Reverse Coherent Information

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← = and ] = Q antidegradable 3 = ρ 1 X ≤ E → i.e., , ;febc lsia communication classical feedback ); Q E | E ↔ φ ← ĺ ĺ i h olwn eak concern remarks following The . = E RBE 2 6= x M 39 G Q ≥ Q 4 4 ( ← uhthat such steprfiainof purification the is ρ . B E Q and x = ) ĸ o all for ĸ 5 Q 1 Λ 0. = (Λ) ρ 5 E G ρ x Q where , B Q ( E I)Increas- (II) . ρ ↔ E noteen- the into ļ ļ = .I i.1 Fig. In ). = ) 2 ρ ρ B ρ E RB nits the ion ty = g . 2 entanglement distribution capacity of quantum channels Reverse entanglement distribution.- A big practical (distributed e-bits per use of the channel) is of crucial im- disadvantage of the previous protocol is that Alice has portance. As for quantum communication, we can also to wait until Bob sends the message bi before apply- b1...bi define four types of assisted (unassisted) capacities for ing i and subsequently sending qubit Ai+1, which entanglement distribution: , , , . As shown greatlyA decreases the transmission rate. A way of avoid- in Fig. 1, all the entanglement{E E→ distributionE← E↔} capacities ing this problem is to simplify the protocol to a single are equivalent to their quantum communication counter- round of classical feedback after Alice has sent all her parts, except for (Λ). qubits A1A2...An through the quantum channel Λ, see Entanglement distributionE← assisted by feedback classi- Fig. 3. We call this familly of simplified protocols reverse cal communication.- The entanglement distribution protocol assisted by classical feedback communication, as A B described in [8], goes as follows. Alice starts preparing 1 ȁ 1 A2 B2 a bipartite entangled state ΨR A1,A2,...,An , where R is a R | Ab ȁ B group of qubits entangled with the qubits Ai sent, one by Ȍ one, through the channel Λ. The first round of the pro- An Bn tocol, see Fig. 2, consists of three steps: i) Alice sends ȁ qubit A through the quantum channel Λ; ii) Bob ap- 1 b plies an incomplete quantum measurement 1 over his received qubit B and communicates the classicalB out- 1 Classical Communication come b1 to Alice. iii) Alice, conditioned on the classical Alice Bob b1 message b1, applies a global quantum operation 1 over the joint system of R and the remaining n A1 qubits − A2A3...An. The next n 1 rounds are a slight modifica- FIG. 3: (color online) A simplification of the general entan- − b1...bi−1 tion of the first one: First, Bob’s measurement glement distribution protocol assisted by classical feedback Bi acts on all his received qubits B1B2...Bi, conditioned on (Fig.2) limits the protocol to a last single round of processing. After Alice has sent all her qubits (A1A2...An) through his previous measurement outcomes b1...bi 1. Second, b1...bi − the quantum channel Λ, Bob applies a collective incomplete Alice’s operation i , acts on all her remaining qubits A measurement among all the qubits B1B2...Bn and commu- RAi+1...An, conditioned on all previous classical communicates the classicalB outcome b to Alice. Finally, conditioned nication messages. By properly choosing Alice’s opera- on the message b, Alice applies the quantum operation b on system R. A

A B 1 1 entanglement distribution protocols, by analogy with the ȁ B1 quantum key distribution scenario [9]. Before the single R Ȍ ȁ post-processing round Alice and Bob’s shared state is n ρR B1,B2,...,Bn = Λ⊗ (ΨR A1,A2,...,An ). (5) ȁ | I⊗ | By properly choosing Alice’s and Bob’s operations both b1 A1 b1 partners extract n ⊳(Λ) e-bits, where ⊳(Λ) is the reverse entanglement≈ distributionE capacity, satisfyingE the Alice Classical Bob inequality ⊳(Λ) (Λ). Communication RemarkE that, in≤E the← particular case where Alice’s inputs are independent and identically distributed, i.e., n ρR A1,A2,...,An = ρ⊗ , the post-processing of the re- | R A FIG. 2: (color online) The first round of the entanglement verse entanglement distribution| protocol is the dynam- distribution protocol assisted by classical feedback consists of ical equivalent of an entanglement distillation protocol three steps: i) Alice sends qubit A1 through the quantum n channel Λ; ii) Bob applies an incomplete quantum measure- over the static resource ρR⊗ B [10]. | ment 1 over his received qubit B1 and communicates the Reverse coherent information capacity.- In what fol- B outcome b1 to Alice; iii) Alice applies a global quantum oper- lows we consider a subset of the reverse entanglement ation b1 over the joint system of R and the remaining n 1 A1 − distribution protocols with a strikingly simple capacity qubits Ai. The next rounds are straightforward extensions of that lower bounds ⊳(Λ). By exchanging the roles of Al- the first one. ice and Bob in theE family of static distillation protocol assisted by one-way classical communication defined in tions and Bob’s incomplete measurements both partners [10], we obtain a new family of static distillation proto- extract n (Λ) units of entanglement (e-bits) at the cols with rate end of the≈ protocol.E← Unfortunately, the calculation of (Λ) is extremely challenging in full generality. IR(ρRB)= S(R) S(RB). (6) E← − 3

By analogy with the quantum key distribution scenario to the class ρ = diag(1 p,p) without loss of general- A − [9], we call the quantity IR(ρRB ) the reverse coherent ity (see appendix). For a given input population p, the information. It is then straightforward to consider a output state is ρB = diag(1 ηp,ηp) and the (reverse) family of entanglement distribution protocols assisted coherent information becomes− by classical feedback with rate IR(Λ,ρA) = IR( Λ( ψ ψ )) = I (ρ ). Optimizing this rate overIρ ⊗ I( ,p) = H(ηp) H((1 η)p), | ih |RA R RB A E − − we define the single-letter reverse coherent information IR( ,p) = H(p) H((1 η)p), (9) (1) E − − capacity R (Λ). SimilarlyE to Eq. (4) we can define a regularized entan- where H(x) is the binary entropy. Optimizing over the input population we obtain (1)( ) and ( ) as func- glement capacity R(Λ) that lowerbounds ⊳(Λ). Inter- E Dη ER Dη estingly, this quantityE can be shown to be additiveE for all tions of the damping parameter η, see Fig. 4. Using the (1) channels, i.e., R = . To do so we only need to prove R 1 0.5 the relation E E 0.8 0.4 I (Λ Λ,ρ 1 2 ) I (Λ,ρ 1 )+ I (Λ,ρ 2 ). (7) R ⊗ A A ≤ R A R A Using the alternative definition of the reverse coherent 0.6 0.3 information I (ρ )= S(BE) S(E), where φ is R RB − | iRBE the purification of ρRB and ρBE, Eq. (7) can be restated 0.4 0.2 as a relation between two von Neumann mutual informa- Quantum Capacity 0.2 0.1 tion quantities: S(B1E1:B2E2) S(E1:E2). This rela- Optimal Input Population tion holds because discarding quantum≥ systems can only 0 0 decrease the mutual information, which results from the 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0 Damping parameter (η) Damping parameter (η) strong-subadditivity of the entropy. The previous proof is strikingly similar to the addi- tivity of the unassisted capacity of degradable channels, FIG. 4: (a) Comparison of ( η) (dashed line) and R( η) E D E D except that it holds for all channels. Since IR(Λ,ρA) is (solid line) as functions of the damping parameter η for additive, it would be extremely interesting if it could be the amplitude damping channel, together with the capac- (1) ity ( η ) = R( η) of the generalized amplitude damp- used to give a definition of (Λ) or ⊳(Λ) similar to E D E D Eq. (4). Unfortunately, this cannotE← be doneE as I (Λ,ρ ) ing channel with a maximally mixed environment (α = 1/2) R A (dotted line). (b) Optimal input population p achieving the does not satisfy the data processing inequality. previous capacities. Despite reverse coherent information capacity restricts the protocols to a very specific subset, its study re- concatenation property of the amplitude damping chan- mains very interesting, as for some channels it achieves nel ( ′ = ′ ) it is easy to prove that the ampli- a remarkable improvements over the unassisted capac- η η ηη tudeD damping◦ D channelD is degradable ( ( )= (1)( )) ity (Λ). To get some intuition on when we may ob- η η for η 1/2 and antidegradable ( ( E)=D 0) forEη D1/2. tainE an improvement, we look at the difference between η We conclude≥ that ( ) outperformsE D ( ) for≤ all η. the coherent information and its reverse counterpart R η η Even more interestingly,E D ( ) remainsE positiveD in the (I (ρ ) I(ρ ) = S(R) S(B)). We see that for R η R RB RB range η 1/2 where ( E) =D 0, see Fig. 4 (a). channels satisfying− S(R) >S(−B) over all inputs, such as η Generalized≤ amplitudeE D damping channel.- Spontaneous the bosonic lossy channel, reverse reconciliation performs emission to an environment at thermal equilibrium leads better than (1). On the other hand, for those channels to the generalized amplitude damping channel , satisfying S(EB) S(R) for all inputs, such as optical am- (η,α) which can be modeled by the Stinespring’s dilationD cir- plifiers or the erasure≥ channel, we obtain (1) . R cuit of Fig. 5. The relaxation operation applies the uni- In the case of the erasure channels is it easyE≥E to see≥E that tary transformation, = ⊳ > > R, which gives an example of strict E↔ E E E separation between ⊳ and . E ER 10 00 Amplitude damping channel.- The amplitude damp- 0 √η √1 η 0 URO =  −  , (10) ing channel describes the process of energy dissipation 0 √1 η √η 0 − − through spontaneous emission in a two-level system. The  00 01  effect of the channel on the input state ρ is (ρ) =   Dη   E0ρE0† + E1ρE1†, where jointly to the input state and the environment. The thermal environment is modeled by inserting half of an entan- 1 0 0 √1 η E0 = , E1 = − , (8) gled state Ψα = √1 α 00 +√α 11 into the second in- 0 √η 0 0 | i − | i | i put of URO. The channel can be seen as the random mix- and 1 η is the probability of spontaneous emission. Gen- ing (η,α) = α (η,0) +(1 α) (η,1) of two limiting cases: eralizing− the results of [11], we can restrict the input state (1)D the amplitudeD damping− channelD when ( ); and D(η,0) 4

(1) that R( (η,α)) > ( (η,α)) for any noise α except for αE= 1D/2, whereE bothD are equal, as shown in Fig. 4. ˮ ˮRXW LQ Unfortunately, we cannot conclude R( η,α) ( (η,α)) for α> 0, as the channels are no longerE D degradable.≥E D Nev-

Ry(ș) ertheless, it is easy to prove that generalized amplitude Relaxation Operation damping channels ( (η,α)) with η 1/2 are antidegrad- ˵ D ≤ ˞ able (ρB = (η/(1 η),α)(ρE )), which shows that for such channels D( −) ( ) = 0 (see Fig. 6). Stinespring’s dilation circuit ER D(η,α) ≥E D(η,α) Added Noise Conclusion.- We reviewed the relation between quantum communication and entanglement distribution capacities, paying special attention to entanglement distribution assisted by classical feedback. By restricting FIG. 5: (color online) Quantum circuit corresponding to the Stinespring’s dilation of the generalized amplitude damping ourselves to realistic protocols with a single final round channel . Alice’s input state ρin and half of an en- of post-processing, we defined the reverse entanglement D(η,α) tangled state Ψα interact through the relaxation operation distribution protocols. A subset of such protocols give an | i URO composed of two CNOT gates and a controlled rotation operational interpretation of the reverse coherent infor- 2 around the y-axis of the Bloch sphere (cos (γ/2) = η). mation, a symmetric counterpart of the coherent information. This allow us to define a new entanglement distribution capacity which is additive and outperforms the (2) a populating channel ( ). We restrict the analy- D(η,1) unassisted capacity for some important channels, such as sis to 0 α 1/2 as for any channel (η,α=1/2+x) with the damping channel and its generalization. optimal≤ input≤ population p there is a symmetricD channel ∗ We acknowledge financial support from the W. M. (η,α=1/2 x) with optimal population 1 p∗ reaching the − Keck Foundation Center for Extreme Quantum Informa- sameD capacity. As before, Alice’s input− can be restricted tion Theory. S.P. acknowledges financial support from to ρ = diag(1 p,p) (see appendix) giving A − the EU (Marie Curie fellowship). S(B) = H(ηp + (1 η)α), (11) − S(AB) = H4(λ1, λ2, λ3, λ4), (12) Appendix: Optimality of the input state where H4 is the Shannon entropy of a 4-dimensional distribution and λj are the four eigenvalues of ρAB, In this appendix we show that the input state ρA = λ1 = α(1 η)(1 p), λ2 = (1 α)(1 η)p, (13) diag(1 p,p) maximizes the (reverse) coherent informa- − − − − tion of− the amplitude damping channel and its general- λ = 1 λ λ 1 2(λ + λ ) + (λ λ )2 /2. 3,4 − 1 − 2 ± − 1 2 2 − 1 ization. The coherent information for degradable chan- h p i nels (amplitude damping) being a concave function im- Optimizing over the input population p we obtain the plies that diagonal input states outperform non-diagonal states [15]. The same argument hold for the reverse co- 0.5 0.7 herent information, this time over all channels. 0.45 (a) (b) 0.6 The optimization of the coherent information for non- 0.4

0.35 0.5 degradable channels, such as the generalized amplitude

0.3 0.4 damping channels, needs a more detailed proof. For 0.25 shake of completeness we present this speciﬁc proof for 0.3 0.2 the (reverse) coherent information for all channels stud- Tolerable Noise 0.15 Optimal Population 0.2 ied in this manuscript. 0.1 0.1 0.05

0 0 1 0.75 0.5 0.25 0 1 0.75 0.5 0.25 0 Dampimg parameter (η) Dampimg parameter (η) Amplitude Damping Channel

FIG. 6: (a) Tolerable thermal noise of the generalized am- The most general input state to the amplitude damp- plitude damping channel (η,α) (minimum α such that the capacity is zero) as a functionD of the damping parameter η ing (AD) channel reads (1) for: ( (η,α)) (dashed line), and R( (η,α)) (solid line). E D E D iφ (b) Input population p achieving the curves of (a). 1 p (1 p)pe− cos θ ρA′ = − − . (1 p)peiφ cos θ p p capacities (1)( ) and ( ). It is easy to show − (14) E D(η,α) ER D(η,α) p 5

A' One valid puriﬁcation of ρA′ (all are equivalent up to a 0 Ry(ϕ) UΦ A' unitary on A) reads

A iφ 0 Ry(Θ) ψ AA′ = 1 p 0 A 0 A′ +√pe 1 A′ [cos θ 0 +sin θ 1 ]A. A | i − | i | i | i | i (15)| i p In Fig. 7 we observe that the state ψφ ′ is gener- | iAA ated from ψ(φ=0) ′ by the local unitary U = I FIG. 7: Quantum circuit generating the bipartite state | iAA φ A ⊗ ψ AA′ . The first rotation Ry(ϕ) generates the quantum 1 0 | i iφ applied just before sending the state A′ state √1 p 0 A′ + √p 1 A′ (sin(ϕ/2) = p). The bipartite 0 e− ′ − | i | i A state is then entangled by the controlled rotation, generat- through the channel. (φ=0) ing ψ AA′ . The phase φ is finally fixed by a last local | i ′ After passage through the channel Alice and Bob en- unitary operation Uφ on A . tangled state reads

2 iφ iφ 1 p + p(1 η)cos θ p(1 η)cos θ sin θ η(1 p)pe− cos θ η(1 p)pe− sin θ − − − − − p(1 η)cos θ sin θ p(1 η) sin2 θ 0 0 ρAB(φ)=  − − p p  , (16) η(1 p)peiφ cos θ 0 pη cos2 θ pη cos θ sin θ −  η(1 p)peiφ sin θ 0 pη cos θ sin θ pη sin2 θ   p   −  p where η is the damping parameter. It is easy to check Coherent Information φ (φ=0) that by applying Uφ† to ρAB we obtain ρAB . Because the von Neumann entropy is invariant under a unitary The coherent information reads I = S(B) S(AB). − transformation Uφ†, we can restrict our study to the case In order to proof that cos θ = 0 maximizes the coherent φ = 0 without loss of generality. information, we calculate the derivative of I,

∂I = p(1 p) sin(2θ)[F (η) F (1 η)] (22) Eigenvalues ∂θ − − − − where The eigenvalues of ρA read x 1 √a F (x)= log − , (23) λA,1(2) = λ (1). (17) √a 1+ √a ± where and a = (1 2xp)2 +4x(1 p)p cos2 θ. In order to ﬁnd the values of−θ maximizing−I, we ﬁrst search for the ex- λ (x) = [1 (1 2xp)2 +4x(1 p)p cos2 θ]/2. (18) trema (∂I/∂θ = 0). The term sin(2θ) (cos2 θ) having ± ± − − p period of π (π/2), we can restrict the study to the do- The eigenvalues of ρAB then read main θ 0, π without loss of generality. The are two cases of∈ pathological { } extrema; Firstly, p = 0 and p = 1 λAB,1(2) = λ (1 η), which correspond to separable input states ( 00 and 11 ± − | i | i λAB,3(4) = 0. (19) respectively) which give I = 0; Secondly, F (η)= F (1 η) giving η = 1/2, i.e., the range limitation of I (the low-− Bob state read est η such that I(η = 1/2) = 0). For 0 < p < 1 and η > 1/2 we have an extremum when sin(2θ) = 0, 1 ηp η(1 p)p cos θ (θ = kπ/2). For θ = 0, π the input state is separa- ρB = − − , (20) { } η(1 p)p cos θ ηp ble ( 0 A ( 0 1 )A′ /√2) and therefore has I = 0 (as p − S(A,| Bi)=⊗S|(Ai)+ ± | Si(B) and S(A) = 0), which is a min- which givesp the eigenvalues imum for η > 1/2. Because a single extremum between two minimums can only be a maximum, we conclude that λB,1(2) = λ (η). (21) θ = π/2 optimizes I. We have then proven that the opti- ± mal input is diag(p, 1 p), as shown in [11]. In the range Now we are ready to calculate the (reverse) coherent in- η < 1/2 the roles of π/− 2 and θ = 0, π are exchanged, formation for a general input state. giving I = 0 as maximum. { } 6

Reverse Coherent Information ical extremum η = 1/2 is now replaced by η = 0, which coincides with the range limitation of the reverse coher-

The reverse coherent information reads IR = S(A) ent information. S(AB). The proof is very similar to the previous result,− where the partial derivative among θ now reads, ∂I = p(1 p) sin(2θ)[F (1) F (1 η)] . (24) Generalized Amplitude Damping Channel ∂θ − − − − For θ = 0, π the initial input being separable, we obtain After passage through the generalized amplitude { } IR = S(B), which is negative. This two extrema being damping (GAD) channel, Alice and Bob entangled state minima,− θ = π/2 remains the maximum. The patholog- reads

Z C ρ = , (25) AB CT W where (1 p)αη + (1 α)(1 p + p(1 η)cos2 θ) p(1 α)(1 η)cos θ sin θ Z = − − − − − − , (26) p(1 α)(1 η)cos θ sin θ p(1 α)(1 η) sin2 θ − − − −

α(1 p)(1 η)+ p(α + (1 α)η)cos2 θ p(α + (1 α)η)cos θ sin θ W = − − − − , (27) p(α + (1 α)η)cos θ sin θ p(α + (1 α)η) sin2 θ, − −

η(1 p)p cos θ η(1 p)p sin θ C = , (28) −0− 0 p p η is the damping parameter and α is related to the thermal noise of the environment. After a long calculation one can show that the eigenvalues of ρAB read 1 λ = 1+ a + bp(1 p)cos2 θ c + dp(1 p)cos2 θ + a + bp(1 p)cos2 θ , AB,1(2) 4 − ± − − q 1 p p λ = 1 a + bp(1 p)cos2 θ c + dp(1 p)cos2 θ a + bp(1 p)cos2 θ . (29) AB,3(4) 4 − − ± − − − p q p where

a = (1 2(1 η)(α + p 2αp))2 (30) − − − b = 4(1 η)(1 4(1 α)α(1 η)) (31) − − − − c = 1 2(1 η)(α + p 2αp)+2(p α)2(1 η)2 (32) − − − − − d = 2(1 η). (33) −

2 In the GAD channel Alice state ρA and its eigenvalues where e = (1 2(pη + α(1 η))) and f =4η. remain the same as in the AD channel. Bob’s state reads − − 1 (ηp + (1 η)α) η(1 p)p cos θ ρB = − − − , (34) η(1 p)p cos θ ηp + (1 η)α − p − which givesp the eigenvalues 1 λ = 1 e + fp(1 p)cos2 θ , (35) B,1(2) 2 ± − h p i 7

Extremum Search

The ﬁrst derivative of the coherent information reads,

∂I p(1 p) 1 = − sin(2θ) Z + Y (i, j) [1 + J(i, j)] , ∂θ 8   i,j=0 X  (36)

where,

f 1 e + fp(1 p)cos2 θ Z = log − − , (37) − e + fp(1 p)cos2 θ " 1+ e + fp(1 p)cos2 θ # − p − p p b i b 2 ( ) 2d Y (i, j) = ( )i + ( )j √a+b cos θ − − . (38) √ 2  − a + b cos θ − √2 c + d cos2 θ ( )i√a + b cos2 θ − −  q  and 1 J(i, j) = log 1 ( )i a + bp(1 p)cos2 θ + ( )j √2 c + dp(1 p)cos2 θ ( )i a + bp(1 p)cos2 θ . 4 − − − − − − − − p q p (39)

In order to ﬁnd the maxima of I as a function of θ, we two extra solutions to the equation ∂I/∂θ=0, as shown search ﬁrst for the extrema of I on the domain θ 0, π . in Fig. 8, except for α = 0, i.e., the AD channel. The ∈{ } As for the AD channel, p = 0 and p = 1 are extrema. The numerical check shows that both solutions θ∗, π θ∗ term sin(2θ) gives us again two sets of extrema; Firstly, are always minima. Therefore we conclude that{ θ =−π/}2 θ = 0, π corresponding to unentangled inputs giving is the maximum, as expected. I ={ 0; Secondly,} the solution θ = π/2, corresponding to the conjectured optimal input, which is the candidate for being the maximum. Unfortunately the complicated Reverse Reconciliation form of the solution of eq. (36) does not preclude the existence of new extrema. After carrying a detailed nu- In the case of reverse coherent information the proof is very similar, we just need to change e and f by e′ = 2 I (1 2p) and f ′ = 4. The maxima and minima remains − 0.015 the same than those of the coherent information, except for the two minima θ∗, π θ∗ which no longer exist, as 0.01 shown in Fig. 9. We{ also observe− } that for θ = 0, π the { } 0.005 reverse coherent information is negative (IR = S(B)), as for the AD channel. − Θ 0.5 1 1.5 2 2.5 3 -0.005 [1] C. E. Shannon, Bell Syst. Tech. J. 27, 379 (1948). -0.01 [2] M. A. Nielsen and I. L. Chuang, Quantum Computation -0.015 and Quantum Information, (Cambridge University Press, Cambridge, 2002). [3] B. Schumacher and M. A. Nielsen, Phys. Rev. A 54, 2629 (1996). FIG. 8: Coherent information I as a function of θ for the [4] D. P. DiVincenzo, P. W. Shor, and J. A. Smolin, Phys. GAD channel with parameters η = 0.62, α = 0.5, and input Rev. A 57, 830 (1998); G. Smith and J. A. Smolin, Phys. population p = 0.25 (solid line) and p = 0.5 (dashed line). Rev. Lett. 98, 030501 (2007); G. Smith and J. Yard, Science 321, 1812 (2008). merical check over a large spectra of values of the pa- [5] S. Lloyd, Phys. Rev. A 55, 1613 (1997); I. Devetak, IEEE rameter p, α and η we have seen that there exist always Trans. Inf. Theory 51, 44 (2005). 8

I [6] I. Devetak and P. W. Shor, Commun. Math. Phys. 256, 287 (2005). 0.2 [7] T. M. Cover and J. A. Thomas, Elements of Information Theory, (Wiley, New Jersey, 2006). 0.1 [8] A. W. Leung, Phys. Rev. A 77, 012322 (2008). [9] F. Grosshans, G. van Assche, J. Wenger, R. Tualle- Θ Brouri, and P. Grangier, Nature (London) 421, 238 0.5 1 1.5 2 2.5 3 (2003). [10] I. Devetak and A. Winter, Phys. Rev. Lett. 93, 080501 -0.1 (2004). [11] V. Giovannetti and R. Fazio, Phys. Rev. A 71, 032314 -0.2 (2005). [12] H. Barnum, E. Knill, and M. A. Nielsen IEEE Trans.Info.Theor. 46, 1317 (2000) [13] C. H. Bennett, D. P. DiVincenzo, and J. A. Smolin, Phys. Rev. Lett. 78, 3217 (1997); FIG. 9: Reverse coherent information IR as a function of θ for the GAD channel with parameters η = 0.75, α = 0.4, and [14] D. Leung, J. Lim, and P. W. Shor, quant-ph/0710.5943. input population p = 0.25 (solid line) and p = 0.5 (dashed [15] M. Wolf and D. Perez-Garcia, Phys. Rev. A 75, 012303 line). (2007) .