Sharp Continuity Bounds for Entropy and Conditional Entropy

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by  Zhihua Chen

SCIENCE CHINA
Physics, Mechanics & Astronomy
1
Vol. No.:
doi:
Sharp Continuity Bounds forEntropy andConditional Entropy
ZhihuaChen1,Zhihao Ma2†, IsmailNikoufar3, andShao-Ming Fei4,5*
7 1DepartmentofAppliedMathematics,ZhejiangUniversityofTechnology,Hangzhou,310014,China;
1 2SchoolofMathematicalSciences,ShanghaiJiaotongUniversity,Shanghai,200240,China;
0 3DepartmentofMathematics,PayameNoorUniversity,19395-3697Tehran,Iran;
2 4SchoolofMathematicalSciences,CapitalNormalUniversity,Beijing100048,China
5Max-Planck-InstituteforMathematicsintheSciences,04103Leipzig,Germany

n
DearEditor, tropyto define thelinear discord. Theyfoundthatthe linear
a
J The von Neumann entropy captures many operational discordhasdeepconnectionwiththeoriginaldiscorddefined
0 quantities in the quantum information theory such as quan- byvonNeumannentropy. Moreover,theygavetheanalytical
1 tum capacity of the communication channel. Von Neumann formulaforarbitrary2⊗d state of the linear discord. How-
entropyiscontinuousandisrepresentedbyFannesinequality, ever,a questionstillremainsopen: if twostates are close, is
]
h whichwasoriginallygivenin[1]. Quantumcorrelationssuch theirlineardiscordalso close to eachother? Inotherwords,
p as entanglement and quantum discord , et al., are important isthelineardiscordcontinuous? Fortheoriginaldiscord,the
t- resourcesinquantuminformationprocessing. Inthelastyear answerisaffirmative,see[10]. Forthelineardiscord,thereis
n enormous progress on the generation, concentration, detec- noansweryet. Henceitisworthwhiletostudythecontinuity
a
u tionandquantificationofentanglementhasbeenachieved[2]. ofconditionallinearentropy.
q Fannes inequality has many applications in the quantum in- We have two aims in this work: first, we study the conti-
[ formation theory, such as the investigation of continuity of nuity estimation of the Re´nyientropy and presenta tight in-
1 entanglementmeasures,includingentanglementofformation, equalityrelatingtheRe´nyientropydifferenceoftwoquantum
v relativeentropyofentanglement,squashedentanglementand states to their trace norm distance, which includes the sharp
8 conditionalentanglementofmutualinformation,andthecon- FannesinequalityforvonNeumannentropyasaspecialcase.
9 tinuity of quantum channel capacities [3]. Recently Fannes Second,westudythecontinuityofconditionallinearentropy,
3
inequality was improved to get a sharp one and it was also andproveausefulpropertyforameasureofthequantumcor-
2
0 generalizedtoTsallisentropy[4,5]. relation:lineardiscord.
1. However, in non-asymptotic settings, the natural quanti- ThevonNeumannentropyofaquantumstateρisdefined
by
0 ties that arise are Re´nyi entropies [6] and the properties of
7 Re´nyi entropieswere also investigatedin many papers, such S(ρ):=−Tr[ρlog2ρ]. (1)
1
as[7]. Re´nyientropieshavemanyapplications,asinthecase For the classical probability distributions, the von Neumann
:
v of one-shot problems, typically arising in cryptographicset- entropyreducestotheShannonentropy,
Xi tings,themin-andmax-entropiesarewidelyused[6]. In[8],
the authors found that Re´nyi-2 entropy was a proper mea- d d
ar sure of information for any multimode Gaussian state, and H(p):=XH(pi)=−Xpilog2pi, (2)
they defined and analyzed the measures of Gaussian entan- i i
glement and quantum correlation by using Re´nyi-2 entropy, where p:=(p)=(p ,p ,...,p )isad-dimensionalprobabil-
i 1 2 d
andfounditspropertiessuchasmonogamy. Inourwork,we ityvector,p >0, d p =1andH(p):=−p log p.
study the continuity property of Re´nyi-α entropy, which in- i Pi i i i 2 i
In[1]Fannesprovedhisfamousinequalityforthecontinu-
cludes Re´nyi-2 entropy as a special case. Our result is also
ityofthevonNeumannentropy,
usefulinstudyingthecontinuityoftheentanglementmeasure
oftheGaussianstateinquantumharmonicsystems. |S(ρ)−S(σ)|62Tlog (d)−2Tlog (2T), (3)
2 2
Ontheotherhand,theauthorsfoundthatTsalli-2entropy
(i.e.,linearentropy)wasnaturaltodefinethemeasureofquan- whereT := ||ρ−σ||1 ishalfofthetracenormdistancebetween
2
tum correlation for the the discrete system [9]. They called thestatesρandσ,||ρ−σ|| =Tr[|ρ−σ|],and|X|:= (X)†(X)
1 p
thismeasureaslineardiscordandusedconditionallinearen- denotesthe absolute value of an operator X. ObviouslyT ∈
1Correspondingauthor(email:†[email protected],*[email protected])
(cid:13)c ScienceChinaPressandSpringer-VerlagBerlinHeidelberg phys.scichina.com link.springer.com
SciChina-PhysMechAstron () Vol. No. -2
[0,1].Theinequality(3)isvalidfor06T 61/2e,whereeis Theorem 3. For bipartite quantumstates ρ and σ , if
AB AB
Euler’snumber.Theinequality(3)isfurtherimprovedtobea ǫ :=||ρ −σ || <1,then
AB AB 1
sharponebyAudenaert[4]:
|D (ρ )−D (σ )|68ǫ+4h (ǫ,1−ǫ). (9)
2 AB 2 AB 2
|S(ρ)−S(σ)|6Tlog (d−1)−H((T,1−T)). (4)
2
In summary, we haveinvestigatedthe continuityof Re´nyi
TheRe´nyientropyisamoregeneralformofthevonNeu- entropyandconditionallinearentropy. Throughthecontinu-
mannentropy, ity of conditionallinear entropy,the continuityof linear dis-
cordhasalsobeenobtained,whichmeansthatthelineardis-
1
H (ρ):= log[Tr(ρα)], α>0, (5) cordvaries as the quantumstate changescontinuously. This
α 1−α fact can guarantee that the errors in state tomograph would
notsignificantlyaffectthe resultof the quantumcorrelations
whenαgoestoone,Re´nyientropybecomesthevonNeumann
inthestate. AsthesharpFannesinequalityisthespecialcase
entropy. InthefollowingweshowthatfortheRe´nyientropy,
ofourtheoremaboutthecontinuityofRe´nyientropy,ourre-
animprovedsharpFannes-typeinequalityexists.
sultscanalsobeusedtoverifythecontinuityofentanglement
Theorem1.Foralld-dimensionalquantumstatesρandσ,
measuresforcontinuousvariablequantumstates.
dα−1
|H (ρ)−H (σ)|6 [1−(1−T)α−(d−1)1−αTα], α>1,
α α 1−α Acknowledgments ThisworkissupportedbytheNSFCun-
(6) dernumber11371247,11275131,11675113and11571313.
1
|H (ρ)−H (σ)|6 [1−(1−T)α−(d−1)1−αTα], α<1,
α α 1−α
(7) 1 M.Fannes,Commun. Math. Phys. 31: 291–294(1973);R.Alicki,M.
whereT is the trace normdistanceof ρand σ. See proofin Fannes,J.Phys.A:Math.Gen.37(2004)L55.
Appendix. 2 MaimaitiW,LiZ,ChesiS,etal,ScienceChinaPhysics,Mechanicsand
WeinvestigatedthecontinuityestimationoftheRe´nyien- Astronomy,58(5),50309(2015).
3 D.Yang,M.Horodecki,Z.D.Wang,AnAdditiveandOperationalEn-
tropy,bypresentinganinequalitywhichrelatestheRe´nyien-
tanglementMeasure:ConditionalEntanglementofMutualInformation,
tropydifferenceoftwoquantumstatestotheirtracenormdis-
Phys.Rev.Lett.101:140501(2008).
tance. In our inequality, equality can be attained for every
4 K.M.R.Audenaert,J.Phys.A:Math.Theor.40(2007)8127.
prescribed value of the trace norm distance. It is direct to 5 AlexeyE.Rastegin,LettersinMathematicalPhysics94(3)(2009).
verifythat for α → 1, our inequality(6) and(7) give rise to 6 M. Tomamichel, PhD thesis, Department of Physics, ETH Zurich,
thesharpFannesinequalityforvonNeumannentropy. Ithas arXiv:1203.2142(2012).
potentialapplicationsininvestigatingthecontinuityofentan- 7 Lin S M, Tomamichel M, Quantum Information Processing, 2015,
14(4):1501-1512.
glementmeasureandmoregeneralcorrelationsformultimode
8 GerardoAdesso,DavideGirolami, AlessioSerafini,Phys. Rev. Lett.
Gaussianstates,sinceRe´nyi-2entropyisaproperinformation
109,190502(2012).
measureforthiskindofstate[8].
9 ZhihaoMa,ZhihuaChen,FelipeFernandesFanchini,Shao-MingFei,
BesidesRe´nyientropy,linearentropyisalsousedtomea- ScientificReports5,10262(2015).
sure quantum correlations, such as linear discord [9], which 10 ZhengjunXi,Xiao-MingLu,XiaoguangWang,YongmingLi,J.Phys.
is defined as the minimal difference of the two conditional A:Math.Theor.44,375301(2011).
linear entropy,beforeand after the localprojectivemeasure- 11 M.M.-Lennert,F.Dupuis,O.Szehr,S.Fehr,M.Tomamichel,J.Math.
ment,D (ρ ) := min(S (A|B)−S (P|B)),whereS (A|B)is Phys.54,122203(2013).
2 AB 2 2 i 2
Pi 12 I.Nikoufar,AdvancesinMathematics259(2014),376-383.
theconditionallinearentropyoftheoriginalstate ρAB, while 13 M.A.NielsenandI.L.Chuang,QuantumComputationandQuantumIn-
S2(Pi|B)istheconditionallinearentropyofthepostmeasure- formation(CambridgeUniversityPress,Cambridge,UK,10thAnniver-
mentstateafterlocalmeasurementP,andtheminimumruns saryEdition,2010).
i
overalllocalprojectionmeasurementsP. Thereforeitisalso 14 K.M.R.Audenaert,J.Math.Phys.48(2007),083507.
i
importantto study the continuityof the linear entropy,espe-
ciallyconditionallinearentropy. Itcanhelpusgettheconti-
nuityofthelineardiscord.
Wecanprovethefollowingconclusion: conditionallinear
entropyiscontinuous,seeproofinAppendix.
Theorem 2. For bipartite quantumstates ρ and σ , if
AB AB
ǫ :=||ρ −σ || <1,thenthefollowinginequalityholds,
AB AB 1
|S (ρ |ρ )−S (σ |σ )|64ǫ+2h (ǫ,1−ǫ). (8)
2 AB B 2 AB B 2
Byusingthemethodof[10], itisstraightforwardtoshow
thatthelineardiscordD (ρ )isalsocontinuous:
2 AB
SciChina-PhysMechAstron () Vol. No. -3
Appendix =S (1− p −T)−S (1−p −T(1−s ))
α 1 α 1 1
ProofofTheorem1: −S (T(1−s )φ)
α 1
Lemma1.1.Foralld-dimensionalquantumstatesρandσ,
,∆.
|H (ρ)−H (σ)|6dα−1|S (ρ)−S (σ)| α>1, (10)
α α α α As S (T(1− s )φ) gets its maximumwhen φ is the uniform
α 1
distribution=(1,1,··· ,1)/(d−2),∆hastheminimum,
|H (ρ)−H (σ)|6|S (ρ)−S (σ)| α<1. (11)
α α α α
whereS (ρ):=[Tr(ρα)−1]. ∆=Sα(1−p1−T)−Sα(1− p1−T(1−s1))
α
TheaboveinequalitiesareobtainedbyusingCauchymean −(d−2)1−αS (T(1−s )).
α 1
valuetheorem.
(c). From
Lemma1.2.Foralld-dimensionalquantumstatesρandσ,
∂∆
|Sα(ρ)−Sα(σ)|61−(1−T)α−(d−1)1−αTα. (12) ∂s =−Tα[(1−p1−T(1−s1))α−1−(d−2)1−α(T(1−s1))1−α]=0,
1
Lemma1.3. Forallprobabilitydistributions p = (p)and weget
i
q=(qi),thefollowinginequalityholds: T(1−s )= (1−p1)(d−2) ≡ω. (15)
1 (d−1)
|Sα(p)−Sα(q)|61−(1−T)α−(d−1)1−αTα, (13) When 0 < T < ω, there is no local minimum of ∆ from
(15). For T(1 − s ) = T, i.e. s = 0, ∆ has a minimum
d 1 1
wSh(eqr)eT== 12[(iPq=1)|αpi−−qq]i|,=Sα(p)S=(qPi)[,(ppi)α>− p0i,]q= P>i S0α(apni)d, i−m(dum−S2)α((1−pα1)S+αT(T))−. SThα(epre1f)o−re(Sdα−(q2))−1−αSSαα((pT))g.etMs oitrseomvienr-,
Piαpi =Pi qPii =1i,Sα(pii):=[(Ppii)αα−pii]. i i dpu1,ewtohethnatp1Sα=(p11−+T(dd−)−1−2)TS, αS(αp(1q))i−s aSαd(epc)regaestisngitsfumncintiiomnuomf
Let λi, i = 1,2,...,d, be the eigenvalues of ρ, one has (1− dT−2)α−(1− (dd−−12)T)α−(d−2)1−αTα.
Sα(ρ) = P[(λi)α−λi] := PSα(λi),whereSα(λi) := [(λi)α− When ω 6 T 6 1− p1, (15) can be satisfied and ∆ gets
i i itsminimum(1− p −T)α−(1−p )α/(d−1)α−1. Therefore
λ]. 1 1
i
S (q)−S (p)getsitsminimum
Firstly, we prove Lemma 1.3. Then Lemma 1.2 is also α α
provedforthediagonalquantumstatesρandσ. (1−p )α
ProofofLemma1.3:Letq= p+δ+−δ−,whereδ+ =(δ+i) −Sα(p1)+Sα(p1+T)+(1−p1−T)α− (d−1)1α−1.
and δ− = (δ−) are two vectors such that δ+ > 0, δ− > 0,
i i i
iLe=mm1,a21,..3..,idn,thδr+ee·cδa−se=sa0cc,oPrdiiδn+igt=othTe.vaWlueesproofvαe:(1α3<) o1f, tThhaendzeerriov.atHiveenocfe,thwehaebnovpe1f=orm1u−laTw,iSthα(rqe)sp−ecStαto(pp)1giestsleistss
16α<2andα>2. minimum1−(1−T)α−(d−1)1−αTα.
(CaseI)α<1: SinceSα(x)−Sα(x−T)isadecreasingfunctionof xand
(a).S (p)isconcave,S (p+δ+−δ−)−S (p)isaconcave 1>1− T ,
α α α d−2
function with respect to δ+. It gets its minimumat a certain
T (d−1)T
point,say,δ+ =e1. 1−(1−T)α 6(1− )α−(1− )α.
ThenT =e1,p=(p ,(1−p )r),q=(p +T,(1−p )r−Ts), d−2 d−2
1 1 1 1
wherer and sared−1dimensionalprobabilityvectorssuch Therefore1−(1−T)α −(d −1)1−αTα 6 (1− T )α −(1−
that p1+T 61,(1−p1)r−Ts>0,andTs=δ−. Wehave (d−1)T)α−(d−2)1−αTαandthen1−(1−T)α−(dd−2−1)1−αTα
d−2
istheminimumofS (q)−S (p).
S (q)−S (p)=S (p +T)−S (p ) α α
α α α 1 α 1 (CaseII)16α<2:
+Sα((1−p1)r−Ts)−Sα((1−p1)r). (a). Sα(p)−Sα(q) = Sα(p)−Sα(p+δ+ −δ−)isconcave
withrespecttoδ+. Takeδ+ =e1.Wehave
(b). Denote(1−p )r−Ts=(1−p −T)η,then
1 1
S (p)−S (q)=S (p )−S (p +T)
α α α 1 α 1
S ((1− p )r−Ts)−S ((1−p )r) (14)
α 1 α 1 +S ((1−p )r)−S ((1−p )r−Ts),
α 1 α 1
=S ((1−p −T)η)−S ((1−p −T)η+Ts).
α 1 α 1
where
SinceS (x)−S (x+y)isconcaveandamonotonouslyin-
α α
creasingfunctionofx,theright-handsideof(14)getsitsmin- Sα((1−p1)r)−Sα((1−p1)r−Ts)
imumatcertainpointofη,say,η=e1.Lets=(s1,(1−s1)φ) =Sα((1−p1−T)η+Ts)−Sα((1−p1−T)η).
withφad−2dimensionalprobabilityvector,wehave
(b).S (x+y)−S (x)isconcavewithrespecttoxforα<2,
α α
S ((1−p −T)η)−S ((1− p −T)η+Ts) S ((1−p −T)η+Ts)−S ((1−p −T)η)getsitsminimum
α 1 α 1 α 1 α 1
SciChina-PhysMechAstron () Vol. No. -4
atη = e1. Let s = (s ,(1− s )φ)withφad−2dimensional Definition WedefinetheTsalliαrelativeentropyas:
1 1
probabilityvector.Weget
1
Tˆ (ρ||σ)= (Tr(ρα)−Tr(ρσα−1)), (16)
S ((1−p −T)η+Ts)−S ((1−p −T)η) α α−1
α 1 α 1
=S (1−p −T(1−s ))+S (T(1−s )φ) here α ∈ (0,∞). When α goes to one, the Tsalli α relative
α 1 1 α 1
−S (1−p −T),∆. entropy becomes the (quantum) relative entropy, S(ρ||σ) =
α 1
−Tr(ρlogσ)−S(ρ).
Whenφ= (1,1,···,1)/(d−2),S (T(1−s )φ)getsitsmini- Lemma 2. The relative entropydefined by (16) is jointly
α 1
mum,and∆getsitsminimum,thatisS (1−p −T(1−s ))− convexforα∈[0,1)∪(1,2].
α 1 1
S (1− p −T)+(d−2)1−αS (T(1−s )). Proof. Thefunction f(t) = 1 (1−tα−1)isaconvexfunc-
α 1 α 1 α−1
(c)From ∂∆ =0,wehavetheformula(15)again. tionforα∈[0,1)∪(1,2]whent>0,then
When0<∂Ts1 <ω,∆hasnolocalminimum.ForT(1−s )=
T,∆hasaminimum(d−2)(1−α)S (T). S (p)−S (q)get1sits g(Lρ,Rσ)= Lρ1/2f(L−ρ1/2RσL−ρ1/2)Lρ1/2
α α α
minimumSα(p1)−Sα(p1+T)+(d−2)1−αSα(T),whichtakes = 1 (L −L2−αRα−1)
theminimumvalue(1− (d−1)T)α−(1− T )α+(d−2)1−αTα α−1 ρ ρ σ
d−2 d−2
at p1 =1− (dd−−12)T. isjointlyconvex(cf.[12]). Itfollowsthat
For ω 6 T 6 1 − p , at T(1− s ) = ω, ∆ gets its min-
1 1
imum ((d1−−1p)1α)−α1 −(1− p1 −T)α. Sα(q)−Sα(p) gets its min- (ρ,σ)7−→hg(Lρ,Rσ)(X),Xi=Tr(X∗g(Lρ,Rσ)(X))
imum Sα(p1) − Sα(p1 + T) − (1 − p1 − T)α + ((d1−−1p)1α)−α1 = is also jointly convex on ρ,σ, where X is an arbitrary oper-
(1−T)α+(d−1)1−αTα−1at p =1−T. Therefore ator and h·,·i is the Hilbert-Schmidt inner product. Taking
1
X =ρα−21 wehave
(1−T)α+(d−1)1−αTα−1
<(1− (d−1)T)α−(1− T )α+(d−2)1−αTα hg(Lρ,Rσ)(ρα−21),ρα2−1i
d−2 d−2 1
and(13)isvalidwhen16α<2. = α−1Tr(ρα−21(Lρ−L2ρ−αRασ−1)(ρα−21))
(CaseIII)α > 2 :FollowingthesamestepasCaseII,we =Tˆα(ρ,σ).
get that S ((1− p )r)−S ((1− p )r−Ts) is concave with
α 1 α 1
Thiscompletestheproof.
respecttos. HenceS ((1−p )r)−S ((1−p )r−Ts)getsits
α 1 α 1
minimuminoneoftheextremepointsofs,say,s=e1. Corollary1. ThelinearentropyS2(ρ)andtheconditional
Letr =(r ,(1−r )φ)withφad−2dimensionalprobability linearentropyS2(A|B)areconcave.
1 1
Lemma 3. (Projective measurements increase entropy)
vector.Then
SupposethatP isacompletesetoforthogonalprojectorsand
i
S ((1−p )r)−S ((1−p )r−Ts) ρ is a density operator. Then the linear entropy of the state
α 1 α 1
=Sα((1−p1)r1)−Sα((1−p1)r1−T),∇. ρ′ :=Pi PiρPiafterthemeasurementwillnotdecrease,thatis,
S (ρ′)>S (ρ).
Since 2 2
Proof.WeknowthatthesquareofHilbert-Schimidtmetric
∂∇ is definedby D2(ρ′,ρ) := Tr(ρ′ −ρ)2, which is nonnegative.
=α(1−p )αrα−1−α(1−p )((1−p )r −T)α−1 >0, 2
∂r 1 1 1 1 1 WehaveTr(ρ′ −ρ)2 =S (ρ′)−S (ρ)>0.
1 2 2
Lemma4. Supposeρ= pρ,where{p}aresomesetof
when (1 − p1)r1 = T, ∇ has the minimum Tα. Therefore Pi i i i
Sα(p)−Sα(q)takes its minimum pα1 −(p1 +T)α +Tα. Be- probabilitiesand{ρi}aredensityoperators. Thenwehavethe
cause pα1 −(p1+T)α+Tα decreaseswiththedecreaseof p1, followingupper bound: S2(ρ) 6 h2(pi)+PpiS2(ρi), where
Sα(p)−Sα(q)takesitsminimumTα+(1−T)α−1atp1 =1−T. h (p):=1− p2. i
Nowwe haveprovedthe inequality(13), i.e. Lemma1.3, 2 i P i
i
namelytheinequality(12)ofLemma1.2forthecasethatboth Proof. TheproofusesthemethodsimilartothevonNeu-
states ρ and σ are diagonal ones. For general ρ and σ, the mannentropycase(seeTheorem11.10of[13]). Fromdirect
inequality can be directly proved by taking into account the calculation and the Cauchy- Schwartz inequality, we get the
factthattheRe´nyientropyisunitaryinvariant[11]. Byusing result.
Lemma1.1,Theorem1isproved. In the following text, we consider the bipartite quantum
ProofofTheorem2: statesonH⊗H,withdbeingthedimensionofHilbertspace
First, we define the relative linear entropy D(ρ||σ) := H.
Tr(ρ2)−Tr(ρσ). ThelinearentropyisthengivenbyS2(ρ) = Lemma5. ForthebipartitequantumstateρAB,thefollow-
−D(ρ||I) = 1 − Tr(ρ2), while the conditional linear entropy inginequalityholds,
is given by S2(ρAB|ρB) := S2(A|B) := −D(ρAB||I ⊗ ρB) = d−1
Tr(ρ2)−Tr(ρ2 ). |S2(A|B)|6 . (17)
B AB d
SciChina-PhysMechAstron () Vol. No. -5
Proof. S (A|B) 6 d−1 comesfromthesubadditivityofthe ǫ)S (ρ ) + ǫS (τ ). From the upper bound S (γ ) 6
2 d 2 B 2 B 2 AB
Tsallisentropy(see[14]),S (ρ )6S (ρ )+S (ρ ). Onthe h (ǫ,1−ǫ)+(1−ǫ)S (ρ )+ǫS (τ ),wegetS (γ |γ )=
2 AB 2 A 2 B 2 2 AB 2 AB 2 AB B
otherhand,wehaveS (A|B)=S (ρ )−S (ρ )>−S (ρ )> S (γ ) − S (γ ) 6 h (ǫ,1 − ǫ) + (1 − ǫ)S (ρ |ρ ) +
2 2 AB 2 B 2 B 2 AB 2 B 2 2 AB B
−d−1,whichcompletestheproof. ǫS (τ |τ ). ThereforeS (ρ |ρ )−S (γ |γ )>−h (ǫ,1−
d 2 AB B 2 AB B 2 AB B 2
Lemma 6. For bipartite quantum states ρ and τ , as- ǫ) − ǫ(S (ρ |ρ ) − S (τ |τ )) > −h (ǫ,1 − ǫ) − 2ǫd−1.
AB AB 2 AB B 2 AB B 2 d
sumethat06ǫ 61,defineγ :=(1−ǫ)ρ +ǫτ ,then Second, as the conditionalentropyis concave, S (γ |γ ) >
AB AB AB 2 AB B
(1 − ǫ)S (ρ |ρ ) + ǫS (τ |τ ), we obtain S (ρ |ρ ) −
d−1 2 AB B 2 AB B 2 AB B
|S2(ρAB|ρB)−S2(γAB|γB)|62ǫ d +h2(ǫ,1−ǫ). (18) S2(γAB|γB) 6 ǫ(S2(ρAB|ρB) − S2(τAB|τB)) 6 2ǫd−d1, which
completestheproof.
Proof. The proof is similar to that of [1]. First, from UsingLemma6,andthemethodof[1],weprovetheresult
the concavity of the entropy, we have S (γ ) > (1 − ofTheorem2.
2 B