Comment on "Witnessed entanglement and the geometric measure of quantum discord"

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by  Swapan Rana

Comment on“Witnessedentanglement andthe geometricmeasure ofquantum discord”
Swapan Rana∗ and Preeti Parashar†
PhysicsandAppliedMathematicsUnit, IndianStatistical Institute, 203BTRoad, Kolkata, India
(Dated:January10,2013)
Inarecentarticle[Phys.Rev.A86,024302(2012)],theauthorshavederivedsomehierarchyrelationsbe-
tween geometric discord and entanglement (measured by negativity and itssquare). Wepoint out that these
resultsareincorrectandgiveanalyticcounterexamples. Wealsodiscussbrieflythereasonforsuchviolations.
PACSnumbers:03.67.Mn,03.65.Ud
WestartwiththedefinitionofGDandnegativityfromRef. m/(m−1)>1andhencethiscasealsoincludesthecasewhen
3 [1]. Foranm⊗n(m≤n)stateρ,theyhavedefined D isnotnormalizedandEq. (2a)istakenasthedefinition
(2)
1 ofnegativity. Nowwewillgiveananalyticexampletoshow
20 D(p)(ρ)=mξ∈iΩnkρ−ξk(pp) (1) thattherearestatesviolatingeventheweakerrelation[6]

n where Ω is the set of zero-discord (or classical-quantum) 2
9 Ja sptantoersmgigvievnenbbyyξkX=k(Pp) p=i|{iiThri[|X⊗†ρXi]pa/n2d}1/kpX.kT(ph)einsetghaetiSvcithyawtteans mm−1D(2) ≥ (mN−21)2 = "λPi<(0mλ−i(ρ1T)A2)# (4)
definedby

] Considerthem⊗mWernerstategivenby
h
p N(ρ)=kρTAk(1)−1 (2a) m−z mz−1
t- =max{0,− min Tr(Wρ)} (2b) ρw = m3−mI+ m3−mF, z∈[−1,1] (5)
n 0≤WTA≤I
a
whereF = |kihl|⊗|lihk|andsetm = 8,z = −1. Now,ifwe
u where ρTA is the partial transposition of ρ with respect to A
q andW isanyoptimalentangledwitness. Attemptingtoprove considerthePmatrixformofρw(incomputationalbasis)asthe
[ stateofa2⊗32system,thelefthandsideofEq. (4)becomes
aconjecturemadein[2], theyhaveclaimed(Eq. 17therein)
1/49whiletherighthandside becomes25/784. Thoughwe
1 thatallbipartitestatessatisfy
haveusedtheformulaforD developedinRef.[7](whichis
v (2)
1 N2 exactfor2⊗nstates),ameasurementincomputationalbasis
6 D(2) ≥ (m−1)2 (3) willyieldtheresult. Anyvalueofz ∈ [−1,−34/43)willalso
9 workwell. We note thatforlargeenough n, the Refs. [3, 8]
1 Wefirstobserveatypothatthetwodefinitionsofnegativity giveenoughintuitionforviolationofthisrelation. Nonethe-
1. inEq. (2)arenotequal,asthequantityinEq. (2b)isthesum less,theanalyticcounterexamplemakesitmoreexplicit.
0 ofabsolutevaluesofnegativeeigenvaluesofρTA whereasthat The authors of Ref. [1] also proposed to take D(1) as a
3 inEq. (2a)isjustdoubleofit. propermeasure of geometric discord and derived the hierar-
1 As a first gapin theirderivation(which could be taken as chyrelation(Eq. 27therein)
:
v anothertypo,though),wenotethattheyhavenotnormalized
Xi D(2) to have maximum value unity, as has been done in the D(1) ≥ N (6)
originalpaper[2]. As a result, if we take Eq. (2a) as defini-
ar tionofnegativity,Eq. (3)isnotnecessarilysatisfiedevenby However,ascanbeseeneasily,thisresultisalsonotcorrect.
thetwo-qubitmaximallyentangledstate(anyoneoftheBell Itiswellknownthatthetracedistancesatisfieskρ−σk(1) ≤2,
states has D(2) = 1/2 whereas N = 1). The importance of forallρandσ[9]. HencewemusthaveD(1) ≤ 2whereasN
normalizationcouldbefoundinRefs. [3,4]. Nowwewilltry cantakevalueupto(m−1)/2(forexampleconsidertheBell
toremoveallthese(possible)typosandshowthattherelation state in m⊗m). Taking the identity matrix as the classical-
(3), whether normalized or not, is always violated by some quantumstate (neednotbeoptimal),weseethattherelation
states. isviolatedby4⊗4Bellstates.
Letusfirstconsiderthecasewhen D isnormalized(tak- As has been pointed out in Ref. [3], the violation for
(2)
ing D = mm−1D(2)) and Eq. (2a) is taken as the defini- D(2) stemsfromthefactthattheHilbert-Schmidtnormisnot
tion of negativity. Then it becomes the original conjecture monotone—D(2)couldbeincreasedordecreasedbyaddingor
(D ≥ N2/(m − 1)2) made in Ref. [2], which we have re- removingafactorizedlocalancilla.Wewouldliketomention
futed recently in Ref. [5]. Note that the normalizing factor thatthetracenormbeingmonotone,doesnotsufferfromthis
problem.ThustheproposaloftakingD asagoodmeasure,
(1)
is interestingand mightbe worth investigating. However,as
we pointedouthere, establishinganyinterrelationshould be
∗[email protected] donecarefully.Anotherpointofconcernregardingtheuseof
†[email protected] the trace norm is thatits analytic calculationis verydifficult
2
and hence the main spirit of usual geometricdiscord will be relation[D > N2/(d−1)]willthenbecorrectforNPTstates
(2)
lost. (thePPTstatestriviallysatisfyequality),animportantpointis
Note added: After submission of this comment to jour- that it can no longer be used to compare geometric discord
nal,wefoundanerratumpostedonarXive[10]inwhichthe andentanglement. Wenotethat,bothN andN2 areentangle-
authors have tried to fix the errors, based on our criticism. mentmonotones,butN/d(ingeneralN/f(d))isnotanentan-
Thoughtheerratumisoutofpurviewofthepresentcomment, glementmonotone,asitmightbeincreasedwithremoval(or
wewouldliketopointoutthatsomefurthermodificationsare addition)oflocalancillarysystems. Thisobservationapplies
needed. First of all, there is nostate ρ forwhich n = d−1 equallywelltotheinterrelationbetweenD andN/d.
(1)
holds;sotherelationmustbeastrictinequality.Althoughthe
[1] T. Debarba, T. O. Maciel and R. O. Vianna, (4)automaticallyimpliesaviolationofEq.(3).
Phys.Rev.A86,024302(2012). [7] S.RanaandP.Parashar,Phys.Rev.A85,024102(2012).
[2] D.GirolamiandG.Adesso,Phys.Rev.A84,052110(2011). [8] T. Tufarelli, D. Girolami, R. Vasile, S. Bose and G. Adesso,
[3] M.Piani,Phys.Rev.A86,034101(2012). arXiv:1205.0251
[4] E.Chitambar,Phys.Rev.A86,032110(2012). [9] The trace norm k.k , being a norm, must satisfy the tri-
(1)
[5] S.RanaandP.Parashar,Phys.Rev.A86,030302(R)(2012). angular inequality: kρ − σk ≤ kρk + kσk = 2. For
(1) (1) (1)
[6] Thisrelationisnot well-justifiedforcomparison, because the many properties and interpretations of the trace norm, we
left hand side varies from 0 to 1 whereas the right hand side refer to A. Gilchrist, N. K. Langford and M. A. Nielsen,
from0to1/4(i.e.,notnormalized).NotethataviolationofEq. Phys.Rev.A71,062310(2005).
[10] T.Debarba,T.O.MacielandR.O.ViannaarXiv1207.1298v3.