Smooth R´enyi Entropies and the Quantum Information Spectrum Nilanjana Datta1,∗ and Renato Renner2,† 1Statistical Laboratory, DPMMS, University of Cambridge, Cambridge CB3 0WB, UK 2Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland (Dated: February 2, 2008) Many of thetraditional results in information theory,such as thechannel coding theorem or the source coding theorem, are restricted to scenarios where the underlying resources are independent and identically distributed (i.i.d.) over a large number of uses. To overcome this limitation, two differenttechniques,theinformationspectrummethod andthesmoothentropyframework,havebeen developed independently. They are based on new entropy measures, called spectral entropy rates and smooth entropies, respectively, that generalize Shannon entropy (in the classical case) and von Neumann entropy (in the more general quantum case). Here, we show that the two techniques are 8 0 closely related. More precisely,thespectral entropyrate can beseen as theasymptoticlimit of the 0 smooth entropy. Ourresults apply to thequantum setting and thusincludethe classical setting as 2 a special case. n a INTRODUCTION Subsequently,Hayashi,Nagaoka,andOgawahavegen- J eralized the information-spectrum method to quantum- 1 Traditional results in information theory, e.g., the mechanical settings. They have applied the method to ] noisy channel coding theorem or the source coding (or study quantum hypothesis testing and quantum source h data compression) theorem, typically rely on the as- coding [16, 18], as well as to determine general expres- p - sumption that underlying resources, e.g., information sions for the optimal rate of entanglement concentration nt sources and communication channels,are “memoryless”. [14] and the classical capacity of quantum channels [13]. a A memorylessinformationsourceis one whichemits sig- The method has been further extended by Bowen and u nals that are independent of each other. Similarly, a Datta [3] and used to obtain general formulae for the q channel is said to be memoryless if the noise acting on optimal rates of various information-theoretic protocols, [ successive inputs to the channel is uncorrelated. Such e.g., the dense coding capacity for a noiseless quantum 1 resourcescanbedescribedbyasequenceofidentical and channel, assisted by arbitrary shared entanglement [4] v independently distributed (i.i.d.) random variables. andtheentanglementcostforarbitrarysequencesofpure 2 [5] and mixed [6] states. Recently, Matsumoto [15] has 8 In reality, however, this assumption cannot generally also employed the information spectrum method to ob- 2 be justified. This is particularly problematic in cryp- 0 tography, where the accurate modeling of the system is tain an alternative (but equivalent) expression for the 1. essential to derive any claim about its security. entanglement cost for an arbitrary sequence of states. 0 Inthepastdecade,twoapproacheshavebeenproposed In a simultaneous but independent development, the 8 independently to overcome this limitation. The infor- necessity to generalize Shannon’s theory became appar- 0 mation spectrum approach was introduced by Han and ent in the context of cryptography. Roughly speaking, : v Verdu´ [11, 12, 28] in an attempt to generalize the noisy one of the main challenges in cryptography is that one Xi channelcodingtheorem. Thisapproachyieldsaunifying needs to deal with an adversary who might pursue an mathematical framework for obtaining asymptotic rate arbitrary (and unknown) strategy. In particular, the ad- r a formulaeformanydifferentoperationalschemesininfor- versarymightintroduceundesiredcorrelationswhich,for mation theory,such as data compression,data transmis- instance, make it difficult to justify assumptions on the sion, and hypothesis testing. The power of this method independence of noise in a communication channel. liesinthefactthatitdoesnotrelyonthespecificnature Bennett, Brassard, Cr´epeau, and Maurer [1] were of the sources or channels involved in the schemes. among the first to make this point explicit, arguing that The main ingredients of this method are new entropy- the Shannon entropy is not an appropriate measure for type measures, called spectral entropy rates, which are the ignorance of an adversary about a (partially secret) defined asymptotically for sequences of probability dis- key. They proposedanalternativemeasurebasedonthe tributions. They can be seen as generalizations of the collisionentropy (i.e., R´enyientropy [24] of order 2) and Shannon entropy, and also inherit many of its proper- anotioncalledspoiling knowledge,whichcanbeseenasa ties, such as subadditivity, strong subadditivity, mono- predecessorofsmoothentropies. Thisapproachhasbeen tonicity, and Araki-Lieb inequalities. They also sat- further investigated by Cachin [7], who also found con- isfy chain rule inequalities. Their main feature, how- nections to other entropy measures, in particular R´enyi ever, is that they characterize various other asymptotic entropies of arbitrary order. information-theoretic quantities, e.g., the data compres- Motivated by the work of Bennett et al. and Cachin, sion rate, without relying on the i.i.d. assumption. smooth R´enyi entropies have been introduced by Ren- 2 ner et al., first for the purely classical case (in [22]), and tral projections A < 0 , A > 0 and A 0 . For { } { } { ≤ } later for the more general quantum regime (in [20, 21]). two operators A and B, we can then define A B as { ≥ } In contrast to the spectral entropy rates, smooth R´enyi A B 0 . The following key lemmas are useful. For { − ≥ } entropies are defined for single distributions (rather a proof of Lemma 1, see [16, 18]. than sequences of distributions). Because of their non- Lemma 1 For self-adjoint operators A, B and any pos- asymptotic nature,they dependonanadditionalparam- itive operator 0 P I the inequality we have eter ε, called smoothness. ≤ ≤ Similarly to the spectral entropy rates, it has been Tr P(A B) Tr A B (A B) (1) (cid:2) − (cid:3) ≤ (cid:2)(cid:8) ≥ (cid:9) − (cid:3) shown that smooth entropies have many properties in Tr P(A B) Tr A B (A B) . (2) common with Shannon and von Neumann entropy (for (cid:2) − (cid:3) ≥ (cid:2)(cid:8) ≤ (cid:9) − (cid:3) Identical conditions hold for strict inequalities in the example, there is a chain rule, and strong subadditivity spectral projections A<B and A>B . holds) [20, 23]. Furthermore, they allow for a quantita- { } { } tive analysis of a broad variety of information-theoretic Lemma 2 Given a state ρ and a self-adjoint operator n tasks—but in contrast to Shannon entropy, neither the ω , for any real γ we have n i.i.d. assumptionnor asymptotics are needed. For exam- Tr ρ 2−nγω ω 2nγ. ple, in the classical regime, it is possible to give a fully n n n (cid:2){ ≥ } (cid:3)≤ general formula for the number of classical bits that can Proof Note that be transmitted reliably (up to some error ε) in one (or Tr ρ 2−nγω (ρ 2−nγω ) 0 finitely many) uses of a classical channel [25]. In the n n n n (cid:2){ ≥ } − (cid:3)≥ quantum regime, they proved very useful in the context Hence, ofrandomnessextraction[20,21],which, inturn, isused 2−nγTr ρ 2−nγω ω Tr ρ 2−nγω ρ for cryptographic applications [8, 9, 10, 27]. In particu- (cid:2){ n ≥ n} n(cid:3) ≤ (cid:2){ n ≥ n} n(cid:3) lar, they are employed for the study of real-worldimple- Trρn =1 (3) ≤ mentationsofcryptographicschemes,wheretheavailable Therefore, resources (e.g., the computational power or the memory size) are finite [26]. Tr ρn 2−nγωn ωn 2nγ. (cid:2){ ≥ } (cid:3)≤ Our aim in this paper is to find connections between the two different approaches described above, by explor- The trace distance between two operators A and B is ing the relationships between spectral entropy rates and given by smooth entropies. We do this in two steps. First, we considerthespecialcasewheretheentropiesarenotcon- A B :=Tr A B (A B) Tr A<B (A B) 1 ditioned on an additional system, in the following called || − || (cid:2){ ≥ } − (cid:3)− (cid:2){ } − (4(cid:3)) thenon-conditionalcase. Then,inasecondstep,wecon- The fidelity of states ρ and ρ′ is defined to be siderthegeneralconditional casewheretheentropiesare conditioned on an extra system. F(ρ,ρ′):=Trqρ12ρ′ρ12. Thetracedistancebetweentwostatesρ andρ′ isrelated to the fidelity F(ρ,ρ′) as follows (see (9.110) of [17]): DEFINITIONS OF SMOOTH ENTROPY AND SPECTRAL ENTROPY RATES 1 ρ ρ′ 1 F(ρ,ρ′)2 2(1 F(ρ,ρ′)) . (5) 2k − k1 ≤p − ≤p − Mathematical Preliminaries We also use the following simple corollary of Lemma 1: Corollary 1 For self-adjoint operators A, B and any Let ( )denotethe algebraoflinearoperatorsacting B H positive operator 0 P I, the inequality on a finite-dimensional Hilbert space . The von Neu- ≤ ≤ H mannentropyofastateρ,i.e.,apositiveoperatorofunit A B ε, 1 || − || ≤ trace in ( ), is given by S(ρ) = Trρlogρ. Through- B H − for any ε>0, implies that out this paper, we take the logarithm to base 2 and all Hilbert spaces considered are finite-dimensional. Tr P(A B) ε. (cid:2) − (cid:3)≤ Thequantuminformationspectrumapproachrequires We also use the “gentle measurement” lemma [19, 29]. the extensive use of spectral projections. Any self- adjointoperatorAactingonafinite-dimensionalHilbert Lemma 3 For a state ρ and operator 0 Λ I, if ≤ ≤ space may be written in its spectral decomposition A = Tr(ρΛ) 1 δ, then ≥ − λ i i. We define the positive spectral projection PoniAi|aish |A 0 := i i, the projector onto ||ρ−√Λρ√Λ||1 ≤2√δ. { ≥ } Pλi≥0| ih | the eigenspace of A corresponding to positive eigenval- The same holds if ρ is only a subnormalized density op- ues. Corresponding definitions apply for the other spec- erator. 3 Definition of spectral divergence rates The spectral generalizations of the von Neumann en- tropy, the conditional entropy and the mutual informa- In the quantum information spectrum approach one tioncanallbeexpressedasspectraldivergencerateswith defines spectral divergence rates, defined below, which appropriate substitutions for the sequence of operators canbe viewedas generalizationsofthe quantum relative ω ={ωn}∞n=1. entropy. b Definition 1 Given a sequence of states ρ = ρ ∞ Definition of spectral entropy rates { n}n=1 and a sequence of positive operators ω = ω ∞ , the b{ n}n=1 qteuramnstuomf tshpeecdtriffalerseunpc-e(inopf-e)rdaitvoerrsgeΠnnce(γrb)at=esρanr−ed2enfiγnωendains HnCo=nsHid⊗ern.a sFeoqrueanncyesoefquHeinlbceertofspsatacteess{ρHn=}∞n{=ρ1n,}w∞n=it1h, withρ beingadensitymatrixactingintheHilbertspace n b D(ρ ω):=inf γ :limsupTr Π (γ) 0 Π (γ) =0 , the sup- and inf- spectral entropy rates are defined bkb n n→∞ (cid:2){ n ≥ } n (cid:3) o aHsnfollows: (6) D(ρkω):=supnγ :linm→i∞nfTr(cid:2){Πn(γ)≥0}Πn(γ)(cid:3)=1o S(ρ)=infnγ :linm→i∞nfTr(cid:2){ρn ≥2−nγIn}ρn(cid:3)=1o (10) b b (7) S(ρb)=sup γ :limsupTr ρ 2−nγI ρ =0 . n n→∞ (cid:2){ n ≥ n} n(cid:3) o b respectively. (11) Although the use of sequences of states allows for im- Here I denotes the identity operator acting in . n n H mensefreedominchoosingthem, thereremainanumber These are obtainable from the spectral divergence rates of basic properties of the quantum spectral divergence as follows [see [3]: rates that hold for all sequences. These are stated and provedin [3]. In the i.i.d. case the sequence is generated S(ρ)= D(ρ I); S(ρ)= D(ρ I), (12) from product states ρ = {̺⊗n}∞n=1, which is used to re- b − b||b b − b||b late the spectral entropy rates for the sequence ρ to the where I = I ∞ is a sequence of identity operators. { n}n=1 entropy of a single state ̺. It isbknown [3] that the spectral entropy rates of ρ are Note that the above definitions of the spectral diver- relatedtothe vonNeumannentropiesofthe statesρ as bn gence rates differ slightly from those originally given in follows: (38) and (39) of [13]. However, they are equivalent, as 1 1 stated in the following two propositions (proved in [3]). S(ρ) liminf S(ρ ) limsup S(ρ ) S(ρ). (13) n n The proofs havebeen included inthe Appendix for com- ≤ n→∞ n ≤ n→∞ n ≤ b b pleteness. Moreover for a sequence of states ρ= ρ⊗n ∞ : { }n=1 Proposition 1 The spectral sup-divergence rate D(ρ ω) b k 1 is equal to S(ρ)= lim S(ρ )=S(ρ). (14) n n→∞n b b D(ρkω)=infnα:linm→s∞upTr(cid:2){ρn ≥enαωn}ρn(cid:3)=0o For sequences of bipartite states ρ = {ρAnB}∞n=1, with (8) ρAnB ∈ B((HA⊗HB)⊗n), the condbitional spectral en- which is thepreviously useddefinition ofthe spectralsup- tropy rates are defined as follows: divergence rate. Hencethetwodefinitions areequivalent. S(AB) := D(ρAB IA ρB); (15) | − | ⊗ Proposition 2 The spectral inf-divergence rate D(ρ ω) S(AB) := D(ρbAB IbA ρbB). (16) k | − | ⊗ is equivalent to b b b Inthe above,IA = IA ∞ andρA = ρA ∞ ,withIA D(ρkω)=supnα:linm→i∞nfTr(cid:2){ρn ≥enαωn}ρn(cid:3)=1o being the idenbtity o{penr}anto=r1actinbg in i{n Hn}A⊗nn=1and ρAn =n (9) TrBρAnB, the partial trace being taken on the Hilbert which is the previously used definition of the spectral inf- space HB⊗n. divergence rate. Despite these equivalences, it is useful to use the defi- Definition of min- and max-entropies nitions (6) and (7) for the divergence rates as they allow the application of Lemmas 1 and 2 in deriving various We start with the definition of non-smooth min- and properties of these rates. max-entropies. 4 Definition 2 ([20]) The min- and max-entropies of a Proof For any constant γ > 0, let us define projection bipartite state ρ relative to a state σ are defined by operators AB B Hmin(ρAB|σB):=−logmin{λ: ρAB ≤λ·IA⊗ρB} Qγn :={ρn <2−nγIn} (18) and and H (ρ σ ):=logTr π (I σ ) , max AB B AB A B | (cid:0) ⊗ (cid:1) Pγ :=I Qγ = ρ 2−nγI . (19) whereπ denotestheprojectorontothesupportofρ . n n− n { n ≥ n} AB AB In terms of these projections, we can write In the special case where the system B is trivial (i.e., 1-dimensional), we simply write H (ρ ) and min A S(ρ)=sup γ :limsupTr Pγρ =0 , (20) Hmax(ρA). These entropies thencorrespondto the usual n n→∞ (cid:2) n n(cid:3) o non-conditional R´enyi entropies of order infinity and b zero, or alternatively as Hmin(ρA)=H∞(ρA)=−logkρAk∞ S(ρ)=supnγ :linm→i∞nfTr(cid:2)Qγnρn(cid:3)=1o, (21) Hmax(ρA)=H0(ρA)=logrank(ρA) , b sinceeachρ inthesequenceρisastate(i.e.,Trρ =1). where denotes the L -norm. n n ∞ ∞ k·k From Proposition 2 and (12)bof S(ρ) it follows that the latter is equivalently given by the expression b Definition of smooth min- and max-entropies S(ρ)=sup γ :limsupTr Pγ(ρ 2−nγI ) =0 , n n→∞ (cid:2) n n− n (cid:3) o Smooth min- and max-entropies are generalizations b (22) of the above entropy measures, involving an additional From (21) it follows that, for any γ < S(ρ) and any smoothness parameter ε 0. For ε = 0, they reduce to δ >0, for n large enough, ≥ b the non-smooth quantities. Tr Qγρ >1 δ. (23) Definition 3 ([20]) For any ε 0, the ε-smooth min- (cid:2) n n(cid:3) − ≥ and max-entropies of a bipartite state ρ relative to a AB For any given α>0, let γ :=S(ρ) α, and let state σ are defined by − B b ργ :=Qγρ Qγ (24) Hε (ρ σ ):= sup H (ρ¯σ ) n n n n min AB| B ρ¯∈Bε min | B e Then using (23) and Lemma 3 we infer that, for n large and enough, Hε (ρ σ ):= inf H (ρ¯σ ) max AB| B ρ¯∈Bε max | B ||ρn−ργn||1 ≤2√δ. (25) where Bε(ρ):= ρ¯ 0: ρ¯ ρ 1 ε,Tr(ρ¯) Tr(ρ) . In other words, for n larege enough, ργ Bε(ρ ) with { ≥ k − k ≤ ≤ } n ∈ n ε=2√δ. Inthefollowing,wewillfocusonthe smoothmin- and e We first prove the upper bound max-entropiesforthecasewhereσ =ρ . Notethatthe B B quantities Hε (ρ B) := max Hε (ρ σ ) and Hmεax(ρAB|B)min:=AmBi|nσBHmεax(ρAσBB|σBm)indefiAnBe|dBin [20] S(ρ)≤εl→im0linm→i∞nf n1Hmεin(ρn) (26) are not studied in this paper. b For n large enough, RELATION BETWEEN NON-CONDITIONAL Hε (ρ ) sup H (ρ ) ENTROPIES min n ≡ ρn∈Bε(ρn) min n H (ργ)= log ργ Relation between S(ρb) and Hmεin(ρ) >≥ nγm=in ne(nS(ρ)− α)kenk∞ (27) − b Theorem 1 Given a sequence of states ρ = {ρn}∞n=1, The last line follows from the inequality ργn < 2−nγIn, where ρn ( n), with n = ⊗n, the inbf-spectral en- and since α is arbitrary, we obtain the deesired bound ∈ B H H H tropy rate S(ρ) is related to the smooth min-entropy as (26). follows: b We next prove the converse, i.e., 1 S(ρb)=εl→im0linm→i∞nf nHmεin(ρn) (17) S(ρ)≥εl→im0linm→i∞nf n1Hmεin(ρn) (28) b 5 Consider an operator ρε Bε(ρ ) for which The third line in (36) is obtained by using the bound n ∈ n −logkρεnk∞ =ρn∈sBuεp(ρn)(cid:2)−logkρnk∞(cid:3). (29) Tr(cid:2)Pnγ(ρn−ρεn)(cid:3)≤ε, which follows from Corollary 1, since ρε Bε(ρ ). We shall also make use of a quantity Υ(ω), defined n ∈ n for any sequence of positive operators ω = ω ∞ as To arrive at the last line of (36) we use Lemma 1 and {b n}n=1 the fact that TrPγ 2nγ, which follows from Lemma 2. follows: b n ≤ Letuschooseγ =α δ/2,foranarbitraryδ >0,with Υ(ω)=sup α:limsupTr ω 2−nαI Πα =0 , α = Υ(ρε) δ/2. The−n both the first and second terms n n→∞ (cid:2){ n ≥ n} n(cid:3) o on the r.h.s−. of (36) goes to zero as n . Therefore, b (30) b ′ → ∞ fornlargeenoughandanyδ >0,inthe limitε 0,we where Παn :=(ωn−2−nαIn). Note that Υ(ω) reduces to must have that → the inf-spectralentropyrateS(ω) givenby (22), if ω is a b sequence of states. b b Tr(Pγρ ) δ′, (37) Bythedefinitionofthesmoothmin-entropy,(28)then n n ≤ follows from Lemma 4 below. which in turn implies that γ S(ρ). ≤ From the choice of the parameters α and γ it follows b that Lemma 4 For any sequence of states ρ= ρ ∞ , and any ε > 0, there exists an n N, such{ tnh}ant=f1or all limΥ(ρε) δ <S(ρ). (38) 0 ∈ b ε→0 − n n0 b b ≥ Butsince δ is arbitrary,we obtainthe inequality(33). 1 S(ρ) log ρε , (31) ≥ n(cid:2)− k nk∞(cid:3) b with ρεn defined by (29). Relation between S(ρb) and Hmεax(ρ) Proof We prove this in two steps. We first prove that for any ε>0 and n large enough, Theorem 2 Given a sequence of states ρ = ρ ∞ , { n}n=1 where ρ ( ), with = ⊗n, the sup-spectral Υ(ρε) 1 log ρε , (32) entropynra∈te BS(Hρ)nis relateHd nto theHsmoothbmax-entropy ≥−n k nk∞ as follows: b b where ρε := ρε ∞ . We then prove that { n}n=1 1 b εl→im0Υ(ρε)≤S(ρ) (33) S(ρb)=εli→m0linm→s∞upnHmεax(ρn) (39) b b For any arbitrary η > 0, let α be defined through the Proof Bydefinition,thesup-spectralentropyrateforthe relation given sequence of states is kρεnk∞ =2−n(α+η). (34) S(ρ)=infnγ :linm→i∞nfTr(cid:2)Pnγρn(cid:3)=1o, (40) Thisimpliestheoperatorinequality,ρε 2−n(α+η)I 0, b and hence ρεn <2−nαIn. n− n ≤ where Pnγ is the projection operator defined by (19). From (40) it follows that, for any γ S(ρ) and any Hence, ≥ δ >0, for n large enough b Tr ρε 2−nαI (ρε 2−nαI )]=0, (35) (cid:2){ n ≥ n} n− n Tr Pγρ >1 δ. (41) Using this, and the definition of Υ(ρε), we infer that (cid:2) n n(cid:3) − α Υ(ρε). Then, using (34) we obtainb the bound For any given α>0, choose γ =S(ρ)+α, and let ≤ b 1 b −nlogkρεnk∞−η ≤Υ(ρε), ργn :=PnγρnPnγ (42) b e which in turn yields (32), since η is arbitrary. Then using (41) and Lemma 3 we infer that, for n large To prove (33) note that enough, 0 ≤ Tr(Pnγρn) ||ρn−ργn||1 ≤2√δ. (43) = Tr(Pγρε)+Tr Pγ(ρ ρε) e ≤ Tr(cid:2)Pnnγ(ρnεn−2−(cid:2)nαnIn)(cid:3)n+−2−nnα(cid:3)TrPnγ +ε anWd ehefinrcsetρepγnro∈veBtεh(eρnb)ouwnidth ε=2√δ. Tr ρε 2−nαI (ρε 2−nαI ) +2−n(α−γ)+ε. ≤ (cid:2){ n ≥ n} n− n (cid:3) limlimsup 1Hε (ρ ) S(ρ) (44) (36) ε→0 n→∞ n max n ≤ b 6 For n large enough, ChooseS(ρ)>β >R. Forsuchachoice,thesecondterm ontherighthandsideof(52)tendstozeroasymptotically Hε (ρ ) = inf H (ρ ) b max n ρn∈Bε(ρn) max n in n. However, the first term does not tend to 1 and we H (ργ) hence obtain the bound (51). ≤ max n = logranek(ργ) (45) n Fromthedefinition(42)ofργ itfeollowsthatrankργ n n ≤ TrPnγ. Hence, e e RELATION BETWEEN CONDITIONAL ENTROPIES Hε (ρ ) logTrPγ max n ≤ n nγ =S(ρ)+α, (46) Consider a sequence of bipartite states ρAB = ≤ where once again we use the boundbTrPnγ ≤ 2nγ. The {ρρAnABB}∞n∞=1,dwenitohteρtAnhBe c∈orBre(cid:0)s(pHoAnd⊗inHgsBe)q⊗une(cid:1)n.ceLoeftbrρedAuBce=d last line of (46) yields the desired bound (44) since α is { n }n=1 b states. arbitrary. ForthesequenceρAB,thesup-spectralconditionalen- To complete the proof of Theorem 2 we assume that tropy rate S(AB) and the inf-spectral conditional en- b 1 | limlimsup Hε (ρ )<S(ρ) (47) tropyrateS(AB),definedrespectivelyby(15)and(16), ε→0 n→∞ n max n can be expresse|d as follows: b and show that this leads to a contradiction. S(AB) = inf γ :liminfTr PγρAB =1 , (53) Let σn,ε be the operator for which | n n→∞ (cid:2) n n (cid:3) o H (σ ):= inf H (ρ ). (48) S(AB) = sup γ :limsupTr PγρAB =0 , (54) max n,ε ρn∈Bε(ρn) max n | n n→∞ (cid:2) n n (cid:3) o Hence, Hε (ρ ) = logrankσ , and the assumption where max n n,ε (47) is equivalent to the following assumption: Pγ := ρAB 2−nγIA ρB . (55) 1 n { n ≥ n ⊗ n} lim lim logrankσn,ε <S(ρ). (49) Here IA denotes the identity operator in ( ⊗n). ε→0n→∞n n B HA b We use the following key properties of Hε (ρ ρ ) Since σ Bε(ρ ), Trσ 1 ε. Let σ0 denote the min AB| B n,ε ∈ n n,ε ≥ − n,ε given by Lemma 5 and Lemma 6 below. projection onto the support of σ . Then n,ε Lemma 5 Letρ andσ bedensityoperators,let∆ Tr σ0 ρ = Tr (ρ σ )+σ σ0 AB B AB (cid:0) n,ε n(cid:1) (cid:2)(cid:0) n− n,ε n,ε(cid:1) n,ε(cid:3) be a positive operator, and let λ R such that = Tr (ρ σ )σ0 +Trσ ∈ (cid:2) n− n,ε n,ε(cid:3) n,ε ρ 2−λ I σ +∆ . AB A B AB Tr ρ σ (ρ σ ) +1 ε ≤ · ⊗ n n,ε n n,ε ≥ (cid:2){ ≤ } − (cid:3) − ≥ −ε+1−ε=1−2ε. (50) Then Hmεin(ρAB|σB)≥λ for any ε≥p8Tr(∆AB). The inequality in the third line follows from Lemma 1. Proof Define Wearriveatthelastinequalityin(50)byusingthebound α :=2−λ I σ AB A B · ⊗ Tr ρ σ (ρ σ ) ε, (cid:2){ n ≤ n,ε} n− n,ε (cid:3)≥− βAB :=2−λ IA σB +∆AB . · ⊗ which arises from the fact that σ Bε(ρ ). n,ε ∈ n and Note, however, that for n large enough, (50) leads to a contradiction, in the limit ε 0. This is because, for T :=α12 β−21 . → AB AB AB any real number R < S(ρ) and any projection π , with n Trπn =2nR, for n large ebnough, we have Let|Ψi=|ΨiABRbeapurificationofρAB andlet|Ψ′i:= T I Ψ and ρ′ :=Tr (Ψ′ Ψ′ ). Tr(π ρ ) 1 c , (51) AB⊗ R| i AB R | ih | n n 0 Note that ≤ − for some constant c0 > 0. The inequality (51) can be ρ′ =T ρ T† proved as follows: AB AB AB AB T β T† ≤ AB AB AB Tr(π ρ ) = Tr π (ρ 2−nβI ) +2−nβTrπ =αAB =2−λ IA σB , n n n n n n · ⊗ (cid:2) − (cid:3) Tr ρ 2−nβI (ρ 2−nβI ) which implies H (ρ′ σ ) λ. It thus remains to be ≤ (cid:2){ n ≥ n} n− n (cid:3) min AB| B ≥ shown that +2−n(β−R) (52) ρ ρ′ 8Tr(∆ ) . (56) k AB− ABk1 ≤p AB 7 We first show that the Hermitian operator Lemma 6 Let ρ and σ be density operators. Then AB B T¯AB := 12(TAB+TA†B) . Hmεin(ρAB|σB)≥λ for any λ R and satisfies ∈ T¯AB ≤IAB . (57) ε=q8Tr(cid:0){ρAB >2−λ·IA⊗σB}ρAB(cid:1) . For any vector φ = φ AB, Proof Let ∆+ and ∆− be mutually orthogonal posi- | i | i AB AB tive operators such that T φ 2 = φT† T φ = φβ−12α β−12 φ k AB| ik hφ|βA−B21βABβ| −i21 φh =| ABφ A2B AB| i ∆+AB−∆−AB =ρAB −2−λ·IA⊗σB . ≤h | AB AB AB| i k| ik Furthermore, let P be the projector onto the support AB wheretheinequalityfollowsfromαAB ≤βAB. Similarly, of ∆+AB, i.e., 1 1 kTA†B|φik2 =hφ|TABTA†B|φi=hφ|αA2BβA−B1αA2B|φi PAB ={ρAB >2−λ·IA⊗σB} . 1 1 φα2 α−1α2 φ = φ 2 ≤h | AB AB AB| i k| ik We then have αw−h1erewthhicehinheoqldusalbiteycafuoslleowthsefrfuomncttihone fτact thτat−1βA−isB1op≤- PABρABPAB =PAB(2−λ·IA⊗σB +∆+AB −∆−AB)PAB AB 7→ − ∆+ eratormonotoneon(0, )(seePropositionV.1.6of[2]). ≥ AB ∞ We conclude that for any vector φ , | i and, hence, 1 T¯ φ T φ +T† φ k AB| ik≤ 2k AB| i AB| ik q8Tr(∆+AB)≤p8Tr(PABρAB)=ε . 1 1 T φ + T† φ φ , ≤ 2k AB| ik 2k AB| ik≤k| ik The assertion now follows from Lemma 5 because which implies (57). ρ 2−λ I σ +∆+ . We now determine the overlapbetween Ψ and Ψ′ , AB ≤ · A⊗ B AB | i | i ΨΨ′ = ΨT I Ψ AB R h | i h | ⊗ | i In the following sections we state and prove the rela- =Tr(Ψ ΨT I )=Tr(ρ T ) . AB R AB AB | ih | ⊗ tions between the conditional spectral entropy rates and Because ρ has trace one, we have the smooth conditional max- and min-entropy. AB 1 ΨΨ′ 1 ΨΨ′ =Tr ρ (I T¯ ) AB AB AB −|h | i|≤Tr−βℜh (|I i T¯ (cid:0) ) − (cid:1) Relation between S(A|B) and Hmεax(ρAB|ρB) AB AB AB ≤ (cid:0) − (cid:1) 1 1 =Tr(βAB)−Tr(αA2BβA2B) Theorem 3 Given a sequence of bipartite states ρAB = ≤Tr(βAB)−Tr(αAB)=Tr(∆AB) . {ρAnB}∞n=1, where ρAnB ∈ B(cid:0)(HA ⊗ HB)⊗n(cid:1), thbe sup- spectralconditionalentropyrateS(AB), definedby(53), | Here, the second inequality follows from the fact that, satisfies because of (57), the operator I T¯ is positive and AB AB − 1 1 ρA1B ≤ βAB. The last inequality holds because αA2B ≤ S(A|B)=εl→im0linm→s∞upnHmεax(ρAnB|ρBn), (58) β2 ,whichisaconsequenceoftheoperatormonotonicity AB of the square root (Proposition V.1.8 of [2]). where Hε (ρAB ρB) is the smooth max-entropy of the max n | n Using (5) and the fact that the fidelity between two stateρAB of the sequence, conditional on thecorrespond- n pure states is given by their overlap,we find ing reduced state ρB. n k|ΨihΨ|−|Ψ′ihΨ′|k1 ≤2p2(1−|hΨ|Ψ′i|) Proof Fromthe definition (53)ofS(A|B) it followsthat 2 2Tr(∆ ) ε . for any γ S(AB) and any δ >0, for n large enough ≤ p AB ≤ ≥ | Inequality (56) then follows because the trace distance Tr(cid:2)PnγρAnB(cid:3)>1−δ, (59) can only decrease when taking the partial trace. where Pγ is defined by (55). n 8 For any given α>0, choose γ =S(AB)+α, and let We arrive at the second last line of (68) using Lemma | 1. The last line of (68) is obtained analogously to (50), ρAn,Bγ :=PnγρAnBPnγ (60) since σnA,Bε ∈Bε(ρAnB). Note,however,that(68)leadstoacontradiction. This Then using (59) and Lemma 3 we infer that, for n large canbe seenasfollows: LetRbe arealnumbersatisfying enough, ρAB Bε(ρAB) with ε=2√δ. Let πAB denote n,γ ∈ n n,γ the projection onto the support of ρAB. Tr πAB(IA ρB) =2nR. n,γ (cid:2) n,ε n ⊗ n (cid:3) We first prove bound It follows from the assumption (67) that, for ε small 1 enough, R<S(AB). Note that limlimsup Hε (ρ ) S(AB). (61) | ε→0 n→∞ n max n ≤ | Tr(πABρAB) n,ε n For n large enough, = Tr πAB(ρAB 2−nγIA ρB) (cid:2) n,ε n − n ⊗ n (cid:3) Hmεax(ρAnB|ρBn) := ρn∈Binε(fρAnB)Hmax(ρAnB|ρBn) +Tr2−nργATBr(cid:2)πnA2,B−εn(IγnAIA⊗ρBnρB)(cid:3)(ρAB 2−nγIA ρB) H (ρAB ρB) ≤ (cid:2){ n ≥ n ⊗ n} n − n ⊗ n (cid:3) ≤ max n,γ| n +2−n(γ−R) = logTr (IA ρB)πAB (cid:0) n ⊗ n n,γ(cid:1) (69) logTr (IA ρB)Pγ ≤ (cid:0) n ⊗ n n(cid:1) Choose S(AB) > γ > R. For such a choice, the second nγ (62) | ≤ term onthe right handside of (69) tends to zero asymp- Thelastinequalityin(62)followsfromLemma2. Hence, totically in n. However, the first term does not tend to for n large enough, 1 and we hence obtain the bound 1Hε (ρAB ρB) γ =S(AB)+α, (63) Tr(πnA,BερAnB)<1−c0, (70) n max n | n ≤ | for some constant c > 0. This contradicts (68) in the 0 and since α is arbitrary, we obtain the desired bound limit ε 0. → (61). To complete the proof of Theorem 3, we assume that Relation between S(A|B) andHmεin(ρAB|ρB) 1 limlimsup Hε (ρAB ρB)<S(AB), (64) ε→0 n→∞ n max n | n | Theorem 4 Given a sequence of bipartite states ρAB = andprovethat this leads to a contradiction. Let σnA,Bε be {ρAnB}∞n=1, where ρAnB ∈ B(cid:0)(HA ⊗ HB)⊗n(cid:1), thbe inf- the operator for which spectral conditional entropy rate S(AB) is related to the | smooth conditional min-entropy as follows: H (σAB ρB)= inf H (ρAB ρB). (65) max n,ε| n ρAB∈Bε(ρAnB) max | n S(AB)= limliminf 1Hε (ρAB ρB) (71) | ε→0 n→∞ n min n | n Hence, Proof Hε (ρAB ρB) = H (σAB ρB) We first prove the bound max n | n max n,ε| n = logTr(cid:0)(InA⊗ρBn)πnA,Bε(cid:1), (66) S(A|B)≥εl→im0linm→i∞nf n1Hmεin(ρAnB|ρBn) (72) where πnA,Bε is the projection onto the support of σnA,Bε. Let σAB be the operator for which n,ε Hence,theassumption(64)isequivalenttothefollow- ing assumption: H (σAB ρB)= max H (ρAB ρB). (73) min n,ε| n ρAB∈Bε(ρAnB) min | n 1 limlimsup logTr πAB(IA ρB) <S(AB). (67) Let us define ε→0 n→∞ n (cid:2) n,ε n ⊗ n (cid:3) | Υε(AB) Note that | := sup α:limsupTr σAB 2−nαIA ρB Πα =0 , Tr(πABρAB) n n→∞ (cid:2){ n,ε ≥ n ⊗ n} n(cid:3) o n,ε n (74) = Tr (ρAB σAB)+σAB πAB (cid:2)(cid:0) n − n,ε n,ε(cid:1) n,ε(cid:3) = Tr(cid:2)(ρAnB −σnA,Bε)πnA,Bε(cid:3)+TrσnA,Bε whAercecoΠrdαnin:=g σtonA,BεD−efi2n−itnioαnInA3⊗ofρBnth.e conditional smooth ≥ Tr(cid:2){ρAnB ≤σnA,Bε}(ρAnB−σnA,Bε)(cid:3)+Tr(cid:2)σnA,Bε(cid:3) min-entropy, that to prove (72), it suffices to prove the ε+1 ε=1 2ε. (68) following lemma: ≥ − − − 9 Lemma 7 For any sequence of bipartite states ρAB = Therefore,fornlargeenoughandanyδ′ >0,inthelimit ρAB ∞ , and any ε > 0, there exists an n N, such ε 0, we must have that { n }n=1 0 ∈ b → that for all n≥n0 Tr(PnγρAnB)≤δ′, (83) S(A|B)≥−n1 log(cid:2){min{λ:σnA,Bε ≤λInA⊗ρBn}}(cid:3), (75) cwhhoiiccheionfttuhrenpiamrapmlieestetrhsaαt γan≤dSγ(Ait|Bfo)l.loHwesntchea,tfrom the with σAB defined by (73). limΥε(AB) δ S(AB), (84) n,ε ε→0 | − ≤ | Proof We provethis lemma in two steps. We first prove andsince δ is arbitrary,we obtainthe inequality (77). that for any ε>0 and n large enough, We next prove the bound Υε(A|B)≥−n1 log(cid:2)min{λ:σnA,Bε ≤λInA⊗ρBn}(cid:3). (76) S(A|B)≤εl→im0linm→i∞nf n1Hmεin(ρAnB|ρBn) (85) Proof of (85): Letδ >0 be arbitrarybut fixed. Thenby We then prove that the definition ofthe inf-spectralconditionalentropyrate S(AB) limΥε(AB). (77) there exists γ ∈R such that | ≥ε→0 | γ >S(AB) δ (86) | − Proof of (76): For any arbitrary η > 0, let α be defined and through the relation limsupTr ρAB 2−nγIA ρB ρAB =0 . (87) min λ:σAB λIA ρB =2−n(α+η). (78) n→∞ (cid:2){ n ≥ n ⊗ n} n (cid:3) { n,ε ≤ n ⊗ n} Inparticular,foranyε>0there existsn N suchthat 0 ∈ Hence, for all n n0. ≥ 1 log min λ:σAB λIA ρB =α+η (79) Tr(cid:2){ρAnB >2−nγ ·InA⊗ρBn}ρAnB(cid:3) − n (cid:2) { n,ε ≤ n ⊗ n}(cid:3) ε2 Tr ρAB 2−nγ IA ρB ρAB < . (88) ≤ (cid:2){ n ≥ · n ⊗ n} n (cid:3) 8 Note that (78) implies that σAB 2−n(α+η)(IA ρB), n,ε ≤ n ⊗ n Using Lemma 6 we then infer that for all n n and hence (σAB 2−n(α+η)IA ρB) 0. This in turn ≥ 0 implies that (nσ,εnA,Bε−−2−nαInA⊗n ρ⊗Bn)n≤0≤and hence Hmεin(ρAnB|ρBn)≥nγ (89) and, hence Tr σAB 2−nαIA ρB (σAB 2−nαIA ρB) =0. (cid:2){ n,ε ≥ n ⊗ n} n,ε − n ⊗ n (cid:3) (80) liminf 1Hε (ρAB ρB) γ . (90) It then follows from the definition (74) of Υε(AB) that n→∞ n min n | n ≥ α Υε(AB). Hence, using (79), we get | Because this holds for any ε>0, we conclude ≤ | 1 limliminf Hε (ρAB ρB) γ >S(AB) δ . (91) 1 log min λ:σAB λIA ρB η Υε(AB), (81) ε→0 n→∞ n min n | n ≥ | − −n (cid:2) { n,ε ≤ n⊗ n}(cid:3)− ≤ | Theassertion(85)thenfollowsbecausethisholdsforany δ >0. which in turn yields (76), since η is arbitrary. Proof of (77): Defining Pγ := ρAB 2−nγIA ρB , n { n ≥ n ⊗ n} note that CONCLUSIONS Tr PγρAB (cid:2) n n (cid:3) So far, the information spectrum approach and the = Tr PγσAB +Tr Pγ(ρAB σAB) (cid:2) n n,ε(cid:3) (cid:2) n n − n,ε (cid:3) smooth entropy framework have been applied within Tr Pγ(σAB 2−nα(IA ρB) pretty different subfields of information theory [30]. In ≤ (cid:2) n n,ε − n ⊗ n (cid:3) +2−nαTr Pγ(IA ρB) +ε the quantum regime, spectral entropy rates have mostly (cid:2) n n ⊗ n (cid:3) been used to characterize information sources, commu- Tr σAB 2−nαIA ρB (σAB 2−nα(IA ρB) ≤ (cid:2){ n,ε ≥ n ⊗ n} n,ε − n ⊗ n (cid:3) nication channels and entanglement manipulations. In +2−n(α−γ)+ε (82) contrast, smooth entropies proved useful in the context of randomness extraction and cryptography. We hope In the above we have made use of Lemma 1, Lemma 2 that our result bridges the gap between these two sub- and Corollary 1. fields. Infact,forthestudyofasymptoticsettingswhere Letuschooseγ =α δ/2,foranarbitraryδ >0,with the underlying resources are available many times, both α=Υε(AB) δ/2. Th−enboththefirstandsecondterms the information-spectrum approach and the smooth en- | − on the right hand side of (82) goes to zero as n . tropy framework can be used equivalently. → ∞ 10 ACKNOWLEDGMENTS α= (ρ ω) δ, γ = (ρ ω) 2δ,we havefromLemma D k − D k − 1, The authors are very grateful to Patrick Hayden for 1n→∞Tr ρ enαω ρ stimulating exchanges, helpful comments and for care- ← (cid:2){ n ≥ n} n(cid:3) fully reading the proofs. They also thank Masahito Tr ρn enγωn (ρn enγωn) ≤ (cid:2){ ≥ } − (cid:3) Hayashi, Robert K¨onig, Jonathan Oppenheim and An- +enγTr ρ enαω ω n n n dreas Winter for interesting discussions. ND acknowl- (cid:2){ ≥ } (cid:3) Tr ρ enγω (ρ enγω ) +e−nδ (95) edges the kind hospitality of McGill University (Mon- ≤ (cid:2){ n ≥ n} n− n (cid:3) treal), where part of this work was done. where Tr ρ enαω ω e−nα holds for any α. n n n (cid:2){ ≥ } (cid:3) ≤ Thuslim Tr ρ enγω (ρ enγω ) =1,where n→∞ n n n n (cid:2){ ≥ } − (cid:3) γ >D(ρ ω), which is a contradiction. APPENDIX k In this Appendix we give the proofs of Proposition 1 and Proposition 2. ∗ Electronic address: [email protected] † Electronic address: [email protected] PROOF OF PROPOSITION 1 [1] C.H.Bennett,G.Brassard,C.Cr´epeau,andU.Maurer, “Generalized privacy amplification,” IEEE Trans. Inf. Proof For any α= (ρ ω)+δ, with δ >0, implies Theory, vol. 41, pp. 1915–1923, 1995. D k [2] R. Bhatia, Matrix Analysis, Springer. 0= lim Tr ρ enαω ρ [3] G.BowenandN.Datta,“Beyondi.i.d.inquantuminfor- n n n n→∞ (cid:2){ ≥ } (cid:3) mation theory,” arXiv:quant-ph/0604013, Proceedings of lim Tr ρ enαω (ρ enαω ) the 2006 IEEEInternational Symposium onInformation n n n n ≥n→∞ (cid:2){ ≥ } − (cid:3) Theory, 2006. 0 (92) [4] G. Bowen andN.Datta,“Quantumcodingtheoremsfor ≥ arbitrarysources,channelsandentanglementresources,” arXiv:quant-ph/0610003, 2006 giving (ρ ω) D(ρ ω), as δ is arbitrary. For the con- D k ≥ k [5] G. Bowen and N. Datta, “ Asymptotic entanglement verse we assume that the inequality is strict, such that manipulation of bipartite pure states,” arXiv:quant- (ρ ω) = D(ρ ω)+4δ for some δ > 0. Then choosing ph/0610199, 2006. D k k α=D(ρ ω)+2δ,γ =D(ρ ω)+δ,wehavefromLemma [6] G. Bowen and N. Datta, “ Entanglement cost for se- k k 1, quences of arbitrary quantum states,” arXiv:0704.1957, 2007. Tr ρ enαω ρ Tr ρ enγω (ρ enγω ) [7] C. Cachin, “Smooth entropyand R´enyientropy,”in Ad- n n n n n n n (cid:2){ ≥ } (cid:3)≤ (cid:2){ ≥ } − (cid:3) vances in Cryptology — EUROCRYPT ’97, LNCS, vol. +enγTr ρ enαω ω (cid:2){ n ≥ n} n(cid:3) 1233, pp.193–208. Springer, 1997, ε +e−nδ (93) [8] I. Damg˚ard, S. Fehr, R. Renner, L. Salvail, and n ≤ C.Schaffner,“Atighthigh-orderentropicuncertaintyre- lation with applications,” in Advances in Cryptology — whereε =Tr ρ enγω (ρ enγω ) andTr ρ n (cid:2){ n ≥ n} n− n (cid:3) (cid:2){ n ≥ CRYPTO 2007,LNCS,vol.4622,pp.360–378. Springer, enαω ω e−nα holds for any α. As the right hand n n 2007. } (cid:3) ≤ side goes to zero asymptotically and since α < (ρ ω) [9] I.Damgaard,S.Fehr,L.Salvail,andC.Schaffner,“Cryp- D k we have a contradiction. tography in the bounded quantum-storage model,” in 46th Annual Symposium on Foundations of Computer Science (FOCS), pp. 449–458, 2005. [10] I.Damg˚ard,S.Fehr,L.Salvail,andC.Schaffner,“Secure PROOF OF PROPOSITION 2 identificationandQKDinthebounded-quantum-storage model,” in Advances in Cryptology — CRYPTO 2007, vol. 4622, pp.342–359. Springer, 2007. Proof For any α=D(ρ ω) δ, with δ >0, implies [11] T. S. Han, Information-Spectrum Methods in Informa- k − tion Theory, Springer-Verlag, 2002. 1 lim Tr ρn enαωn ρn [12] T. S. Han and S. Verdu, “Approximation theory of out- ≥n→∞ (cid:2){ ≥ } (cid:3) putstatistics,”,IEEETrans.Inform.Theory,vol.39,pp. lim Tr ρ enαω (ρ enαω ) n n n n 752–772, 1993. ≥n→∞ (cid:2){ ≥ } − (cid:3) [13] M. Hayashi and H. Nagaoka, “General formulas for ca- =1 (94) pacity of classical–quantum channels,” IEEE Trans. In- form. Theory, vol. 49, pp.1753–1768, 2003. giving (ρ ω) D(ρ ω), as δ is arbitrary. For the con- [14] M.Hayashi,“Generalformulasforfixed-lengthquantum D k ≥ k verse we assume that the inequality is strict, such that entanglement concentration,”IEEE Trans. Inform. The- (ρ ω) = D(ρ ω)+4δ for some δ > 0. Then choosing ory, Vol. 52, No. 5, 1904-1921, 2006. D k k