Potential Capacities of Quantum Channels 2

Home , Classical capacity

arXiv:1505.00907v5 [quant-ph] 18 Jan 2016 hne o ie aki h etpsil context. possible best the usefulnes in these of task for given limit bounds a ultimate upper for the provide channel quantify and way which operational quantities, an in nels hjag301,Cia mi:[email protected]. Email: China. Univ 310018, Jiliang China Zhejiang Information, Quantum ( for Laboratory Bellaterra and ES-08193 Barcelona, de Autònoma Universitat ITRADYN:PTNILCPCTE FQATMCHANNELS QUANTUM OF CAPACITIES POTENTIAL YANG: AND WINTER Breoa,San mi:[email protected]. Email: Spain. (Barcelona), E inputs Barcelona, Autònoma de Fenòmens Quàntics product Universitat possible using is just counterpart. than it classical information channels, more no the transmit for has to is inputs quantum which entangled theory transmit entanglement, Employing to Shannon to useful quantum is due in it to Non-additivity channel, related information. questions quantum whet answer as a such to given ones, qualitative [11], difficult simple cf. some it even capacities, impossible, makes approach This variables numerical number [12]. head-on have infinite an a we over making where optimization an (“regularization”), perform uses to gro channel over of formula optimization known involves numbers capacities best the the calculate in and be of to science, center to the information at much quantum known is which in is are [10], [9], quantities capacities [8], [7], relevant channel non-additive The quantum single-lette wher complicated. of famous theory, more Shannon’s status information capacity by the private classical expressed formula, the to is and capacity contrast a [5] the In for [4], [6]. [2] defined [3], [1], [5], be capacity capacity be classical can quantum the to that the them is capacities among information channel, several quantum of are type there what sent, on Depending faithfully. T diiiy nageet aaadchannel. Hadamard entanglement, additivity, bet hrceieis“oeta”version. “potential” its discu characterize activa also to We be able becaus capacities. channel; capacity, cannot three quantum contextual it the environment-assisted other so-called all then any for however channel, by holds noisy, noiseless channel conclusion is noiseless the channel the a to into if is fo that it show that close we Emp implying equal. bounds additive, are upper capacity capacity strongly these plain private and is and potential Hadamar channels, channel quantum a these Hadamard the to a that channel the prove the of of then “lift” terms and first in quan channel we the capacity, bound for private bound For upper a and capacities: an analogues establish To potential “plain” give formation. of some their we entanglement on as capacity, compute bounds classical to upper the hard study as thus be We to seem ities aucitdt 6Jnay2016. January 16 date Manuscript ogYn swt ´sc eoia nomc´ Fenòmens i Informació Teòrica: F´ısica with is Yang Dong nra itri ihIRAadFıiaT`rc:Informac Teòrica: F´ısica and ICREA with is Winter Andreas notntl,ecp o e sltdcss oeta c potential cases, isolated few a for except Unfortunately, Abstract ne Terms Index aaiyo os hne o rnmtiginformation the transmitting find for to channel is noisy a theory of information capacity in problem central HE W introduce —We oeta aaiiso unu Channels Quantum of Capacities Potential qatmcanl oeta aaiy non- capacity, potential channel, —quantum .M I. oeta capacities potential OTIVATION fqatmchan- quantum of nra itradDn Yang Dong and Winter Andreas -89 Bellaterra S-08193 acln) Spain Barcelona), riy Hangzhou, ersity, eare we e Quàntics loying fa of s sthe ss terest apac- wing tum ´ i ió her, this ted d e r r . , , . noapretoe ial eedwt umr n open and summary a with V. end Section we activat bounds in be Finally questions upper cannot one. channel give perfect imperfect or a an into it that prove evaluate and poten IV it, of Section for notion in the and introduce we capacity III, furth environment-assis Section channels, the In capacities. and degradable entanglement-assisted of of the additivity formulas more about regularized results capacit private the the and known review quantum, basic (classical, we capacities some three particular, state In and facts. definitions notation, much. introduce s too help the this side cannot on In but possibility. bounds zero-capacity help this upper can proper exclude provide entanglement will to a we capacities of work, potential this assistance In noiseles a the channel. like behave by on could channel superactivation, channel noisy ha by a seems that Encouraged guess it question. might channels, this diffic zero-capacity is answer it the to Since all channel?” characterize side to zero-capacity suitable a of hti Cnaniycanl hs unu aaiyis capacity quantum whose channel, extreme? noisy other the a at ≤ “Can sense is this in That capacit help low entanglement of regime Could the when advantage, exhibit capacities entanglement Superactivation of jointly. channel regime used the are of incr channels sum an two the the i.e., above superadditivity, capacity a to the differe rise provide of give entangling can can generally uses uses more channel but channel en- advantage, different that dramatic channel across shows side inputs capacity zero-capacity channel tangled another quantum of of o Superactivation positive-capacity assistance a the becomes under channel zero-capacity a that iesoa.Rcl htaqatmstate quantum a that Recall dimensional. otxulcanl?W nrdc h oeta aaiyt other capacity any notion. potential with this the combination capture introduce formally transmit in We to used channels? channel is contextual a it Wh of question: when capability following possible information the maximum ask are to the channels natural is contextual is other it what uti So on available. the depends since channel channel, the the of characterize adequately ch not quantum a does of capacity the together. that implies used super-additivity, are they when ca transmit rate but positive cannot individually, at used that transmit are classical channels they in when quantum information impossible two quantum are exist there that superactivat [13]; effects is phenomenon of extreme An theory. kinds information all t to open channels door quantum different between inputs Entangled hsppri tutrda olw.I eto Iwe II Section In follows. as structured is paper This easm htalHletsae,denoted spaces, Hilbert all that assume We ueatvto a lob ehae na lentv way alternative an in rephrased be also can Superactivation of broadly more and superactivation, of phenomenon The log d − δ eoepretynieesudrteassistance the under noiseless perfectly become , I N II. TTO N PRELIMINARIES AND OTATION ρ salna operator linear a is H r finite are , ) and y), annel one s ense, ease lity tial ion ted er- ult he ed ne rd nt at y. o n e s 1 , . WINTER AND YANG: POTENTIAL CAPACITIES OF QUANTUM CHANNELS 2

BE A B on satisfying ρ 0 and Tr ρ =1. A quantum channel is a where ρ = Uρ U †, U is the isometry of , and S(ρ ) completelyH positive≥ and trace preserving (CPTP) linear map, S(ρE)= S(ρB) S(ρRB) is known as coherentN information− from an input system A to output system B (we shall generally [16], [17], [18], [19].− use the same names for the underlying Hilbert spaces). From The private capacity of the quantum channel is given by the Stinespring dilation theorem [14], we know that for a [5], [6] N 1 (1) n (channel , there always exist an isometry U : A ֒ B E P ( ) = lim P ( ⊗ ), (7 N → ⊗ n n for some environment space E, i.e. U †U = 11, such that N →∞ N (1) (ρ) = TrE UρU †. The complementary channel of , which with P ( ) defined as weN denote c, is the channel that maps fromN the input N N P (1)( )= max (I(T : B) I(T : E)), system A to the environment system E, obtained by taking N pt,ρt − { } the partial trace over system B rather than the environment: TBE T A c with respect to ρ = pt t t Uρt U †. (ρ) = TrB UρU †. Since the Stinespring dilation is unique t | ih | ⊗ upN to a change of basis of environment E, c is well- X defined up to unitary operations on E. A quantumN channel Now we recall the definition of a degradable channel and has another representation known as the Kraus representation: its properties on quantum and private capacities.

(ρ) = i KiρKi†, where Ki are called Kraus operators N ABC Definition 1 A channel is called degradable [20] if it satisfying i Ki†Ki = 11. Given a multipartite state ρ , we N c A P ABC can simulate its complementary channel , i.e. there is a write ρ = TrBC ρ for the corresponding reduced state. N c P A degrading CPTP map such that = . The von Neumann entropy is defined as S(A)ρ = S(ρ ) = D D ◦ N N A A Tr ρ log ρ . The conditional von Neumann entropy of A Lemma 2 (Devetak/Shor [20]) If and are degradable − AB B given B is defined as S(A B) = S(ρ ) S(ρ ), the channels, then their single-letter quantumN capacityM is additive: | A B− AB mutual information I(A : B) = S(ρ )+ S(ρ ) S(ρ ), Q(1)( )= Q(1)( )+ Q(1)( ). and the conditional mutual information as I(A −: B C) = N ⊗M N M S(ρAC )+ S(ρBC ) S(ρABC ) S(ρC ). When there| is no − − Lemma 3 (Smith [21]) If a quantum channel is a degrad- ambiguity as to which state is being referred, we simply write able channel, then its quantum capacity is equalN to its private A S(A)= S(ρ ). capacity, and both are given by the single-letter coherent We review the regularization formulas of the three principal information: Q( )= P ( )= Q(1)( )= P (1)( ). capacities: the classical, quantum, and private capacity. N N N N The classical capacity of a quantum channel is the rate at We furthermore recall two other capacities: The which one can reliably send classical information through a entanglement-assisted classical capacity of [22], which is quantum channel, and is given by [1], [2], the capacity for transmitting classical informationN through the 1 n channel with the help of unlimited prior entanglement shared C( ) = lim χ ⊗ , (1) n n N →∞ N between the sender and the receiver and which is given by with the Holevo capacity χ( ) defined as the simple and beautiful formula N CE( ) = max I(R : B), (8) χ( )= max S pi (φi) piS (φi) . (2) N ρA N pi,φi N − N { } i ! i X X where I(R : B)= S(ρR)+ S(ρB) S(ρRB) is the quantum Note the elementary rewriting of the Holevo capacity as mutual information of the state ρRB− = (id )( φ φ RA), follows, known as the MSW identity [15]: with a purification φ RA of ρA. And the environment-assisted⊗ N | ih | B BE | i χ( ) = max S(ρ ) EF (ρ ), (3) quantum capacity, which refers to active feed-forward of A N ρ − classical information from the channel environment E to the BE A where ρ = Uρ U †, U is the Stinespring isometry of , receiver B [23], [24], is given by BE N and EF (ρ ) is the entanglement of formation of the bipartite A A BE QA( ) = max min S(ρ ),S (ρ ) . (9) state ρ defined as N ρA N BE B BE BE n o EF (ρ ) = min piS(φ ), s.t. ρ = pi φi φi . i | ih | III. POTENTIAL CAPACITIES i i X X (4) Notice that the formulas for C, P and Q all are regularized The quantum capacity of a quantum channel is the rate at expressions due to the non-additivity of their respective single- which one can reliably send quantum information through a letter quantities, χ, P (1) and Q(1). quantum channel, and is given by [3], [4], [5], We call a real function f( ) on the set of channels weakly- n N 1 (1) n additive if f( ⊗ ) = nf( ) for all n 1, and strongly- Q( ) = lim Q ( ⊗ ), (5) N N ≥ N n n N additive if f( )= f( )+ f( ) for any channels →∞ N ⊗M N M N with Q(1)( ) defined as and . Obviously, if f is strongly-additive, then it is also N weakly-additiveM but not vice versa; and example of this is (1) A RA Q ( )= max S (φ ) S id (φ ) , given by the environment-assisted capacity Q . Further- N φ RA N − ⊗ N A( ) | i (6) more, for fixed f, we call a channel strongly-additiveN , if = max(S(ρB ) S(ρE)), N ρA − for all other channels , f( )= f( )+ f( ). M N ⊗M N M WINTER AND YANG: POTENTIAL CAPACITIES OF QUANTUM CHANNELS 3

(1) From their expression as regularizations, or directly from Proof: We prove the claim for fp ( ); the proof for ( ) N the definition, one can directly deduce that the capacities fp∞ ( ) is similar. Namely, for an arbitrary channel , C( ), Q( ) and P ( ) are weakly-additive. Furthermore, it N T N N N f (1)( ) f (1)( ) is known that neither Q( ) nor P ( ) are strongly-additive; N ⊗M⊗T − T C( ) is believed to be notN strongly-additive,N though this has = f (1)( ) f (1)( ) N N ⊗M⊗T − M ⊗ T not been proved so far. The single-letter quantities χ( ), + f (1)( ) f (1)( ), Q(1)( ), P (1)( ) are not even weakly-additive. N M ⊗ T − T N N sup f (1)( ) f (1)( ) Due to their non-additivity, the capability to transmit in- ≤ N ⊗ S − S formation through a quantum channel does not only depend S h i + sup f (1)( ) f (1)( ) , on the channel itself, but also on any contextual channel M ⊗ S − S with which it can be combined. So the standard capacity S h i = f (1)( )+ f (1)( ). cannot uniquely characterize the utility of the channel. It is p N p M natural to consider the maximal possible capability to transmit Maximization over concludes the proof. T information when it is used in combination with any other contextual channels. We introduce the potential capacity to Lemma 6 The potential capacity is upper bounded by the describe this notion. It describes the potential capability that potential single-letter capacity, more precisely can be activated by a proper contextual channel. Since the (1) ( ) ( ) (1) f ( ) f ∞ ( ) f ∞ ( ) f ( ). three capacities share the same property, we define the notion N ≤ N ≤ p N ≤ p N in a unified way. Proof: The first “ ” comes from Eq. (10) by taking n =1 ≤ In the following definitions, we assume a super-additive and the second “ ” from Eq. (11) by taking as a fixed ≤ M function f, i.e. f( ) f( )+ f( ) for any channels state channel. For the third “ ”, consider the following chain N⊗M ≥ N ( M) ≤ and , so that the regularization f ∞ is given by of inequalities: N M ( ) ( ) ( ) 1 n 1 n f ∞ ( ) f ∞ ( ) f ∞ ( ) = sup f( ⊗ ) = lim f( ⊗ ). (10) n n n N ⊗M − M N n N →∞ N 1 n n 1 n = lim f( ⊗ ⊗ ) lim f( ⊗ ), ( ) n n n n By its definition, f ∞ is always weakly-additive, and f( ) →∞ N ⊗M − →∞ M ( ) N ≤ 1 n n n f ∞ ( ). = lim f( ⊗ ⊗ ) f( ⊗ ) , N n n →∞ N ⊗M − M Definition 4 For a channel , the potential capacity associ- 1 (1) n lim f ( ⊗ ), N n n p ated to f is defined as ≤ →∞ N 1 (1) (1) ( ) ( ) ( ) lim nfp ( )= fp ( ), fp∞ ( ) := sup f ∞ ( ) f ∞ ( ) , (11) ≤ n n N N N N ⊗M − M →∞ M h i where the first inequality uses the definition of the potential ( ) where f ∞ ( ) is the regularization of f. N single-shot capacity and the second one the sub-additivity. Similarly, the potential single-letter capacity is defined as Hence we have f (1)( ) := sup [f( ) f( )] , (12) (1) ( ) ( ) (1) p N N ⊗M − M f ( ) f ∞ ( ) fp∞ ( ) fp ( ). M N ≤ N ≤ N ≤ N where f (1)( )= f( ) is the single-letter function. N N This notion has been introduced before in [25, Sec. VII], Remark Notice that all capacities and their single-letter for the case of f = Q(1), under the name of “quantum value formulations are super-additive, and that the single-letter form added capacity.” is a lower bound of the regularized form. However, their “potential” counterparts have the reverse relation; this was Note that we have (always assuming super-additivity of f) glimpsed in [25] without any further investigation. f (1)( )= f( ) iff is strongly additive. p N N N IV. FIVE CONCRETE POTENTIAL CAPACITIES Eq. (11) is difficult to calculate because of the unlimited Now we can turn to five concrete examples. We start with dimension of the contextual channel and the regularization the entanglement-assisted capacity, which presents a trivial function f . Eq. (12) looks simplerM but still suffers from the ∞ case: Namely, CE is known to be strongly-additive [26], i.e., unlimited dimension problem. So we would like to provide up- for all channels and , CE( )= CE ( )+CE( ). per bounds for them. Before that we will show that the notion N M N⊗M N M Thus, CE equals its own regularization and in turn its own “potential” is intrinsically sub-additive and a little surprising potential capacity: ( ) (1) ( ) (1) fact: fp∞ ( ) fp ( ) though f ∞ ( ) f ( ). N ≤ N N ≥ N ( ) CE( )= CE∞ ( ) = (CE)p( ). (1) N N N Lemma 5 For any super-additive f, both fp ( ) and The next subsection presents the slightly more interesting ( ) N fp∞ ( ) are sub-additive, i.e. case of Q , which is not additive, but it has a single-letter N A f (1)( ) f (1)( )+ f (1)( ), formula. For this case we are still able to evaluate (QA)p( ) p N ⊗M ≤ p N p M in a simple single-letter formula, but for the subsequent C,NP ( ) ( ) ( ) f ∞ ( ) f ∞ ( )+ f ∞ ( ). p N ⊗M ≤ p N p M and Q we will only be able to give upper bounds. WINTER AND YANG: POTENTIAL CAPACITIES OF QUANTUM CHANNELS 4

A. Potential environment-assisted capacity By Lemma 6, we have

There are two types of channels with QA( )=0, χ( ) C( ) Cp( ) χp( ). (15) which we will use to activate a givenM , on one hand,M those N ≤ N ≤ N ≤ N N with one-dimensional input system, on the other those with To give non-trivial bounds on χp( ), we invoke the following N one-dimensional output system. Their Stinespring isometries previous result. are ′ ′ Lemma 9 (Yang et al. [27]) For a mixed four-partite state V : C B E , 1 φ B E , 1 ′ ′ B1B2E1E2 , −→ ⊗ 7−→ | i′ ′ ρ B E V : A′ C E′, ψ 1 (W ψ ) , 2 2 B1B2:E1E2 B1:E1 B2:E2 −→ ⊗ | i 7−→ ⊗ | i EF (ρ ) G(ρ )+ EF (ρ ), (16) where W is an isometry. Using these, we show the following ≥ 2 BE simple result: where the fuction G(ρ ) is defined as B:E BE G(ρ ) := min piC (ρ ), (17) BE i Theorem 7 For any channel : A B, pi,ρ ← i i N → { } X A A BE B (QA)p( ) = max max S(ρ ),S (ρ ) , with C (σ )= S(σB) min rj S(σj ), A ← − Pj N ρ N { } n o A where Pj ranges over POVMs on E, i.e. Pj 0 and = max log A , max S (ρ ) . { } BE B 1 ≥ | | ρA N Pj = 11, rj = Tr(11 Pj )σ , and σ = TrE(11 j ⊗ j rj ⊗ Proof: First, for “ ”: By tensoring with a channel Pj )σBE . P B:E B:E : A B of the above≥ type having zero environment- Furthermore, G(ρ ) is faithful, meaning G(ρ )=0 ′ ′ BE Massisted capacity,→ i.e. either where the only input state iff ρ is separable. M1 has zero entropy, or 2 where the only output state has zero entropy. In this wayM we can bump up either the output Theorem 10 For a channel with Stinespring isometry U, ′ ′ N entropy S (ρAA ) , or the input entropy S(ρAA ) by B BE χp( ) max S(ρ ) G(ρ ) , (18) an arbitraryN amount, ⊗M without changing the respective other. N ≤ ρA − Thus indeed, BE A where ρ = Uρ U †. A A (QA)p( ) QA( ) max S(ρ ),S (ρ ) . N ≥ N ⊗M ≥ N Proof: Using the MSW identity, Eq. (3), and Lemma 9, In the other direction, consider ann arbitrary channelo . we have the following chain of identities and inequalities: M Then we have, χ( ) N ⊗M (QA)p( ) = sup QA( ) QA( ), = max S(B1B2) EF (B1B2 : E1E2), N N ⊗M − M ρA1A2 − M ′ ′ AA AA sup max min S(ρ ),S (ρ ) S B S B G B : E E B : E , AA′ max ( 1)+ ( 2) ( 1 1)+ F ( 2 2) ≤ ρ N ⊗M ρA1A2 M ≤ − n ′ ′ o A A n o +max S(ρ ), S (ρ ) , = max S(B1) G(B1 : E1) + S(B2) EF (B2 : E2) , − − M ρA1A2 − − S nA A ,S B B , o n o sup max′ max ( ′) ( ′) ≤ ρAA | | max S(B1) G(B1 : E1) + max S(B2) EF (B2 : E2) , M ≤ ρA1 − ρA2 − sup max maxS(A),S(B) , A =maxS(B1) G(B1 : E1) + χ( ). ≤ ρ A M ρ 1 − M and we are done. By definition of χp, the claim follows. In [28], a channel is perfect when its capacity is log dout. B. Potential classical capacity In the general case, the input space may have the different In this section, we study the potential classical capacity dimension from the output space. It is obvious that the capacity and its relation to the single-letter Holevo capacity, and most of the channel is upper-bounded by min log din, log dout . importantly establish an upper bound via a specific entangle- Here we call a channel perfect if its capacity{ is equal to} ment measure. This bound is used to prove that an imperfect log dmin with dmin = min din, dout and we prove the quantum channel cannot be activated into a perfect one by any following corollary. { } other contextual channel. Corollary 11 If a quantum channel is not perfect for Definition 8 Specializing Definition 4 to the case f C, we transmitting classical information in theN single-letter sense, ≡ obtain the potential classical capacity then it cannot be activated to the perfect one by any contextual channel: Cp( ) = sup C( ) C( ) , (13) N N ⊗M − M M χ( ) < log dmin = χp( ) < log dmin. and likewise the potential Holevo capacity N ⇒ N Proof: Suppose the potential capacity of the channel is χp( ) = sup χ( ) χ( ) . (14) N N ⊗M − M χp( ) = log dmin. N M WINTER AND YANG: POTENTIAL CAPACITIES OF QUANTUM CHANNELS 5

In the case of d = dout = d, from Cp( ) χp( ) The symmetric side-channel assisted quantum capacity, min N ≤ N ≤ maxρA [S(ρB) G(ρBE )], we know that there is an input state Qss, introduced and investigated in [25], is obtained by A − BE A ρ such that for ρ = Uρ U †, we have S(ρB) = log d and restricting the above optimization to channels that are M G(ρBE )=0. symmetric, i.e. both degradable and anti-degradable, which is Since G is faithful (see Lemma 9), this means that ρBE is a special subclass of zero-capacity channels. Unlike Q, Qss separable, which amounts to EF (ρBE )=0. From the MSW is additive and has many other nice properties, and from the identity, Eq. (3), we obtain that χ( ) = log d, which means definition and the above, we have (cf. [25, Sec. VII]) N the channel is perfect in the single-letter sense. (1) Qss( ) Qp( ) Qp ( ). (22) In the case of dmin = din = d, suppose Cp( ) = log d = N ≤ N ≤ N A 1 1 BEN A S(A) = S(BE), where ρ = d 1 and ρ = Uρ U †. How do we establish the upper bound for the potential quan- From Lemma 27 in the Appendix, we obtain log d = S(A)= tum capacity? The idea is channel simulation inspired by the Cp( ) maxρA [S(ρB) G(ρB:E)] maxρA S(A) = log d. NA ≤1 − ≤ approach to obtain an upper bound for the quantum capacity: So ρ = 11 is the optimal input to achieve maxρ [S(ρB) d A − If the channel can be simulated by another channel ↑ G(ρBE )]. This means G(ρB:E) = S(B) S(BE) for the N N BE − using pre- and post-processing, i.e. = ↑ with state ρ . Also from Lemma 27, we know that EF (B : E)= N T ◦ N ◦ S suitable CPTP maps and , then clearly Q( ) Q( ↑). S(B) S(BE), meaning that χ( ) = log d. S T N ≤ N − N We call ↑ a lifting of . Furthermore, if the channel ↑ is degradable,N then its quantumN capacity is given by the singleN - Remark In [28], it is shown that if χ( ) < log dout, then N letter capacity Q(1) , and obtain a single-letter upper C( ) < log dout. Notice that Holevo capacity is the capacity ( ↑) N bound for Q( ). ThisN was observed and exploited before when the codewords are restricted to product states. That is to N say if the capacity when using product state encoding cannot under the name of “additive extensions” [29]. achieve the possibly maximal quantity log dout, then it cannot From inequality (21) and the definition of potential single- either when using entangled state encoding. In other words, an letter quantum capacity, we see that we should try to lift the imperfect channel cannot be activated to a perfect one by itself. channel to a strongly additive one, because then we get even Corollary 11 is stronger in two points. One is that it covers an upper bound for the potential quantum capacity, and in fact the case din < dout where [28] says nothing about. Indeed the potential single-letter quantum capacity! it is not immediately to obtain so we need the Appendix to However it is not enough to lift the channel to a degradable deal with this case. The other point is that Corollary 11 asserts one, because we learn from the superactivation phenomenon an imperfect channel cannot be activated to a perfect one by that its single-letter quantum capacity is not strongly additive. any channel. The reasoning for dmin = dout is almost the But an even narrower class of degradable channels, called same as that in [28] but for dmin = din we need more. Here Hadamard channels, satisfies the required property. we emphasiz that we use the particular entanglement measure G(ρBE ) while other entanglement measures may be employed Definition 13 A Hadamard channel (HC) [30] is a to prove the result in [28]. quantum channel whose complementary channel Nc is an entanglement-breaking channel (EBC) [31], where Nc can be expressed as N C. Potential quantum capacity c ˜ ˜ In this section, we move on to the potential quantum (ρ)= φi φi ψi ρ ψi , N | ih |h | | i i capacity and study its relations to the single-letter quantity X Q(1)( ). In [25, Sec. VII], this had been introduced under the ˜ ˜ N in which i ψi ψi = 11 is a POVM. Such channels are said name of “quantum value added capacity”, and our Lemma 5 to be entanglement-breaking| ih | because the output state id already been observed in that case. Here, we establish an c(ρRA)Pis separable for any state ρRA. ⊗ upper bound in terms of the entanglement of formation of the N channel, and finally prove that an imperfect quantum channel The isometry of the Hadamard channel is of the form cannot be activated into a perfect one by any other contextual (up to local unitary operation on E) N channel. B E A V = i φi ψ˜i , (23) | i | i h | Definition 12 Specializing Definition 4 to the case f Q, i ≡ X we obtain the potential quantum capacity from which we see that the Hadamard channel can simulate its complementary channel c by the operationN of first Qp( ) = sup Q( ) Q( ) , (19) N measuring in the basis i and then preparing the state φi N N ⊗M − M | i | i M according to the outcome of the measurement. Thus Hadamard and the potential single-letter quantum capacity channels are special degradable channels. Q(1)( ) = sup Q(1)( ) Q(1)( ) . (20) p N N ⊗M − M M Proposition 14 (Cf. Bradler et al. [32], Wilde/Hsieh [33]) (1) By Lemma 6, we have If is a Hadamard channel, then Q is strongly additive: Q(1)N( ) = Q(1)( ) + Q(1)( ) for any contextual (1) (1) N ⊗M N M Q ( ) Q( ) Qp( ) Q ( ). (21) channel . N ≤ N ≤ N ≤ p N M WINTER AND YANG: POTENTIAL CAPACITIES OF QUANTUM CHANNELS 6

Proof: The “ ” part is trivial and we only need to prove Hadamard channel ↑ are i Ki , and one way to write the “ ” part. Suppose≥ the isometry of the Hadamard channel the channel is as N {| i⊗ } ≤ is V : A1 ֒ B1 E1, of the form (23), and the isometry N → ⊗ ↑(ρ)= i i KiρKi†, ⊗| of is W : A2 ֒ B2 E2. The output, for an input state N | ih M → ⊗ i φ RA1A2 , is X | i which we call the canonical lifting. Its quantum capacity is V W φ RA1A2 = V φ RA1 B2E2 , B ⊗ | i | i Q( ↑)= max piS(ρ ), B1 E1 A1 RA1 B2E2 i = i φi ψ˜i φ , N φ RA | i | i h | | i | i i i X X s.t. pi = Tr(11 Ki) φ φ (11 Ki)†, B1 E1 RB2E2 = √pi i φi ψi . ⊗ | ih | ⊗ | i | i | i 1 i ρi = TrR(11 Ki) φ φ (11 Ki)†. X pi ⊗ | ih | ⊗ The coherent information is thus, Now take the minimum over all different Kraus representations S(B1B2) S(E1E2) (which after all we are free to choose), to obtain the best bound − = S(B1)+ S(B2 B1) S(E1) S(E2 E1), from this particular family of canonical liftings. | − − | As a result, the quantum capacity of the optimal canonical S(B1) S(E1)+ S(B2 Y ) S(E2 Y ), ≤ − | − | lifting is equal to the entanglement of formation of the original = S(B ) S(E )+ pi(S(B )ψ S(E )ψ ), 1 − 1 2 i − 2 i channel, which is defined as i X RB (1) (1) EF ( ) := max min piE(φi), √pi φi = (11 Ki) φ , Q ( )+ Q ( ), RA N φ Ki | i ⊗ | i ≤ N M { } i | i X where we use the isometry V : i B1 i Y i Z and | i 7→ | i | i and where E(ϕ) is the entropy of entanglement of the bipartite the fact that S(A B) S(A C) if there is an operation pure state ϕ. B C | ≤ | B C → satisfying ρAC = id → (ρAB): S(B2 B1) The following lemma is implied by the proof of [35, E ⊗B E1 Y | ≤ S(B2 Y ) from the operation → (ρ) = TrE V ρV † and Lemma 13], though not explicitly stated there. | E Y E1 S(E2 E1) S(E2 Y ) from the operation → (ρ) = | Y ≥ E | E i ρ i φi φi 1 . ih | | i | ih | Lemma 16 (Berta et al. [35]) With the above notation, the PRemark The above proposition 14 is a special case of a following minimax formula holds: more general fact [32, Lemma 4], [33, Thm. 3 & Lemma min max piE(φi)= max min piE(φi), RA RA 2]; cf. also [20, App. B, Lemma 7] for the special case Ki φ φ Ki { } | i i | i { } i of “generalized dephasing channels” (also known as Schur X X where the infimum is taken over all Kraus representations of multipliers). the channel . An immediate corollary is the following. N Now we obtain an upper bound on potential quantum Corollary 15 The potential quantum capacity (and potential capacity in terms of the entanglement of formation of the single-letter quantum capacity) of a Hadamard channel is channel: equal to its single-letter quantum capacity: N Theorem 17 For a general channel , we have the following (1) (1) N Q ( )= Qp( )= Q( )= Q ( ). upper bound on the potential quantum capacity: p N N N N (1) Thus we have reduced our task to finding a good lifting of Qp( ) Qp ( ) EF ( ). a given channel to a Hadamard channel. The question how to N ≤ N ≤ N find the optimal one is of interest in itself and we will discuss Proof: Lifting the channel to the optimal canonical the general method elsewhere [34]. For our present purposes, Hadamard channel, we get (1) (1) there is a rather straightforward way to lift a channel to a Q ( ) Q ( ↑ ), N ⊗M ≤ N ⊗M Hadamard channel. Namely, choose Kraus operators for as (1) (1) (1) N =Q ( ↑)+ Q ( )= EF ( )+ Q ( ), (ρ) = KiρKi†. Then a Stinespring isometry for can N M N M N E A B N be written as U = i K → . where the first inequality comes from simulation, the first P i | i i Let us define a new channel, the lifting ↑, via its isometry equality from the strong additivity (Proposition 14), and the P N ′ ′ second equality from Lemma 16. A BB E E B A B V → ⊗ := i i K → , | i | i i Analogous to the classical capacity, we call a channel i X perfect if its quantum capacity is equal to log dmin with where the environment system is still E, but the receiver has d = min din, dout . Before we have the similar corollary, min { } now BB′, and B′ holds a coherent copy of E. As we now we recall a result in [36]. give a copy of E to the channel output, the output of the complementary channel of ↑ will be completely decohered Lemma 18 (Schumacher/Westmoreland [36]) For a pure in the i basis, so the complementaryN channel is EBC, hence bipartite state φRA , let the system A be transmitted through {| i} | i RB ↑ is Hadamard, as desired. The Kraus operators of the lifted a channel : A B and the joint output state is ρ = N N → WINTER AND YANG: POTENTIAL CAPACITIES OF QUANTUM CHANNELS 7

RA id (φ ). If EF (R : B) = S(R), then there exists a be maximal without it already being the single-letter private quantum⊗ N operation : B A such that id (ρRB )= φRA. capacity. Especially we prove that the private capacity of D → ⊗D Hadamard channels is strongly additive. Corollary 19 If a quantum channel is not perfect for transmitting quantum information, thenN its potential single- Deﬁnition 20 Specializing Deﬁnition 4 to the case f P , ≡ letter quantum capacity is not maximal, either: we obtain the potential private capacity

(1) (1) Pp( ) = sup P ( ) P ( ) , (24) Q ( ) < log dmin = Qp ( ) < log dmin. N N ⊗M − M N ⇒ N M In particular, if a quantum channel is not perfect for trans- and the potential single-letter private capacity mitting quantum information, then it cannot be activated to a P (1)( ) = sup P (1)( ) P (1)( ) . (25) perfect one by any zero-quantum-capacity channels. p N N ⊗M − M M Proof: Suppose that the potential quantum capacity is By Lemma 6, we have (1) (1) log dmin, then from the upper bound in Theorem 17 we know P ( ) P ( ) Pp( ) Pp ( ). (26) that E ( ) = log d . N ≤ N ≤ N ≤ N F min As for in the quantum case, we aim to lift channels to In theN case of d = d , from Lemma 18, we know that min in strongly additive ones, so as to obtain single-letter upper the channel operation can be perfectly corrected by a suitable bounds on the potential private capacity. Indeed, we can extend operation acting on B that means the channel is already Proposition 14 to the private capacity: perfect. D In the case of dmin = dout, denoting the isometry of the Proposition 21 If is a Hadamard channel, then P (1) is channel : A B by U : A ֒ B E, there exists hence strongly additive: PN(1)( )= P (1)( )+ P (1)( ) for N RA → → ⊗ an input φ such that the output state plus the environment any contextual channel N ⊗M. N M RBE| i RA RB is φ = U φ , where ρ = id (φRA) satisfying M E |(ρiRB )= S(|B)i = log d. This implies⊗ that N ρBE is a product Proof: The “ ” part is trivial and we only need to prove F ≥ state. From the Uhlmann theorem [37], we know there is a the “ ” part. Suppose the isometry of the Hadamard channel ≤ unitary V : R R R such that is V : A1 ֒ B1 E1, of the form (23) as before, and the → 1 ⊗ 2 N → ⊗ isometry of is W : A2 ֒ B2 E2. For the input state RBE RA R1B R2E V φ = V U φ = φ φ , MA1A2 → ⊗ | i ⊗ | i | i1 ⊗ | i2 ensemble pt,ρt , we construct the classical-quantum (cq) { } A1A2 1 R1 B R2 E state with the reference system R that puriﬁes each ρ , = λj i ei j ej , t √ | i | i | i | i i,j d T RA1A2 X pt t t φt φt , R B t | ih | ⊗ | ih | where i 1 , ei are the bases in the Schmidt decom- X {| i |R iB } R E position of φ 1 , and j R2 , e E for φ 2 . So the which is mapped by V W to 1 j 2 ⊗ | i RA {| i 1| i R}1 R2| i B E input state V 11 φ = i j U †( ei ej ) T RB1E1B2E2 ⊗ | i i √d | i | i | i | i pt t t φt φt . will give a product state as output. Now from this input we t | ih | ⊗ | ih | P X construct a new input Here,

RA 1 R1 R2 B E RB1E1B2E2 B1 E1 RB2 E2 ψ = i 0 U †( ei e0 ), φt = √qi t i φi ψi t . (27) | i √ | i | i | i | i | i | | i | i | | i i d i X X yielding the output Using the isometry i B1 i Y i Z , and the notation | i 7→ | i | i 1 F (X) T = ptF (Xt), we get RA R1 R2 B E | U ψ = i 0 ei e0 , | i √d| i | i | i | i I(T : B1B2) I(T : E1E2) i P − X = S(B B ) S(E E ) (S(TB B ) S(T E E )), which is the desired output of product state between B and 1 2 − 1 2 − 1 2 − 1 2 E. This means Q(1)( ) = log d, i.e. is noiseless already. = S(B1) S(E1)+ S(B2 B1) S(E2 E1) N N − | − | (S(B1B2 T ) S(E1E2 T )), Remark We know that the channel can be very entangled − | − | S(B1) S(E1)+ S(B2 Y ) S(E2 Y ) even though its quantum capacity is zero. It is very difficult to ≤ − | − | (S(B B T ) S(E E T )), characterize channels with zero quantum capacity. So it seems − 1 2| − 1 2| = S(B ) S(E ) + [I(T : B ) I(T : E )] Y that it is hard to say whether a noisy channel can be activated 1 − 1 2 − 2 | into a noiseless one under the assistance of zero-quantum- + (S(TB2 Y ) S(T E2 Y )) (S(B1B2 T ) S(E1E2 T )), capapcity channels. However from the notion of potential | − | − | − | = S(B1) S(E1) + [I(T : B2) I(T : E2)] Y quantum capacity, we can answer this question in the negative. − − | + (S(B Y T ) S(E Y T )) (S(B B T ) S(E E T )), 2 | − 2 | − 1 2| − 1 2| = S(B ) S(E ) + [I(T : B ) I(T : E )] Y D. Potential private capacity 1 − 1 2 − 2 | + [S(B Y ) S(E Y ) S(B B )+ S(E E )] T, In this section, we repeat the analysis of the preceding 2 − 2 − 1 2 1 2 | P (1)( )+ P (1)( ) subsection, but for the potential private capacity. We shall ≤ N M show that, as for Q, the potential private capacity cannot + [S(B Y ) S(E Y ) S(B B )+ S(E E )] T, 2 − 2 − 1 2 1 2 | WINTER AND YANG: POTENTIAL CAPACITIES OF QUANTUM CHANNELS 8 where the first inequality comes from S(B B ) S(B Y ) channel. Thus, the notion of lifting to a Hadamard channel 2| 1 ≤ 2| and S(E2 E1) S(E2 Y ), which hold for the same reason once more yields and outer bound on the potential capacity in the proof| of≥ Prop. 14.| Next we show that each term in the region of the achievable triples (q,c,e). average over T is non-positive. Indeed, we evaluate the term We have studied potential capacities only for the basic in the pure state of the form (27) and notice that S(E2Y )= quantities, and one (QA) for which we could calculate the S(E1E2Y ). Then we have potential capacity exactly. For most capacities, we may assume that it will be prohibitive to calculate the potential version as S(B Y ) S(E Y ) S(B B )+ S(E E ) 2 − 2 − 1 2 1 2 well as its plain version, so we have to be content with bounds. = I(R : Y E E ) 0, − | 1 2 ≤ In the domain of zero-error information theory, other exact which concludes the proof. characterizations of some potential capacities are known [38]. In fact, in [38], differences between the capacity of K( An immediate corollary is the following. ) and another parameter for , which represents a moreN⊗ Mgeneral value V ( ) of the channel,M were considered, where Corollary 22 The potential private capacity (and the poten- K is superadditiveM and V ( ) K( ). Then, tial single-letter private capacity) of a Hadamard channel M ≥ M (1) N is equal to its private capacity, which in turn equals Q ( ): V ∗( ) = sup K( ) V ( ) N N N ⊗M − M (1) (1) M P ( )= Pp( )= P ( )= Q ( ). p N N N N is a kind of amortized value of , the rationale being that the gain from the “profit” K( N ) has to be offset by Theorem 23 For any channel , we have the upper bound N ⊗M (1) N the “price” V ( ) of the borrowed resource. Of particular Pp( ) Pp ( ) EF ( ). M N ≤ N ≤ N interest is the case of an economically fair pricing V = V ∗, i.e. of a situation where the same value V ( ) = V ∗( ) Proof: Notice that for the Hadamard channel, the poten- N N tial capacity is equal to the quantum capacity and the rest is quantifies the price of resource when we have to borrow it, the same as the quantum case, i.e. the proof of Theorem 17. as well as the amortized value in a suitable context. In the setting of zero-error capacities, specifically K = log α (with the independence number α), this has been shown to hold true As in the quantum case, we obtain the following immediate for V = log ϑ (the Lovász number). For Shannon theoretic corollary: capacities, i.e. K C,P,Q or similar, the existence and possible characterization∈ { of a} value V that yields a fair value Corollary 24 If a quantum channel is not perfect for trans- V = V ∗ is perhaps one of the most intriguing questions raised mitting private information, then it cannot be activated to the by the notion of potential capacities. For instance, for the perfect one by any contextual channels. quantum capacity, it turns out that its symmetric side-channel assisted version Qss is such a fair price. Namely, for a given V. DISCUSSIONANDOPENQUESTIONS channel and any contextual channel , N M E have introduced potential capacities of quantum Q(1)( ) Q( ), W channels as “big brothers” of the plain capacities, to N ⊗M ≤ N ⊗M Qss( )= Qss( )+ Qss( ), capture the degree of non-additivity of the latter in the most ≤ N ⊗M N M favorable context. By bootstrapping strongly additive channels, thus i.e. those whose plain capacity equals its potential version, (1) sup Q ( ) Qss( ) we were able to give some general upper bounds on various N ⊗M − M M potential capacities. While the potential concept makes sense sup Q( ) Qss( ) Qss( ). ≤ N ⊗M − M ≤ N for any capacity, here we focused on a few examples, basically M the principal channel capacities C, Q and P. Our central result On the other hand, restricting to symmetric channels in M is that a noisy channel cannot be activated into a noiseless one this optimization, for which Qss( )=0, we attain equality by any contextual channel. This result holds for the classical, asymptotically by definition of theM symmetric side-channel quantum, and private capacity, and improves upon previous assisted quantum capacity. The same reasoning can be applied statements. Notice that in the notion of potential capacity, a to the private capacity and the symmetric side-channel assisted PPT-entanglement-binding channel may have positive poten- version Pss [21], so we have proved the following. tial quantum capacity. So it is tempting to speculate whether all entangled channels have positive potential quantum capacity. Theorem 25 For any channel , This is a big open question deserving of study in the future. N (1) Looking beyond capacities and at the tradeoff between Qss( ) = sup Q ( ) Qss( ), N N ⊗M − M different resources, note that Hadamard channels that served us M = sup Q( ) Qss( ), so well in the treatment of the potential quantum and classical N ⊗M − M M capacities, also allow for a single-letter formula for the qubit- (1) Pss( ) = sup P ( ) Pss( ), cbit-ebit tradeoff region [33]; in fact, Thm. 3 and Lemma 2 of N N ⊗M − M M that paper show that this region is strongly additive regarding = sup P ( ) Pss( ), the tensor product of a Hadamard channel with any other N ⊗M − M M WINTER AND YANG: POTENTIAL CAPACITIES OF QUANTUM CHANNELS 9 and equality is asymptotically attained by symmetric channels where the first “ ” comes from the concavity of the condi- . tional entropy and≤ the second “ ” from Lemma 26 above. M ≤ If S(B) S(BE) = G(BE), then S(Bi) S(BEi) = If the quantum and private capacities have alternative fair − BE − BE C (BEi) for each ρi . From Lemma 26, the state ρi has pricing schemes along these lines or if Qss (Pss) are unique, ← the property EF (BEi) = S(Bi) S(BEi). Then S(B) and whether there is an analogous statement for C or χ, remain − − S(BE) EF (BE) piEF (BEi) = pi[S(Bi) open. ≤ ≤ i i − S(BEi)] = S(B) S(BE) and the proof ends. − P P ACKNOWLEDGMENTS In fact, the constraint is so sharp that we can even learn the We thank Ke Li, Graeme Smith and John Smolin for structure of the bipartite state. Proposition 28 explains this and helpful discussions, and Runyao Duan, Marius Junge and is also of independent interest. Mark Wilde for interesting remarks, on the concept of potential capacity and potential capacity regions, respectively. Proposition 28 A state ρ in the finite dimensional Hilbert Part of this work was done during the programme Math- BE space satisfies S(B) S(BE)= E (BE), if and ematical Challenges in Quantum Information (MQI) at the B E F only ifH it is⊗ of H the form − Isaac Newton Institute in Cambridge, whose hospitality was L R gratefully acknowledged, and where DY was supported by a BE Bi Bi E ρ = piρ φ , i ⊗ i Microsoft Visiting Fellowship. DY’s work is supported by the i M ERC (Advanced Grant “IRQUAT”) and the NSFC (Grant No. R Bi E 11375165). AW’s work is supported by the European Com- where φi are pure states and the system B is decomposed mission (STREP “RAQUEL”), the European Research Council into the direct sum of tensor products (Advanced Grant “IRQUAT”), the Spanish MINECO (projects B = BL BR . FIS2008-01236 and FIS2013-40627-P), with the support of H H i ⊗ H i i FEDER funds, as well as by the Generalitat de Catalunya M CIRIT, project 2014-SGR-966. Proof: Suppose the optimal realization of EF (B : E) XBE is the ensemble px, ψx , and construct the state ρ = APPENDIX {BE | i} x px x x ψx . From the condition S(B) S(BE) = | ih |⊗ − B Here we analyze the structure of the state satisfying S(B) EF (B : E), we get S(B) S(BE) = pxS(ψ ). This P − x x S(BE) = G(B : E) or S(B) S(BE) = E (B : E−). condition can be expressed as I(X : E B) = 0, where F P| The general relation among these− three quantities is S(B) I(X : E B) = S(XB) + S(EB) S(XBE) S(B) is − | − − S(BE) G(B : E) EF (B : E), the first “ ” comes from the conditional quantum mutual information. From [39], we Lemma≤ 26 and the second≤ “ ” from Lemma≤ 9. Obviously know that I(X : E B)=0 if and only if the state ρXBE can ≤ | the condition S(B) S(BE)= EF (B : E) implies S(B) be decomposed as − − S(BE) = G(B : E). Lemma 27 asserts that the latter also L R XBE XBi Bi E ρ = qiρ ρ , implies the former. i ⊗ i i M Lemma 26 For a mixed state ρBE, S(B) S(BE) where the system B is decomposed into the direct sum of − ≤ C (ρBE), and equality holds iff there exists a unitary on tensor products ← BE BL BRE B such that UBρ UB† = ρ φ , where B = B E ⊗ H and φ R is pure. B = BL BR . BL BR H H i ⊗ H i H ⊗ H i Proof: Consider the purification φRBE of the state ρBE. M BE Thus we have Then C (ρ )= S(B) EF (R : B) and S(BE)= S(R). ← − BL BRE From the inequality E R : B S R , we arrive at S B BE i i F ( ) ( ) ( ) ρ = qiρi ρi . BE ≤ − ⊗ S(BE) C (ρ ). i When≤ the← equality holds, this amounts to E (R : B) = M F BRE RE i S(R). From the relation C (ρ )= S(R) E (R : B)=0, If all the states ρi are pure, then we are done. In general, F R RE R ←E − Bi E we get that ρ = ρ ρ . From Uhlmann’s theorem [37], some of ρi may be mixed. Apply the condition S(B) ⊗ RBE RBL BE − there exists a unitary UB such that UBφ UB† = φ S(BE) = EF (B : E) to the structured state ρ , we get BRE ⊗ R R R φ . Tracing out R concludes the proof. qi(S(Bi ) S(Bi E)) = qiEF (Bi : E). Since S(B) − R − Bi E BE S(BE) EF (B : E) is true for all of the components ρi , Lemma 27 For a state ρ , S(B) S(BE) G(B : E). If P ≤ R RP R − ≤ we arrive at S(Bi ) S(Bi E) = EF (Bi : E) for each i. S(B) S(BE)= G(B : E), then S(B) S(BE)= EF (B : So we can use the argument− again and get that the structure − − R E). Bi :E of each state ρi is of the direct sum of tensor products. BR E Proof: Suppose that the optimal realization of G(B : E) If some of the new states ρ i(j) are mixed, we repeat the BE i(j) is the state ensemble pi,ρ . Then, { i } argument for these states. In each iteration, the dimension is BE reduced because of the direct sum of tensor products. Since S(B) S(BE) pi[S(Bi) S(BEi)] piC (ρi ), − ≤ − ≤ ← i i system B is a finite dimensional Hilbert space, the iteration X X WINTER AND YANG: POTENTIAL CAPACITIES OF QUANTUM CHANNELS 10

BR E i(··· ) [30] C. King, K. Matsumoto, M. Nathanson, and M. B. Ruskai, “Properties ends after finitely many steps when all the states ρi( ) are pure. After renumbering the labels, we get the desired··· of Conjugate Channels with Applications to Additivity and Multiplica- R tivity”, Markov Processes and Related Fields 13(1-2):391-423 (2007). Bi E decomposition where all the states φi are pure. [31] M. Horodecki, P. W. Shor, and M. B. Ruskai, “Entanglement Breaking Channels”, Rev. Math. Phys. 15(6):629-641 (2003). REFERENCES [32] K. Bradler, P. Hayden, D. Touchette, and Mark M. Wilde, “Trade-off capacities of the quantum Hadamard channels”, Phys. Rev. A 81:062312 [1] A. S. Holevo, “The Capacity of the Quantum Channel with General (2010). Signal States”, IEEE. Trans. Inf. Theory 44(1):269-273 (1998). [33] M. M. Wilde and M.-H. Hsieh, “The quantum dynamic capacity formula [2] B. Schumacher and M. D. Westmoreland, “Sending classical information of a quantum channel”, Quantum Inf. Process. 11(6):1431-1463 (2012). via noisy quantum channels”, Phys. Rev. A 56(1):131-138 (1997). [34] D. Yang, “Upper Bounds for Capacities of Quantum Channels ”, in [3] S. Lloyd, “Capacity of the noisy quantum channel”, Phys. Rev. A preparation (2015). 55(3):1613-1622 (1997). [35] M. Berta, F. G. S. L. Brandão, M. Christandl, and S. Wehner, “Entangle- [4] P. W. Shor, “The quantum channel capacity and coherent information”, ment Cost of Quantum Channels”, IEEE Trans. Inf. Theory 59(10):6779- MSRI Workshop on Quantum Computation (2002). 6795 (2013). [5] I. Devetak, “The Private Classical Capacity and Quantum Capacity of a [36] B. Schumacher and M. D. Westmoreland, “Entanglement and perfect Quantum Channel”, IEEE Trans. Inf. Theory 51(1):44-55 (2005). quantum error correction”, J. Math. Phys. 43(9):4279-4285 (2002). [6] N. Cai, A. Winter, and R. W. Yeung, “Quantum Privacy and Quantum [37] A. Uhlmann, “The ‘Transition Probability’ in the State Space of a ∗- Wiretap Channels”, Probl. Inf. Transm. 40(4):318-336 (2004). Algebra”, Rep. Math. Phys. 9:273-279 (1976). [7] M. Hastings, “Superadditivity of communication capacity using en- [38] A. Ac´ın, R. Duan, A. B. Sainz, and A. Winter, “A new property of tangled inputs”, Nature Physics 5:255-257 (2009); arXiv[quant- the Lovász number and duality relations between graph parameters”, ph]:0809.3972. submitted; arXiv:1505.01265 (2015). [8] D. P. DiVincenzo, P. W. Shor, and J. A. Smolin, “Quantum-channel [39] P. Hayden, R. Jozsa, D. Petz, A. Winter, “Structure of states which capacity of very noisy channels”, Phys. Rev. A 57(2):830-839 (1998). satisfy strong subadditivity of quantum entropy with equality”, Commun. [9] K. Li, A. Winter, X. Zou, and G. Guo, “Private Capacity of Quantum Math. Phys. 246(2):359-374 (2004). Channels is Not Additive”, Phys. Rev. Lett. 103:120501 (2009). [10] G. Smith and J. A. Smolin, “Extensive Nonadditivity of Privacy”, Phys. Rev. Lett. 103:120503 (2009). [11] T. S. Cubitt, D. Elkouss, W. Matthews, M. Ozols, D. Pérez-Garc´ıa, and S. Strelchuk, “Unbounded number of channel uses are required to see quantum capacity”, Nat. Commun. 6:7739 (2015). [12] D. Elkouss and S. Strelchuk, “Superadditivity of private information for any number of uses of the channel”, Phys. Rev. Lett. 115:040501 (2015). [13] G. Smith and J. T. Yard, “Quantum Communication with Zero-Capacity Channels”, Science 321:1812-1815 (2008). [14] W. F. Stinespring, “Positive functions on C*-algebras”, Proceedings of the American Mathematical Society 6(2):211-216 (1955). [15] K. Matsumoto, T. Shimono, and A. Winter, “Remarks on Additivity of the Holevo Channel Capacity and of the Entanglement of Formation”, Commun. Math. Phys. 246:427-442 (2004). [16] B. Schumacher, “Sending entanglement through noisy quantum channels”, Phys. Rev. A 54(4):2614-2628 (1996). [17] B. Schumacher and M. A. Nielsen, “Quantum data processing and error correction”, Phys. Rev. A 54(4):2629-2635 (1996). [18] H. Barnum, M. A. Nielsen, and B. Schumacher, “Information transmission through a noisy quantum channel”, Phys. Rev. A 57(6):4153-4175 (1998). [19] H. Barnum, E. Knill, and M. A. Nielsen, “On quantum fidelities and channel capacities”, IEEE Trans. Inf. Theory 46(4):1317-1329 (2000). [20] I. Devetak and P. W. Shor, “The Capacity of a Quantum Channel for Simultaneous Transmission of Classical and Quantum Information”, Commun. Math. Phys. 256(2):287-303 (2005). [21] G. Smith, “Private classical capacity with a symmetric side channel and its application to quantum cryptography”, Phys. Rev. A 78:022306 (2008). [22] C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal, “Entanglement-Assisted Capacity of a Quantum Channel and the Re- verse Shannon Theorem”, IEEE Trans Inf. Theory 48(10):2637-2655 (2002). [23] J. A. Smolin, F. Vestraete, and A. Winter, “Entanglement of assistance and multipartite state distillation”, Phys. Rev. A 72:052317 (2005). [24] A. Winter, “On Environment-Assisted Capacities of Quantum Chan- nels”, Markov Processes and Related Fields 13(1-2):297-314 (2007). [25] G. Smith, J. A. Smolin, and A. Winter, “The Quantum Capacity With Symmetric Side Channels”, IEEE Trans. Inf. Theory 54(9):4208-4217 (2008). [26] C. Adami and N. J. Cerf, “von Neumann capacity of noisy quantum channels”, Phys. Rev. A 56(5):3470-3483 (1997). [27] D. Yang, M. Horodecki, R. Horodecki, and B. Synak-Radtke, “Irre- versibility for All Bound Entangled States”, Phys. Rev. Lett. 95:190501 (2005). [28] F. G. S. L. Brandão, J. Eisert, M. Horodecki, and D. Yang, “Entangled Inputs Cannot Make Imperfect Quantum Channels Perfect”, Phys. Rev. Lett. 106:230502 (2011). [29] G. Smith and J.A. Smolin, “Additive extensions of a quantum channel”, in: Proc. IEEE Information Theory Workshop, Porto, 5-9 May 2008, pp. 368-372 (2008); arXiv:0712.2471.