ2,λ3,λ4 befoureigen-
nalpaperdefiningthenotionofentanglementtodescribethe valuesofρTB inthesmall-to-la{rgeorder[ρTB}andρTA share
] AB AB AB
h correlationsbetweentwoparticles. Entanglement,aquantum thesameeigenvalues]. ThenthecriterionforEPRsteeringis
p statewhichcannotbeseparated,isindeedtheessentialentity givenby
- thatevaluateswhetheraquantuminformationprocessingcan
nt be accomplished in quantum level. The more entanglement S =λ1+λ2−(λ1−λ2)2 <0, (1)
a
is, themoreprowessoftheresourcehas. Variouscriteriafor when(1)issatisfied,thenEPRsteeringexists.
u
q quantitativewitnessesofentanglement [3–5]havebeenpro- Example1.—Thenonmaximalentangledstate
[ posed in recent decades. Generally speaking, entanglement
cos2θ 0 0 sinθcosθ
measuresaremostlyasfunctionsofdensityoperator.
4  0 0 0 0 
v EPR steering, like entanglement, was originated from ρ1 = 0 0 0 0 (2)
93 Ssihstreo¨ndcinygbeert’ws ereepnlyqutoantthuemEmPeRchpaanraicdsoaxntdolroecflaelcrteathliesmin,caonnd- sinθcosθ 0 0 sin2θ 
6
was formalizedby Wiseman, Jones, and Doherty [6]. In the withθ [0,π/4]alwaysviolatestheCHSHinequalityaswell
4 ∈
. steeringscenario,forapureentangledstateheldbytwosep- assteeringinequalitygivenin Ref.[9] exceptθ = 0. Inthis
2 aratedobserversAliceandBob,Bob’squbitcanbe“steered” case, we have λ sinθcosθ,sin2θ,sinθcosθ,cos2θ ,
1 ∈ {− }
into different ensembles of states although Alice has no ac- andthesteerabilitycriteriongives
1
1 cesstothequbit. AlicetriestoconvinceBobthattheyshare 1
: two systems in an entangled state. If the systems are actu- = sin2θ(1+2sin2θ), (3)
v S −2
allyentangled,quantummechanicspredictsthat,byperform-
i
X ingdifferentmeasurementsonhersystem,Alicecanremotely hencedetectsallthesteering.
Example2.—Theisotropicstate
r prepare different states for Bob’s system. EPR steering is
a
commonlydetectedbytheviolationofEPR-steeringinequali- 1
tiesintheformofcorrelations[7–16]. Althoughmanyefforts ρ2 = Vρ0+(1 V)
− 4
have been devoted to the investigationsof EPR steering, the 1+V 0 0 V
EPR-steering inequalities in the literatures are not effective  04 1−V 0 02 
enoughfortwo-qubitsystems. Therefore,itisnotpossibleto = 0 04 1−V 0 , (4)
observetheEPRsteeringforsomestates,especiallyformixed  V 0 04 1+V
states. For EPR steering, entanglement is necessary but not  2 4 
sufficient. Byresortingtopartialtransposeofdensityopera- whereρ0 = 21(|00i+|11i)(h00|+h11|)is themaximalen-
tor,entanglementcanbecertified. Thisisunderstandablethat tangled state, and 1 is the four-by-four identity matrix. It
the densityoperatorcontainsallthe informationof thestate. has been known that the state has the steering in the region
It is hence reasonableto anticipate a criterion based entirely V (1/2,1], and no steering in V [0,1/2]. In this case,
ondensitymatrixforEPRsteeringwitness. we∈have λ 1−3V,1+V,1+V,1+∈V , and the steerability
∈ { 4 4 4 4 }
Inthiswork,weproposeacriteriontodetectEPRsteering criteriongives
of an arbitrary two-qubit density matrix ρAB. The criterion 1
canbeobtainedfromtheconstraintsontheeigenvaluesofpar- = (2V 1)(1+V), (5)
S −2 −
2
hencedetectsthecriticalvalueVcr = 21. 1.0
Example3.—TheBell-diagonalstate
0.8
ρ3 = V Ψ+ Ψ+ +(1 V)χ+ χ+
| ih | − | ih | 0.6
V2 0 0 V2 F 2-settingandCHSH
0 1−V 1−V 0 0.4 3-setting
= 0 1−2V 1−2V 0 (6) 10-setting
 2 2  0.2
V 0 0 V
2 2
0.0
violatesthesteeringinequalityin Ref.[9] exceptV = 1. In
2 0.0 0.5 1.0 1.5
thiscase, wehaveλ V 1,1 V,1,1 , andthe steer-
∈ { − 2 2 − 2 2} Θ
abilitycriteriongives
= (1 2V)2, (7)
FIG.1: (Coloronline)Quantumpredictionsofsteeringinequalities.
S − −
Theregionabovethebluelineissteerabledetectedbythetwo-setting
whichrecoversthesameresult.
steeringinequalityaswellasCHSHinequality.Theregionabovethe
Example 4.—The nonmaximal entangle state with color greenandredlinesarerespectivelysteerabledetectedbythethree-
noise andten-settingsteeringinequalities.
V cos2θ+ 1−V 0 0 V sinθcosθ
2
 0 0 0 0  with g = 4/9 for γ [0,2√2/3]. It violates the 10-
ρ4 = (8) ∈
0 0 0 0 setting steering inequality in Ref. [9] for γ 0.2564. In
 V sinθcosθ 0 0 V sin2θ+ 1−V this case, we have λ 1 (1 1+3≥24γ2), 1 (1 +
 2  ∈ {36 − 36
1+324γ2),4/9,1/2 ,andthesteerpabilitycriteriongives
with θ [0,π/4] always violates CHSH inequality as well }
∈ p
as the steering inequality in Ref. [9] exceptV = θ = 0. In 17
Vthicsocsa2sθe),,w1e(1ha+veVλc∈os{2−θ)V,siannθdcothseθ,sVteesrianbθilcitoysθc,ri12te(r1io−n S = 324 −γ2, (14)
2 } whichpredictsthecriticalvalue
gives
√17
= V2sin22θ, (9) γcr = 0.2291. (15)
S − 18 ≃
whichrecoversthesameresult.
Example 5.—The maximally entangled mixed state Example7.—Thestate
(MEMS) 1−cosθF 0 0 sinθF
2 2
g(γ) 0 0 γ/2 ρ7 = 0 cosθF 0 0 , (16)
ρ5 = 00 1−20g(γ) 00 00 , (10)  sin20θF 00 00 1− 1+0c2osθF
γ/2 0 0 g(γ)
  withF [0,1]andθ [0,π/2].Theconcurrenceofthestate
∈ ∈
isgivenby
with g(γ) = 1/3 for γ [0,2/3] and g(γ) = γ/2 for
∈
γ [2/3,1]. It violates the 10-settingsteering inequality in
Re∈f. [9] for γ 0.6029. In the case of γ [0,2/3], we = 2F(1 F(1+cosθ) 1)sin2 θ, (17)
≥ ∈ C r −| − | 2
haveλ 1−√1+9γ2,1,1,1+√1+9γ2 ,andthesteerability
∈ { 6 3 3 6 } which vanishes only when F = 0 or θ = 0. In this case,
criteriongives λ Fsin2 θ,Fsin2 θ,Fcos2 θ,1 Fcos2 θ ,wehave
∈ {− 2 2 2 − 2}
= 1 (16 9γ2 8 1+9γ2), (11) λ1 =−λ2 =−Fsin2 2θ,andthesteerabilitycriteriongives
S 36 − −
p θ
= 4F2sin4 , (18)
whichpredictsthecriticalvalue S − 2
2 which predicts that steering always exists. We compare the
γcr = 3q2(6−√33)≃0.4765. (12) aboveresultwiththeviolationofthesteeringinequalitiesand
CHSHinequality(seeFig.1).
Example6.—Thestate
Anytwo-qubitstatecanbewritteninthefollowingform
g 0 0 γ/2 3
1
 0 1/2 g 0 0  ρ = (1 1+~σA u 1+1 ~σB v+ β σA σB),
ρ6 = − , (13) AB 4 ⊗ · ⊗ ⊗ · ij i ⊗ j
0 0 0 0 iX,j=1
 
γ/2 0 0 1/2 (19)
 
3
where u and v are Bloch vectors for particles A and B, re-
spectively; β are some real numbers. Particularly, we take
ij
u = (0,0,r), v = (0,0,s)and β = c δ , then we obtain
ij i ij
thefive-parameterX-stateas
3
1
ρAB = 4(1⊗1+r σ3A⊗1+1⊗sσ3B+ ciσiA⊗σiB).
Xi=1
(20)
InFig.2andFig.3wecomparethedetectiveabilityofsteer-
abilitycriterionandtheten-settingsteeringinequality.
Insummary,thecriterionisproposedduetonumericalob-
servation,whichworksefficientlyfordetectingEPRsteering
oftwo-qubitdensitymatrix. Similar toPPT criterionforde-
tectingentanglement,thesteerabilitycriterionmayalsowork
asanecessaryconditionfordemonstratingsteerabilityoftwo
qubits. Itwouldbesignificanttoderivethesteerabilitycrite-
rionfromanalyticapproach,suchaspositivemaps,andthen
placeitonafirmerfoundation.
FIG.2:(Coloronline)Comparisonofdetectiveabilityofsteerability J.L.C. is supported by National Basic Research Program
criterionandthesteeringinequalityforr =s=0. Thetetrahedron (973Program)ofChinaunderGrantNo. 2012CB921900and
is the set of all two-qubit states. Four vertices represent four Bell NSF of China (Grant Nos. 10975075 and 11175089). This
states,respectively. ThestateswithS =0locateontheredsurface,
work is also partly supportedby National Research Founda-
inside which are states with S > 0, outside which are states with
tionandMinistryofEducation,Singapore(GrantNo. WBS:
S < 0. The green points represent the states that violate the ten-
R-710-000-008-271).
setting steering inequality. Numerical result shows that the green
pointsarealwayslocatedinthevolumebetweentheredsurfaceand
thepolytope(tetrahedrron)definedbythefourvertices(Bellstates).
∗ Electronicaddress:[email protected]
† Electronicaddress:[email protected]
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