Entropy Bound for the Classical Capacity of a Quantum Channel Assisted by Classical Feedback

Dawei Ding∗, Yihui Quek†, Peter W. Shor‡, and Mark M. Wilde§ ∗Stanford Institute for Theoretical Physics, Stanford University, Stanford, California 94305, USA, [email protected] †Information Systems Laboratory, Stanford University, Stanford, California 94305, USA, [email protected] ‡Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA, [email protected] §Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA, [email protected]

Abstract—We prove that the classical capacity of an arbitrary strong converse statement in [6], [7]. Bowen et al. proved quantum channel assisted by a free classical feedback channel is that the capacity of an entanglement-breaking channel for bounded from above by the maximum average output entropy sending classical messages is not increased by a free classical of the quantum channel. As a consequence of this bound, we conclude that a classical feedback channel does not improve feedback channel [8], and this result was strengthened to a the classical capacity of a quantum erasure channel, and by strong converse statement in [9]. Ref. [10] discussed several taking into account energy constraints, we conclude the same inequalities relating the classical capacity assisted by classical for a pure-loss bosonic channel. The method for establishing the feedback to other capacities. At the same time, it is known that aforementioned entropy bound involves identifying an informa- in general there can be an arbitrarily large gap between the tion measure having two key properties: 1) it does not increase under a one-way local operations and classical communication unassisted classical capacity and the classical capacity assisted channel from the receiver to the sender and 2) a quantum channel by classical feedback [11]. from sender to receiver cannot increase the information measure Our aim here is to go beyond [8] to establish an upper bound by more than the maximum output entropy of the channel. This on the classical capacity of an arbitrary, not just entanglement- information measure can be understood as the sum of two terms, breaking, quantum channel assisted by a classical feedback with one corresponding to classical correlation and the other to entanglement. channel. Due to the fact that a quantum feedback channel is a stronger resource than a classical feedback channel, an I.INTRODUCTION immediate consequence of Bowen’s result [2] is that the A famous result of Shannon is that a free feedback channel entanglement-assisted capacity is an upper bound on the does not increase the capacity of a classical channel for classical capacity assisted by classical feedback. However, communication [1]. That is, the feedback-assisted capacity is since a quantum channel can, in general, establish quantum equal to the channel’s mutual information. Shannon’s result entanglement [12]–[14] and entanglement can increase capac- indicates that the mutual information formula for capacity ity [3]–[5], in such cases it may appear difficult to establish is particularly robust, in the sense that, a priori, one might an upper bound on this capacity other than the entanglement- consider a feedback channel to be a strong resource for assisted capacity. Our main result is that the latter is actually assisting communication. possible: we prove here that the maximum output entropy of With the rise of quantum information theory, several re- a quantum channel is an upper bound on its classical capacity

arXiv:1902.02490v2 [quant-ph] 15 Jul 2019 searchers have found variations and generalizations of Shan- assisted by classical feedback. As a generalization of this non’s aforementioned result, in the context of communication result, we find that the maximum average output entropy is over quantum channels. For example, Bowen proved that the an upper bound on the same capacity for a channel that is a capacity of a quantum channel for sending classical messages, probabilistic mixture of other channels. when assisted by a free quantum feedback channel, is equal The approach that we take for establishing the aforemen- to the channel’s entanglement-assisted capacity [2], which tioned bounds is similar in spirit to approaches used to bound is in turn equal to the mutual information of a quantum other assisted capacities or protocols [15]–[18]. We identify channel [3]–[5]. This result indicates that the mutual infor- an information measure that has two key properties: 1) it mation of a quantum channel is robust, in a sense similar does not increase under a free operation, which in this case to that mentioned above. The result also indicates that the is a one-way local operations and classical communication best strategy, in the limit of many channel uses, is to use the (1W-LOCC) channel from the receiver to the sender, and 2) quantum feedback channel once in order to establish sufficient a quantum channel from sender to receiver cannot increase shared entanglement between the sender and receiver, and the information measure by more than the maximum output to subsequently employ an entanglement-assisted communi- entropy of the channel. This information measure can be cation protocol [3]–[5]. Bowen’s result was strengthened to a understood as the sum of two terms, with one corresponding to classical correlation and the other to entanglement. W We organize the rest of the paper as follows. Section II provides a formal definition of a protocol for classical com- A’  A’  A’  A’ munication over a quantum channel assisted by classical F  A B F  A B F  A B feedback. Section III discusses explicitly how to purify such N N N Ŵ    a protocol, which is an important conceptual step for our B’ B’ B’ analysis. Section IV introduces our key information measure and several important supplementary lemmas regarding it. Fig. 1. A protocol for classical communication over three uses of a quantum Section V then employs this information measure and the sup- channel NA→B , when assisted by a classical feedback channel. plementary lemmas to establish the maximum output entropy bound for classical capacity assisted by classical feedback. We 0 apply this bound to the erasure channel and pure-loss bosonic Bob processes his systems B1B1 with the decoding chan- 1 channel in Section VI. We conclude in Section VII. nel 0 0 and Alice acts with the encoding channel B1B1→F1B2 2 D A0 F →A0 A , resulting in the state II.PROTOCOLFORCLASSICALCOMMUNICATIONOVERA E 1 1 2 2 QUANTUMCHANNELASSISTEDBYCLASSICALFEEDBACK (2) 2 1 (1) ω 0 0 ( 0 0 0 0 )(ρ 0 0 ). WA A B A F1→A A2 B1B →F1B WA B B 2 2 2 ≡ E 1 2 ◦ D 1 2 1 1 1 To begin with, let n, M N, let ε [0, 1], and let E 0. (5) ∈ ∈ ≥ Let A→B be a quantum channel, and let H be a Hamiltonian This process iterates n 2 more times, resulting in the N − acting on the input system A of A→B. An (n, M, H, E, ε) following states: N protocol for classical communication over a quantum channel (i) (i) ρWA0 B B0 Ai→Bi (ωWA0 A B0 ), (6) A→B consists of n uses of the quantum channel A→B, i i i ≡ N i i i N N along with the assistance of a classical feedback channel from (i+1) ωWA0 A B0 the receiver Bob to the sender Alice, in order for Alice to send i+1 i+1 i+1 ≡ i+1 i (i) one of M messages to Bob such that the error probability is ( 0 0 0 0 )(ρ 0 0 ), (7) A Fi→A Ai+1 BiBi→FiBi+1 WA BiB no larger than ε. Furthermore, the average state at the input of E i i+1 ◦ D i i each channel use should have energy no larger than E, when for i 2, . . . , n 1 . The final decoding (measurement) channel∈ {n −results} in the following state: taken with respect to the Hamiltonian H. B B0 →Wˆ D n n In more detail, the protocol consists of an initial classical– n (n) 0 0 0 ρ ˆ (TrA 0 )(ρ 0 0 ). (8) quantum state σF B , with F0 classical and B quantum, of W W n BnB →Wˆ WAnBnBn 0 1 1 ≡ ◦D n the form Figure 1 depicts the above protocol for n = 3.

X f0 0 For an (n, M, H, E, ε) protocol, the following is satisfied σF0B = p(f0) f0 f0 F0 σB0 . (1) 1 | ih | ⊗ 1 f 1 0 Φ ˆ ρ ˆ ε, (9) 2 W W − W W 1 ≤ It also involves n encoding channels, with each one denoted 1 PM i where ΦW Wˆ M m=1 m m W m m Wˆ is by A0 F →A0 A for i 1, . . . , n , as well as n decoding ≡ | ih | ⊗ | ih | i−1 i−1 i i the maximally classically correlated state. Note that E ∈ { } i channels, with each of them denoted by 0 0 for 1 BiB →FiB Φ ρ = Pr Wˆ = W W D i i+1 2 W Wˆ W Wˆ 1 , where here denotes i 1, . . . , n 1 . Note that all F systems are classical the uniform− random variable{ 6 corresponding} to the message ∈ { − } because the feedback channel is constrained to be a classical choice and Wˆ denotes the random variable corresponding channel. So this means that each decoding channel is a quan- ˆ to the classical value in the register W of the state ρW Wˆ . tum instrument. The final decoding is denoted by n . B B0 →Wˆ Furthermore, the following energy constraint applies as well: D n n We now detail the form of such a protocol. It begins with n 1 X (i) Alice preparing the following classical–quantum state: Tr HωA E, ωA ω , (10) { } ≤ ≡ n Ai M i=1 1 X m ρWA0 m m W ρA0 , (2) which limits the energy of the average input state. 0 M 0 ≡ m=1 | ih | ⊗ III.PURIFIED PROTOCOL m for some set ρA0 m of quantum states. The global initial { 0 } Our goal is to bound the rate of such a protocol. With this state is then ρ 0 σ 0 . Alice then performs the encoding WA0 F0B1 1 ⊗ in mind, we can devise a protocol that simulates the above channel A0 F →A0 A and the state becomes as follows: E 0 0 1 1 one. It consists of purifying each step of the above protocol (1) 1 and Bob keeping a copy of the classical feedback, such that ω 0 0 0 0 (ρ 0 σ 0 ). (3) WA A B A F0→A A1 WA F0B 1 1 1 ≡ E 0 1 0 ⊗ 1 at each time step, conditioned on the value of the message in W and the feedback in the existing systems labeled by F , Alice transmits the A1 system through the first use of the the state is pure. To be clear, we go through the steps of the channel A1→B1 , resulting in the following state: N purified protocol. In order to simplify notation, we let Aˆ be (1) (1) 0 ρWA0 B B0 A1→B1 (ωWA0 A B0 ). (4) a joint system throughout, referring to both the original A 1 1 1 ≡ N 1 1 1 system as well as a purifying system, and we take the same simulation protocol, we also consider an isometric channel ˆ N convention for B. The initial state of Alice is as follows: A→BE that simulates the original channel A→B as follows: U = Tr N . N 1 M A→B E A→BE X m N Thus, the various◦U states involved in the purified protocol are ρW Aˆ m m W ψ ˆ , (11) 0 ≡ M | ih | ⊗ A0 m=1 as follows. The global initial state is ρW Aˆ σF F 0Bˆ . Alice 0 ⊗ 0 0 1 m m 1 where ψ is a purification of ρ 0 , such that tracing over a performs the enlarged encoding channel ˆ ˆ and the Aˆ A A0F0→A1A1 0 m m 0 U subsystem of ψ gives ρ 0 . The initial state of Bob is as state becomes as follows: Aˆ0 A0 follows: (1) 1 ω (ρ σ 0 ). (15) ˆ ˆ 0 ˆ ˆ W Aˆ0 F0F Bˆ1 W A1A1B1F0 A0F0→A1A1 0 X f0 ≡ U ⊗ σ 0 ˆ p(f0) f0 f0 F f0 f0 F 0 ϕ , (12) F0F B1 0 0 Bˆ 0 ≡ | ih | ⊗ | ih | ⊗ 1 Alice transmits the A1 system through the first use of the f0 extended channel N , resulting in the following state: A1→B1E1 f0 f0 U where ϕ ˆ is a purification of σB0 , such that tracing over a B1 1 (1) N (1) f f ρ = (ω ). 0 0 0 ˆ ˆ 0 A1→B1E1 ˆ ˆ 0 (16) subsystem of ϕ gives σ 0 , and he keeps an extra copy F0 W A1B1B1E1F0 W A1A1B1F0 Bˆ1 B1 U i of the classical data. Let ˆ ˆ denote an isometric ˆ UAi−1Fi−1→AiAi Bob processes his systems B1B1 with the enlarged i 1 channel extending the encoding channel A0 F →A0 A , decoding channel and Alice acts with i−1 i−1 i i B Bˆ →F F 0Bˆ E V 1 1 1 1 2 for i 1, . . . , n . Since the system Fi is classical, for the enlarged encoding channel 2 , resulting ∈ { } i Aˆ1F1→Aˆ2A2 i 1, . . . , n 1 , the decoding channel B B0→F B0 can (2) U ∈ { − } D i i i i+1 in the state ω ( 2 W Aˆ A Bˆ E F 0F 0 Aˆ F →Aˆ A be written explicitly as 2 2 2 1 0 1 ≡ U 1 1 2 2 ◦ 1 (1) X i,f ˆ 0 ˆ )(ρ 0 ). This process iterates i i B1B1→F1F B2 W Aˆ1B1Bˆ1E1F 0 0 = 0 0 f f , (13) V 1 0 BiB →FiB B B →B i i Fi D i i+1 D i i i+1 ⊗ | ih | n 2 more times, resulting in the following states: fi − (i) N (i) i,fi ρ (ω ), W Aˆ B Bˆ Ei [F i−1]0 Ai→BiEi W Aˆ A Bˆ Ei−1[F i−1]0 such that B B0→B0 fi is a collection of completely pos- i i i 1 0 ≡ U i i i 1 0 {D i i i+1 } P i,fi (i+1) itive maps such that the sum map 0 0 is trace ω fi BiB →B ˆ ˆ i i 0 i i+1 W Ai+1Ai+1Bi+1E1[F0 ] i,f D ≡ i (i) preserving. Let V ˆ ˆ be a linear map such that tracing i+1 i BiBi→Bi+1 ( 0 )(ρ i i−1 ), Aˆ F →Aˆ A BiBˆi→FiF Bˆi+1 W Aˆ B Bˆ E [F ]0 i,fi i,fi † U i i i+1 i+1 ◦ V i i i i 1 0 over a subsystem of V ˆ ˆ ( )[V ˆ ˆ ] gives the BiBi→Bi+1 · BiBi→Bi+1 i,fi for i 2, . . . , n 1 . The final enlarged original map 0 0 , and define the map BiB →B ∈ { n− } D i i+1 decoding channel ˆ ˆ ˆ results in VBnBn→Bn+1W i,fi i,fi i,fi † the following state: ρ ˆ ˆ ˆ n n−1 0 (τ ˆ ) V τ ˆ [V ] . W AnBn+1WE1 [F0 ] B Bˆ →Bˆ BiBi B Bˆ →Bˆ BiBi B Bˆ →Bˆ ≡ V i i i+1 ≡ i i i+1 i i i+1 n (ρ(n) ). Note that we recover B Bˆ →Bˆ Wˆ W Aˆ B Bˆ En[F n−1]0 Then we define the enlarged decoding channel V n n n+1 n n n 1 0 each state of the original protocol from Section II by i as B Bˆ →F Bˆ F 0 V i i i i+1 i performing particular partial traces.

i X i,fi f f f f 0 . B Bˆ →F Bˆ F 0 ˆ ˆ i i Fi i i Fi IV. INFORMATION MEASURE FOR ANALYSIS OF PROTOCOL V i i i i+1 i ≡ VBiBi→Bi+1 ⊗ | ih | ⊗ | ih | f i The key information measure that we use to analyze this Note that this enlarged decoding channel keeps an extra copy protocol is as follows: 0 of the classical feedback value for Bob in the register Fi . The final decoding channel in the original protocol is equivalent to I(W ; CF )τ + S(C WF )τ , (17) | a measurement channel, and thus can be written as where τWFC is a classical–quantum state of the form n X w (τB B0 ) = Tr Λ 0 τB B0 w w ˆ , B B0 →Wˆ n n BnBn n n W X w,f D n n { }| ih | τWFC = p(w, f) w w W f f F τ . (18) w | ih | ⊗ | ih | ⊗ C w,f w where Λ 0 w is a POVM. We enlarge it as follows in the BnBn simulation{ protocol:} The first term in (17) represents the classical correlation between system W and systems CF , while the second term n (τ ˆ ) = represents the average entanglement between the system C of B Bˆ →Bˆ Wˆ BnBn V n n n+1 the state τ w,f and a purifying reference system. X q w q w C ΛB B0 τ ˆ ΛB B0 w w ˆ , (14) n n BnBn n n ⊗ | ih |W We now establish some properties of the information mea- w sure in (17). Let us first recall the following lemma from [19]: where the meaning of the notation is that the map Lemma 1: Let φAB be a pure bipartite state, and let q w q w x ΛB B0 ( ) ΛB B0 acts nontrivially on the subsystems p(x), ϕA0B0 be an ensemble of pure bipartite states obtained n n · n n { } 0 from φAB by means of a 1W-LOCC channel of the form BnBn in the original protocol and trivially on all other B subsystems, while mapping all B systems to a system X x x A→A0 B→B0 x x X , (19) Bˆn+1 large enough to accommodate all of them. In the x U ⊗ V ⊗ | ih | x 0 0 where B→B0 x is a collection of completely positive trace = I(W ; B B F )ω + S(B B WF )ω (26) {V } x x x † | | non-increasing maps with 0 ( ) = V 0 ( )[V 0 ] 0 0 0 B→B B→B B→B = S(B B F )ω S(B B WF )ω + S(B B WF )ω (27) x V · · | − | | and A→A0 x is a collection of isometric channels, so that 0 {U } = S(B B F )ω S(B)ω. (28) | ≤ x 1 x x ϕ 0 0 ( 0 0 )(φAB), (20) A B ≡ p(x) UA→A ⊗ VB→B All inequalities follow from definitions and applying chain x x rules for mutual information and entropy. The final inequality p(x) Tr ( 0 0 )(φAB) . (21) ≡ { UA→A ⊗ VB→B } follows because conditioning does not increase entropy. 0 Then the following inequality holds S(B)φ S(B X)τ , for P x ≥ | V. MAXIMUMOUTPUTENTROPYBOUND τXA0B0 p(x) x x X ϕ 0 0 . ≡ x | ih | ⊗ A B The above lemma leads to the following one, which is the Now that we have identified a quantity that does not increase statement that the quantity in (17) is monotone with respect under 1W-LOCC from Bob to Alice and cannot increase by to 1W-LOCC channels: more than the output entropy of a channel under its action, we Lemma 2: Let τW F AB be a classical–quantum state, with can use these properties to establish the following upper bound classical systems WF and quantum systems AB pure when on the rate of a feedback-assisted communication protocol: conditioned on WF , and let AB→A0B0X be a 1W-LOCC Theorem 4: (n, M, H, E, ε) M For an protocol for classical channel of the form in (19). Then the following holds communication over a quantum channel A→B assisted by N 0 0 classical feedback, of the form described in Section II, the I(W ; BF )τ +S(B WF )τ I(W ; B FX)θ +S(B WFX)θ, | ≥ | following bound applies where θWFA0B0X AB→A0B0X (τW F AB). ≡ M 0 (1 ε) log M n sup S( (ρ)) + h (ε). (29) Proof. The inequality I(W ; BF )τ I(W ; B FX)θ holds 2 2 ≥ − ≤ · ρ:Tr{Hρ}≤E N from data processing. In more detail, consider that θWFB0X is equal to Proof. Let us consider the purified simulation of a given ( ) (n, M, H, E, ε) protocol, as given in Section III. We start with X x x = TrA0 ( A→A0 B→B0 )(τW F AB) x x X ˆ log2 M = I(W ; W ) (30) x U ⊗ V ⊗ | ih | Φ ˆ X x I(W ; W )ρ + ε log2 M + h2(ε), (31) = 0 (τWFB) x x X , (22) VB→B ⊗ | ih | ≤ x where we have applied the condition in (9) and standard x where the last equality follows because each map 0 entropy inequalities. Continuing, we find that UA→A is trace preserving. So the state θ 0 can be under- WFB X ˆ stood as arising from the action of the quantum instrument I(W ; W )ρ P x n−1 0 n−1 0 0 x x X on the state τWFB, and since this is I(W ; B Bˆ [F ] ) (n) + S(B Bˆ [F ] W ) (n) x VB→B ⊗ | ih | n n 0 ρ n n 0 ρ a channel from B to B0X, the data processing inequality ≤ | (32) 0 applies so that I(W ; BF )τ I(W ; B FX)θ. The inequality ˆ n−1 0 ˆ n−1 0 0 ≥ = I(W ; BnBn[F0 ] )ρ(n) + S(BnBn [F0 ] W )ρ(n) S(B WF )τ S(B WFX)θ follows from an application of | | ≥ | h ˆ 0 ˆ 0 i Lemma 1, by conditioning on the classical systems WF . I(W ; B1F ) (1) + S(B1 F W ) (1) (33) − 0 ω | 0 ω The following lemma places an entropic upper bound on ˆ n−1 0 ˆ n−1 0 = I(W ; BnBn[F ] ) (n) + S(BnBn [F ] W ) (n) the amount by which the quantity in (17) can increase by the 0 ρ | 0 ρ h ˆ 0 ˆ 0 i action of a channel A→B: I(W ; B1F ) (1) + S(B1 F W ) (1) N − 0 ω | 0 ω Lemma 3: Let τW F AB0 be a classical–quantum state of the n following form: X ˆ i−1 0 ˆ i−1 0 + I(W ; Bi[F ] ) (i) + S(Bi [F ] W ) (i) 0 ω | 0 ω X w,f i=2 τW F AB0 = p(w, f) w w W f f F τ 0 . (23) | ih | ⊗ | ih | ⊗ AB h ˆ i−1 0 ˆ i−1 0 i w,f I(W ; Bi[F ] ) (i) + S(Bi [F ] W ) (i) . − 0 ω | 0 ω Then The first inequality follows from data processing and non- 0 0 negativity of entropy. The first equality follows because I(W ; BB F )ω + S(BB WF )ω ˆ 0 ˆ 0 I(W ; B1F ) (1) + S(B1 F W ) (1) = 0 for the initial state 0 | 0 0 ω 0 ω [I(W ; B F )τ + S(B WF )τ ] S(B)ω, (24) (1) | ω ˆ ˆ 0 (there is no classical correlation between W and − | ≤ W A1A1B1F0 0 0 ˆ 0 ˆ where ωWFBB A→B(τW F AB ). B1F0, and the state on system B1 is pure when conditioned ≡ N 0 Proof. Consider that on F0W ). The last equality follows by adding and subtracting the same term. Continuing, we find that the quantity in the I(W ; BB0F ) + S(BB0 WF ) ω ω last line above is bounded as 0 | 0 [I(W ; B F )τ + S(B WF )τ ] − | ˆ n−1 0 ˆ n−1 0 0 0 I(W ; BnBn[F ] ) (n) + S(BnBn [F ] W ) (n) = I(W ; BB F )ω + S(BB WF )ω 0 ρ 0 ρ ≤ h | i 0 |0 ˆ 0 ˆ 0 [I(W ; B F )ω + S(B WF )ω] (25) I(W ; B1F ) (1) + S(B1 F W ) (1) − | − 0 ω | 0 ω n X ˆ i−2 0 is a weak converse bound. Going forward from here, it would + I(W ; Bi−1Bi−1[F ] ) (i−1) 0 ρ be good to find strong converse and tighter bounds on the i=2 ˆ i−2 0 classical capacity assisted by classical feedback. + S(Bi−1Bi−1 [F ] W )ρ(i−1) | 0 Acknowledgements. We acknowledge discussions with Xin Wang, Patrick h ˆ i−1 0 ˆ i−1 0 i I(W ; Bi[F ] ) (i) + S(Bi [F ] W ) (i) Hayden, and Tsachy Weissman. DD is supported by a National Defense − 0 ω | 0 ω n Science and Engineering Graduate Fellowship. YQ is supported by a Stan- X ˆ i−1 0 ˆ i−1 0 = I(W ; BiBi[F ] ) (i) + S(BiBi [F ] W ) (i) ford Graduate Fellowship and a National University of Singapore Overseas 0 ρ | 0 ρ i=1 Graduate Scholarship. PWS is supported by the NSF under Grant No. CCF- h ˆ i−1 0 ˆ i−1 0 i 1525130 and through the NSF Science and Technology Center for Science of I(W ; Bi[F0 ] )ω(i) + S(Bi [F0 ] W )ω(i) (34) − | Information under Grant No. CCF0-939370. MMW acknowledges NSF grant n X no. 1350397. DD and MMW thank God for all His provisions. 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Wilde, “Amortized entanglement of a quantum inputs. Applying Theorem 5, we conclude that the classical channel and approximately teleportation-simulable channels,” Journal capacity of the erasure channel assisted by classical feedback of Physics A, vol. 51, no. 3, p. 035303, January 2018. is equal to (1 p) log d, so that classical feedback does not [18] M. Berta, C. Hirche, E. Kaur, and M. M. Wilde, “Amortized channel − 2 divergence for asymptotic quantum channel discrimination,” August increase the classical capacity of the erasure channel. 2018, arXiv:1808.01498. [19] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, VII.CONCLUSION “Mixed-state entanglement and quantum error correction,” Physical Review A, vol. 54, no. 5, pp. 3824–3851, November 1996. Our main result is that the maximum average output entropy [20] V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H. Shapiro, and of a quantum channel is an upper bound on its classical H. P. Yuen, “Classical capacity of the lossy bosonic channel: The exact capacity assisted by classical feedback. Note that the bound solution,” Phys. Rev. Lett., vol. 92, no. 2, p. 027902, January 2004. APPENDIX A We can then bound the sum over entropies as follows: MAXIMUMAVERAGEOUTPUTENTROPYBOUNDFOR n X PROBABILISTICMIXTUREOFCHANNELS S(Bi Zi) (i) nS(B Z)ρ (40) | ρ ≤ | In this appendix, we provide a simple proof of Theorem 5. i=1 X z The main idea behind the proof is to observe that any = n pZ (z)S( (ω)) (41) N feedback-assisted protocol of the form discussed in Section II, z X z which is for communication over a probabilistic mixture n sup pZ (z)S( (ρ)). (42) P z channel A→B = pZ (z) , has a simulation of the ≤ ρ:Tr{Hρ}≤E z N N z NA→B following form: The first inequality is by concavity of conditional entropy, 1) Before the ith use of the channel A→B in the feedback- and the conditional entropy is defined on the averaged channel N P z assisted protocol, Bob selects a random variable Zi inde- output state over uses ρ pZ (z) z z (ω), BZ ≡ z | ih | ⊗ N pendently according to the distribution pZ . He transmits ω = 1 Pn ω(i). The second equality is by definition A n i=1 Ai Zi over the classical feedback channel to Alice. of conditional entropy. The third inequality follows from 2) Each channel use A→B from the original protocol is N optimizing over states that satisfy the energy constraint in (10). replaced by a simulation in terms of another channel This concludes the proof of Theorem 5. AZ0→B, which accepts a quantum input on system AM and a classical input on system Z0. Conditioned 0 on the value z in system Z , the channel AZ0→B applies z to the quantum system A.M Thus, if NA→B the random variable Z pZ is fed into the input 0 ∼ system Z of AZ0→B, then the channel AZ0→B is M M indistinguishable from the original channel A→B. N 3) Alice feeds a copy of the classical random variable Zi into the ith use of the channel AZ0→B. 4) All other aspects of the protocolM are executed in the same way as before. Namely, even though it would be an advantage to Alice to modify her encodings and Bob to modify later decodings based on the realizations of Zi, they do not do so, and they instead blindly operate all other aspects of the simulation protocol as they are in the original protocol. Our goal now is to establish the inequality in Theorem 5, relat- ing the n, M, E, ε parameters of the original (n, M, H, E, ε) protocol by using the above simulation. The main observation to make from here is that the same proof from Lemma 3 gives the following bound:

0 0 I(W ; BB FZ)ω + S(BB WFZ)ω 0 | 0 [I(W ; B FZ)τ + S(B WFZ)τ ] S(B Z)ω, (36) − | ≤ | where ωWFZBB0 is the following state:

ωWFZBB0 AZ0→B(τWFZZ0AB0 ) (37) ≡ M τWFZZ0AB0 ≡ X w,f,z p(w, f, z) w, f, z, z w, f, z, z W,F,Z,Z0 τ 0 . (38) | ih | ⊗ AB w,f,z This follows by grouping Z with F , but then discarding only F and B0 at the end of the proof. We then apply this bound, and the same reasoning in the proof of Theorem 4, except that the variables Z0,...,Zi are grouped together with the feedback i−1 0 variables [F0 ] and then the same reasoning in (32)–(34) applies. At this point, we invoke (36) and find that n X (1 ε) log M S(Bi Zi) (i) + h2(ε). (39) − 2 ≤ | ρ i=1