Communication over a quantum broadcast channel with cooperation between the receivers is considered. The first form of cooperation addressed is classical conferencing, where receiver 1 can send classical messages to receiver 2. Another cooperation setting involves quantum conferencing, where receiver 1 can teleport a quantum state to receiver 2. When receiver 1 is not required to recover information and its sole purpose is to help the transmission to receiver 2, the model reduces to the quantum primitive relay channel. The quantum conferencing setting is intimately related to quantum repeaters as the sender, receiver 1, and receiver 2 can be viewed as the transmitter, the repeater, and the destination receiver, respectively. We develop lower and upper bounds on the capacity region in each setting. In particular, the cutset upper bound and the decode-forward lower bound are derived for the primitive relay channel. Furthermore, we present an entanglement-formation lower bound, where a virtual channel is simulated through the conference link. At last, we show that as opposed to the multiple access channel with entangled encoders, entanglement between decoders does not increase the classical communication rates for the broadcast dual.

Attenuation in optical fibers poses a great challenge for long-distance quantum communication protocols, including both current applications, such as quantum key distribution,1 and future implementation of the quantum Internet2 and quantum networks in general.3 Quantum repeaters have been proposed as a potential solution where the distance is divided into smaller segments with quantum repeaters at the intermediate stations.4 In its simplest form, the process begins with using quantum communication and entanglement distillation to prepare two pairs of qubits at maximally entangled states, namely, |ΦAP1 between the sender and the repeater and |ΦP2B between the repeater and the receiver. At the next stage, the repeater teleports the quantum state of P1 onto B, thus swapping the entanglement such that A and B are now entangled at a distance twice that of the initial entangled pairs. Experimental implementation of the elementary building blocks for quantum repeaters has recently been considered by van Loock et al.5 in platforms based on quantum dots,6 trapped ions,7 and color centers in diamond.8,9 Here, we will give an information-theoretic perspective that can be associated with such a network.

The cross-disciplinary field of quantum information processing and communication is rapidly evolving in both practice and theory.10–17 Quantum information theory is the natural extension of the classical theory. Nevertheless, this generalization reveals astonishing phenomena with no parallel in classical communication.18 For example, pairing two memoryless quantum channels, each with zero quantum capacity, can result in a nonzero quantum capacity.19 This property is referred to as super-activation. It should be noted that super-activation has also been demonstrated in recent years for classical channels in advanced settings, such as secure message transmission over a wiretap channel with a jammer20 and identification over a discrete memoryless channel with feedback.21 Nevertheless, super-activation does not occur in the fundamental model of a classical one-way memoryless channel.

Communication over quantum channels can be separated into different tasks and categories. For classical information transmission, a regularized (“multi-letter”) formula for the capacity of a quantum channel without assistance was established by Holevo22 and Schumacher and Westmoreland.23 Although the calculation of such a formula is intractable, in general, it provides computable lower bounds, and there are special cases where the capacity can be computed exactly. The reason for this difficulty is that the Holevo information is not necessarily additive.24 A similar difficulty occurs with the transmission of quantum information. A regularized formula for the quantum capacity is given in Refs. 25–28, while a computable formula is obtained in the special case where the channel is degradable or less noisy.29 Quantum communication can also be used for the purpose of entanglement generation.28,30,31

Another scenario of interest is when the transmitters and receivers are provided with entanglement resources a priori.32–34 While entanglement can be used to produce shared randomness, it is a much more powerful aid.3,35,36 In particular, super-dense coding37 is a well-known communication protocol where two classical bits are transmitted using a single use of a noiseless qubit channel and a maximally entangled pair that is shared between the transmitter and the receiver. Thereby, transmitter–receiver entanglement assistance doubles the transmission rate of classical messages over a noiseless qubit channel. The entanglement-assisted capacity of a noisy quantum channel was fully characterized by Bennet et al.38,39 in terms of the quantum mutual information. In the other direction, i.e., using information measures to understand quantum physics, the quantum mutual information plays a role in investigating the entanglement structure of quantum field theories.40–43 

There are communication settings where entanglement resources can even increase the capacity of a classical channel. In particular, Leditzky et al.44 recently showed that entanglement between two transmitters can strictly increase the achievable rates for a classical multiple access channel. The channel construction in Ref. 44 is based on a pseudo-telepathy game45 where quantum strategies guarantee a certain win and outperform classical strategies, extending ideas by Nötzel46 and Quek and Shor.47 Entanglement assistance has striking effects in different communication games and their security applications as well.44,48–52 Furthermore, entanglement can assist in the transmission of quantum information. Given a classical channel with transmitter–receiver entanglement resources, qubits can be sent at half the rate of classical bits by employing the teleportation protocol.53 

Quantum broadcast and multiple access channels were studied in various settings, as in Refs. 54–77. Yard et al.54 derived the superposition inner bound and determined the capacity region for the degraded classical–quantum broadcast channel. By the monogamy property of quantum entanglement,78 the sender’s system cannot be in a maximally entangled state with both receivers simultaneously. However, different forms of entanglement can be generated. In particular, Yard et al.54 characterized the entanglement-generation rates for GHZ states. Wang et al.57 used the previous characterization to determine the capacity region for Hadamard broadcast channels. Dupuis et al.58,59 developed the entanglement-assisted version of Marton’s region for users with independent messages. Bosonic broadcast channels were considered in Refs. 79–83. The quantum broadcast and multiple access channels with confidential messages were recently considered in Refs. 84–87. An equivalent description of the super-activation phenomenon19 is that there exists a broadcast channel such that the sum-rate capacity with full cooperation between the receivers is positive, while the capacities of the marginal channels are both zero.

Savov et al.88,89 derived a partial decode-forward lower bound for the (non-primitive) classical–quantum relay channel, where the relay encodes information in a strictly causal manner. Recently, Ding et al.90 generalized those results and established the cutset, multihop, and coherent multihop bounds for the classical–quantum relay channel. Communication with the help of environment measurement can be modeled by a quantum channel with a classical relay in the environment.91 Considering this setting, Smolin et al.92 and Winter93 determined the environment-assisted quantum capacity and classical capacity, respectively. Savov et al.88 further discussed future research directions of interest (see Sec. V in Ref. 88) and pointed out that quantum communication scenarios over the relay channel may have applications for the design of quantum repeaters (see also Ref. 90). Our aim is to fulfill this prevision.

In this paper, we consider quantum broadcast channels in different settings of cooperation between the decoders. Using those settings, we provide an information-theoretic framework for quantum repeaters. The first form of cooperation that we consider is classical conferencing, where receiver 1 can send classical messages to receiver 2. This can be viewed as the quantum version of the classical setting by Dabora and Servetto94 (see also Ref. 95). We provide a regularized characterization for the classical capacity region of the quantum broadcast channel with classical conferencing and a single-letter formula for Hadamard broadcast channels.57 Next, we consider quantum conferencing, where receiver 1 can teleport a quantum state to receiver 2. We develop inner and outer bounds on the quantum capacity region with quantum conferencing, characterizing the trade-off between the communication rates Q1 and Q2 to receiver 1 and reciever 2, respectively, and the conferencing capacity CQ,12. The case where receiver 1 is not required to recover information and its sole purpose is to help the transmission to receiver 2 reduces to the model of the primitive relay channel,96 for which the decode-forward lower bound and cutset upper bound follow as a consequence. In addition, we establish an entanglement-formation lower bound, where a virtual channel is simulated through the conference link, following the results of Berta et al.97 on quantum channel simulation.

The quantum conferencing setting is intimately related to quantum repeaters as the sender, receiver 1, and receiver 2 can be viewed as the transmitter, the repeater, and the destination receiver, respectively, in the repeater model. In particular, the sender can employ quantum communication to receiver 1 (the repeater) in order to prepare a maximally entangled pair |ΦAP1, which consists of nQ1 entangled bits (ebits). Given entanglement between the receivers, we also have a maximally entangled pair |ΦP2B, which consists of nCQ,12 ebits, shared between the repeater and the destination receiver. Then, the repeater can swap his entanglement by using the classical conferencing link to teleport the state of P1 onto B, thus swapping the entanglement such that A and B are now entangled. Hence, our results provide an information-theoretic analysis, characterizing the achievable rates of ebits that can be generated in each stage. As our model includes direct transmission from A to B as well, our results exhibit the trade-off between repeaterless communication and communication via the repeater. Other relay channel models for quantum repeaters can also be found in Refs. 98–102.

At last, we compare entanglement cooperation for the multiple access channel and the broadcast channel. The duality between the multiple access channel and the broadcast channel has emerged as a prominent tool in the study of wireless communication systems.103–105 We show that as opposed to the multiple access channel with entangled transmitters,44 entanglement between the receivers cannot enlarge the classical capacity region of a broadcast channel. That is, the broadcast dual to the multiple access channel property by Leditzky et al.44 does not hold. Therefore, our result reveals a fundamental asymmetry and demonstrates the limitations of the duality between the broadcast channel and the multiple access channel.

This paper is organized as follows. In Sec. II, we begin with the basic definitions. In Sec. III, we present three coding scenarios for the quantum broadcast channel with cooperation between the receivers. In particular, we consider classical communication when the receivers share entanglement resources a priori (Subsection III A), classical communication over the quantum broadcast channel with a classical conference link from receiver 1 to receiver 2 (Subsection III B), and quantum communication when receiver 1 can teleport a quantum state to receiver 2 via conferencing (Subsection III C). The quantum primitive relay channel is presented as a special case as well. Our results for classical conferencing are given in Sec. IV. Next, our main results on quantum conferencing are derived in Sec. V for the quantum broadcast channel (Secs. V A and V B) and the quantum primitive relay channel (Sec. V C). Section V is concluded with the resulting observations on the quantum repeater. In Sec. VI, we show that the broadcast dual to the multiple access channel property by Leditzky et al.44 does not hold, as entanglement between receivers cannot enlarge the classical capacity region. We conclude with a summary and discussion in Sec. VII.

We use the following notation conventions. Script letters X,Y,Z, are used for finite sets. Lowercase letters x, y, z, … represent constants and values of classical random variables, and uppercase letters X, Y, Z, … represent classical random variables. The distribution of a random variable X is specified by a probability mass function (pmf) pX(x) over a finite set X. We use xj = (x1, x2, …, xj) to denote a sequence of letters from X. A random sequence Xn and its distribution pXn(xn) are defined accordingly.

The state of a quantum system A is a density operator ρ on the Hilbert space HA. A density operator is an Hermitian, positive semidefinite operator, with unit trace, i.e., ρ = ρ, ρ ⪰ 0, and Tr(ρ) = 1. The state is said to be pure if ρ = |ψ⟩⟨ψ| for some vector |ψHA, where ⟨ψ| is the Hermitian conjugate of |ψ⟩. In general, a density operator has a spectral decomposition,

ρ=xXpX(x)|ψxψx|,
(1)

where X={1,2,,|HA|}, pX(x) is a probability distribution over X, and {|ψx}xX forms an orthonormal basis of the Hilbert space HA. A measurement of a quantum system is any set of operators {Λj} that forms a positive operator-valued measure (POVM), i.e., the operators are positive semi-definite and jΛj=1, where 1 is the identity operator.32 According to the Born rule, if the system is in state ρ, then the probability of the measurement outcome j is given by pA(j) = Tr(Λjρ). The trace distance between two density operators ρ and σ is ρσ1, where F1=Tr(FF).

Define the quantum entropy of the density operator ρ as H(ρ) ≜ −Tr[ρ log(ρ)], which is the same as the Shannon entropy associated with the eigenvalues of ρ. Consider the state of a pair of systems A and B on the tensor product HAHB of the corresponding Hilbert spaces. Given a bipartite state σAB, define the quantum mutual information as

I(A;B)σ=H(σA)+H(σB)H(σAB).
(2)

Furthermore, conditional quantum entropy and mutual information are defined by H(A|B)σ = H(σAB) − H(σB) and I(A;B|C)σ = H(A|C)σ + H(B|C)σH(A,B|C)σ, respectively. The coherent information is then defined as

I(AB)σ=H(A|B)σ.
(3)

A pure bipartite state is called entangled if it cannot be expressed as the tensor product of two states in HA and HB. The maximally entangled state between two systems of dimension D is defined by |ΦAB=1Dj=0D1|jA|jB, where {|jA}j=0D1 and {|jB}j=0D1 are respective orthonormal bases. Note that I(A;B)|Φ⟩⟨Φ| = 2 log(D) and I(AB)|Φ⟩⟨Φ| = log(D).

The entanglement of formation of a joint state ρAB is defined as97,106

EF(ρAB)infpX(x),|ψABxH(A|X)ρ,
(4)

where the infimum is over all pure state decompositions ρAB=xpX(x)|ψABxψABx|.

A quantum broadcast channel maps a quantum state at the sender system to a quantum state at the receiver systems. Here, we consider a channel with two receivers. Formally, a quantum broadcast channel is a linear, completely positive, trace-preserving map NAB1B2 corresponding to a quantum physical evolution. We assume that the channel is memoryless. That is, if the systems An = (A1, …, An) are sent through n channel uses, then the input state ρAn undergoes the tensor product mapping NAnB1nB2nNAB1B2n. The marginal channel NAB1(1) is defined by

NAB1(1)(ρA)=TrB2NAB1B2(ρA)
(5)

for receiver 1, and similarly, NAB2(2) for receiver 2. One may say that NAB1B2 is an extension of NAB1(1) and NAB2(2). We will consider a broadcast channel with conferencing where receiver 1 can transmit classical information to receiver 2 using a noiseless communication link of capacity C12. We will denote this classical communication channel by CCGG, where G and G′ represent the registers that store the conference message transmitted from receiver 1 and received at receiver 2, respectively. The transmitter, receiver 1, and receiver 2 are often called Alice, Bob 1, and Bob 2.

A quantum broadcast channel has a Kraus representation

NAB1B2(ρA)=jNjρANj
(6)

for some set of operators Nj such that jNjNj=1.

Remark 1.
The classical broadcast channel is the special case where the input and the outputs can be represented by classical random variables X and Y1, Y2, respectively, while the Kraus operators are Nx,y1,y2=PY1Y2|X(y1,y2|x)|y1,y2x| for some probability kernel PY1Y2|X and orthonormal bases {|x⟩}, {|y1, y2⟩}. Therefore, given an input xX, the output state of a classical broadcast channel is
NXY1Y2Cl(|xx|)=(y1,y2)Y1×Y2PY1Y2|X(y1,y2|x)|y1,y2y1,y2|.
(7)

We will also be interested in the following special cases.

Definition 1
(Degraded broadcast channel and Hadamard broadcast channel57). A quantum broadcast channel NAB1B2 is called degraded if there exists a degrading channel PB1B2 such that the marginals satisfy the following relation:
NAB2(2)=PB1B2NAB1(1).
(8)
In this case, we say that N(2) is degraded with respect to N(1). A quantum–classical–quantum degraded channel NAY1B2H is called a Hadamard broadcast channel.

Intuitively, if a broadcast channel is degraded, then the output state of receiver 2 is a noisy version of that of receiver 1. A Hadamard broadcast channel NAY1B2H can be viewed as a measure-and-prepare channel where the marginal channel NAY1(1) acts as a measurement device, while the degrading channel PY1B2 corresponds to state preparation. In this case, the marginal quantum channel NAB2(2) of receiver 2 is said to be entanglement-breaking.107 

Next, we define a broadcast channel with degradable marginals. Every point-to-point quantum channel MAB has an isometric extension UABEM(ρA)=UρAU, also called a Stinespring dilation, where the operator U is an isometry, i.e., UU=1 (see Ref. 108, Sec. VII). System E is often associated with the decoder’s environment or with a malicious eavesdropper in the wiretap channel model.28 The channel M̂AE(ρA)=TrB(UρAU) is called the complementary channel for MAB.

Definition 2

(Degradable marginals). A point-to-point quantum channel MAB is called degradable if there exists an isometric extension such that the complementary channel M̂AE is degraded with respect to MAB. In other words, the channel to the environment is degraded with respect to the channel to the receiver. We say that the quantum broadcast channel NAB1B2 has degradable marginals if both marginals NAB1(1) and NAB2(2) are degradable.

Examples of degradable quantum channels include the erasure channel and the dephasing channel.29 

We consider different broadcast scenarios with cooperation between the decoders, where the transmitted information can be classical or quantum, with entanglement resources or without, and when conferencing between the receivers is available or not.

First, we consider a broadcast channel where receiver 1 and receiver 2 share entanglement resources. We denote their entangled systems by SB1 and SB2, respectively.

Definition 3.

A (2nR0,2nR1,n) classical code for the quantum broadcast channel NAB1B2 with degraded message sets and entangled decoders consists of the following:

  • Two index sets [1:2nR0] and [1:2nR1], corresponding to the common message for both users and the private message of user 1, respectively.

  • An encoding map FM0M1An, where M0 and M1 are classical registers that store the common and private messages, respectively.

  • A pure entangled state ΨSB1,SB2.

  • Two decoding POVMs: {ΛSB1B1nm0,m1} for receiver 1 and {ΓSB2B2nm0} for receiver 2, where the measurement outcome mk is an index in [1:2nRk] for k = 0, 1.

We denote the code by (F,Ψ,Λ,Γ).

The communication scheme is depicted in Fig. 1. The sender Alice has the systems An, and the receivers Bob 1 and Bob 2 have the systems B1n,SB1 and B2n,SB2, respectively. Alice chooses a common message m0[1:2nR0] that is intended for both users and a private message m1[1:2nR1] for Bob 1 and stores them in the classical registers M0 and M1, respectively. She encodes the messages by applying the encoding map FM0M1An, which results in an input state ρAnm0,m1=FM0M1An(m0,m1), and transmits the systems An over n channel uses of NAB1B2. Hence, the output state is

ρB1nB2nm0,m1=NAB1B2n(ρAnm0,m1).
(9)

Bob 1 receives the channel output systems B1n, combines them with his entangled system SB1, and applies the POVM {ΛSB1B1nm0,m1}. Bob 1 then obtains from the measurement outcome an estimate of the message pair (m̂0,m̂1)[1:2nR0]×[1:2nR1]. Similarly, Bob 2 finds an estimate of the common message m̃0[1:2nR0] by performing a measurement using {ΓSB2B2nm0} on the output systems B2n and his entangled system SB2. The conditional probability of error of the code, given that the message pair (m0, m1) was sent, is given by

Pe|m0,m1(n)(F,Ψ,Λ,Γ)=1Tr[(ΛSB1B1nm0,m1ΛSB1B2nm0)ρB1nB2nm0,m1].
(10)
FIG. 1.

Classical coding for a quantum broadcast channel NAB1B2 with shared entanglement between the decoders and degraded message sets. The quantum systems of Alice, Bob 1, and Bob 2 are marked in red, blue, and purple, respectively. Bob 1 and Bob 2 share entanglement resources in the systems SB1 and SB2, respectively. Alice encodes the messages m0 and m1 by applying the encoding map FM0M1An to the respective registers M0 and M1, which store the messages. Then, she transmits the systems An over the broadcast channel. The decoder D1 of Bob 1 receives the channel output systems B1n and estimates the common and private messages by performing a decoding measurement on the systems B1n and SB1, using a POVM {ΛSB1B1nm0,m1}. Similarly, the decoder D2 of Bob 2 estimates the common message by measuring a POVM {ΓSB2B2nm0} on B2n and SB2.

FIG. 1.

Classical coding for a quantum broadcast channel NAB1B2 with shared entanglement between the decoders and degraded message sets. The quantum systems of Alice, Bob 1, and Bob 2 are marked in red, blue, and purple, respectively. Bob 1 and Bob 2 share entanglement resources in the systems SB1 and SB2, respectively. Alice encodes the messages m0 and m1 by applying the encoding map FM0M1An to the respective registers M0 and M1, which store the messages. Then, she transmits the systems An over the broadcast channel. The decoder D1 of Bob 1 receives the channel output systems B1n and estimates the common and private messages by performing a decoding measurement on the systems B1n and SB1, using a POVM {ΛSB1B1nm0,m1}. Similarly, the decoder D2 of Bob 2 estimates the common message by measuring a POVM {ΓSB2B2nm0} on B2n and SB2.

Close modal

A (2nR0,2nR1,n,ε) classical code satisfies Pe|m0,m1(n)(F,Ψ,Λ,Γ)ε for all (m0,m1)[1:2nR0]×[1:2nR1]. A rate pair (R0, R1) is called achievable with entangled decoders if for every ɛ > 0 and sufficiently large n, there exists a (2nR0,2nR1,n,ε) code. The classical capacity region is defined as the set of achievable pairs (R0, R1) with entangled decoders.

One may also consider the broadcast channel with independent messages, i.e., when the common message m0 is replaced by a private message m2 that is intended for Bob 2, in which case Bob 1 is not required to decode this message. In general, the capacity region with independent messages can be larger than with degraded message sets.

FIG. 2.

Classical coding for a quantum broadcast channel NAB1B2 with conferencing and degraded message sets. The quantum systems of Alice, Bob 1, and Bob 2 are marked in red, blue, and purple, respectively. There is a conferencing link between the decoders, which allows Bob 1 to send a conferencing message to Bob 2 at a rate C12. Alice encodes the messages m0 and m1 by applying the encoding map FM0M1An to the respective registers M0 and M1, which store the messages. Then, she transmits the systems An over the broadcast channel. The decoder D1 of Bob 1 receives the channel output systems B1n and performs a measurement using the POVM {ΛB1nm0,m1,g}. Bob 1 then obtains from the measurement outcome an estimate of the message pair and a conference message. Next, Bob 1 sends the conference message g to Bob 2. Given the conference message g, the decoder D2 of Bob 2 chooses a measurement POVM {ΓB2n|gm0} to perform on the channel output systems B2n, producing an estimate of the common message as the measurement outcome.

FIG. 2.

Classical coding for a quantum broadcast channel NAB1B2 with conferencing and degraded message sets. The quantum systems of Alice, Bob 1, and Bob 2 are marked in red, blue, and purple, respectively. There is a conferencing link between the decoders, which allows Bob 1 to send a conferencing message to Bob 2 at a rate C12. Alice encodes the messages m0 and m1 by applying the encoding map FM0M1An to the respective registers M0 and M1, which store the messages. Then, she transmits the systems An over the broadcast channel. The decoder D1 of Bob 1 receives the channel output systems B1n and performs a measurement using the POVM {ΛB1nm0,m1,g}. Bob 1 then obtains from the measurement outcome an estimate of the message pair and a conference message. Next, Bob 1 sends the conference message g to Bob 2. Given the conference message g, the decoder D2 of Bob 2 chooses a measurement POVM {ΓB2n|gm0} to perform on the channel output systems B2n, producing an estimate of the common message as the measurement outcome.

Close modal

Another form of cooperation between the decoders involves conferencing. We consider a broadcast channel where receiver 1 can transmit information to receiver 2 using a classical conferencing link of capacity C12.

Definition 4.

A (2nR0,2nR1,n) classical code for the quantum broadcast channel NAB1B2 with degraded message sets and conferencing consists of the following:

  • Three index sets [1:2nR0], [1:2nR1], and [1:2nC12], corresponding to the common message for both users, the private message of user 1, and the conference message, respectively.

  • An encoding map FM0M1An, where M0 and M1 are classical registers that store the common and private messages, respectively.

  • A decoding POVM {ΛB1nm0,m1,g} for receiver 1, where the measurement outcome is a triplet of indices (m0, m1, g) in [1:2nR0]×[1:2nR1]×[1:2C12].

  • A collection of decoding POVMs {ΓB2n|gm0}, g[1:2C12], for receiver 2, where the measurement outcome is an index in [1:2nR0].

We denote the code by (F,Λ,Γ).

The communication scheme is depicted in Fig. 2. The sender Alice has the systems An, and the receivers Bob 1 and Bob 2 have the systems B1n and B2n, respectively. Alice chooses a common message m0[1:2nR0] and a private message m1[1:2nR1] for Bob 1 and stores them in the classical registers M0 and M1, respectively. She encodes the messages by applying the encoding map FM0M1An, which results in an input state

ρAnm0,m1=FM0M1An(m0,m1),
(11)

and transmits the systems An over n channel uses of NAB1B2. Hence, the output state is

ρB1nB2nm0,m1=NAB1B2n(ρAnm0,m1).
(12)

Bob 1 receives the channel output systems B1n and applies the POVM {ΛB1nm0,m1,g}. Bob 1 then obtains from the measurement outcome an estimate of the message pair (m̂0,m̂1)[1:2nR0]×[1:2nR1] and a conference message g[1:2nC12]. Next, Bob 1 sends the conference message g to Bob 2. Given the conference message g, Bob 2 chooses a POVM {ΓB2n|gm0} to perform on the channel output systems B2n, producing an estimate of the common message m̃0[1:2nR0] as the measurement outcome. The conditional probability of error of the code, given that the message pair (m0, m1) was sent, is given by

Pe|m0,m1(n)(F,Λ,Γ)=1g=12nC12Tr[(ΛB1nm0,m1,gΛB2n|gm0)ρB1nB2nm0,m1].
(13)

A (2nR0,2nR1,n,ε) classical code satisfies Pe|m0,m1(n)(F,Λ,Γ)ε for all (m0,m1)[1:2nR0]×[1:2nR1]. A rate pair (R0, R1) is called achievable with conferencing if for every ɛ > 0 and sufficiently large n, there exists a (2nR0,2nR1,n,ε) code. The classical capacity region RCl(N) is defined as the set of achievable pairs (R0, R1) with conferencing.

Remark 2.

The setting above is the quantum version of the classical broadcast channel with cooperating decoders, by Dabora and Servetto.94 The main motivation involves a sensor network, where an external transmitter (B1) wants to download data, such as network configuration into the network (see Ref. 94, Subsection I A). The model can be viewed as a combination of the broadcast channel and the primitive relay channel.96 In this context, the term “conferencing” indicates cooperation between two different users, whereas a relay channel109 consists of a single user and a helper (see Definition 6).

Remark 3.

The conferencing link can be described as a bit-pipe,95 i.e., a noiseless link, from receiver 1 to receiver 2, through which information is transmitted at a constant rate C12.

Next, we consider the case where the messages are quantum. Furthermore, given entanglement between the decoders, the classical conference link can be used to transfer quantum information from receiver 1 to receiver 2 using the teleportation protocol. This is thus equivalent to a conferencing link with quantum capacity CQ,12=12C12. In other words, given entanglement resources, the conferencing bit-pipe of capacity C12 can be transformed into a conferencing qubit-pipe of capacity CQ,12=12C12 (see Remark 3). Note that due to the no-cloning theorem, the encoder cannot transmit a quantum message to both receivers; thus, we consider two private messages.

Definition 5.

A (2nQ1,2nQ2,n) quantum code for the quantum broadcast channel NAB1B2 with independent messages and conferencing consists of the following:

  • A quantum message state ρM1M2, where M1 and M2 are quantum systems that store the private messages of user 1 and user 2, respectively. The dimension of each system is given by |HMk|=2nQk for k = 1, 2.

  • An encoding map FM1M2An.

  • A decoding map DB1nM̂1G1 for receiver 1, where G is a quantum register of dimension 2nCQ,12 that stores the conference message from receiver 1 to receiver 2.

  • A decoding map DGB2nM̃02 for receiver 2.

We denote the code by (F,D1,D2).

The communication scheme is depicted in Fig. 3. The sender Alice has the systems M1, M2, and An; Bob 1 has the systems B1n, G, and M̂1; and Bob 2 has the systems B2n, G′, and M̂2. Alice encodes the quantum state of the message systems M1 and M2 by applying the encoding map FM1M2An, which results in the input state

ρAn=FM1M2An(ρM1M2),
(14)

and transmits the systems An over n channel uses of NAB1B2. Hence, the output state is

ρB1nB2n=NAB1B2n(ρAn).
(15)

Bob 1 receives the channel output systems B1n and applies the decoding map DB1nM̂1G1, which results in

ρM̂1GB2n=DB1nM̂1G1(ρB1nB2n).
(16)

The reduced state of M̂1 is Bob 1’s estimate of the original state of his private message system M1. Next, Bob 1 sends the conference message from G to G′ using the noiseless conference link idGG; hence, ρM̂1GB2n=idGG(ρM̂1GB2n)=ρM̂1GB2n. Bob 2 receives the channel output systems B2n and the conference message in G′ and applies the decoding map DGB2nM̂22 such that M̂2 is his estimate of his private message. The estimated state is then given by

ρM̂1M̂2=DGB2nM̂22(ρM̂1GB2n),
(17)

and the estimation error is given by

e(n)(E,D1,D2,ρM1,M2)=12ρM1M2ρM̂1M̂2.
(18)

A (2nQ1,2nQ2,n,ε) quantum code satisfies e(n)(F,D1,D2,ρM0M1)ε for all ρM0,M1. A rate pair (Q1, Q2) is called achievable with conferencing if for every ɛ > 0 and sufficiently large n, there exists a (2nQ1,2nQ2,n,ε) code. The quantum capacity region RQ(N) is defined as the set of achievable pairs (Q1, Q2) with conferencing.

The setting of a broadcast channel with conferencing is closely related to that of a primitive relay channel.96 

Definition 6.

A primitive relay channel NAB1B2relay is a broadcast channel with conferencing when user 1 does not send information, i.e., Q1 = 0. Alice, Bob 1, and Bob 2 are then called the source, relay, and destination receiver, respectively. A quantum rate Q2 > 0 is called achievable for the primitive relay channel if (0, Q2) is achievable for the broadcast channel with conferencing. The quantum capacity CQ(Nrelay) is defined as the supremum of achievable rates for the primitive relay channel.

Bob 1 is called a relay in this setting because his only task is to help the transmission of information to Bob 2 (see Fig. 4). The channel is called “primitive” since it is a simplified version of the (non-primitive) relay channel109 where information is received and encoded at the relay in a strictly causal manner.

FIG. 3.

Quantum coding for a broadcast channel NAB1B2 with conferencing and private messages. The quantum systems of Alice, Bob 1, and Bob 2 are marked in red, blue, and purple, respectively. Given entanglement between the decoders, the classical conference link can be used to transfer quantum information from Bob 1 to Bob 2 using the teleportation protocol. This is thus equivalent to a conferencing link with quantum capacity CQ,12=12C12. Alice encodes the quantum state of the message systems M1 and M2 by applying the encoding map FM1M2An. Then, she transmits the systems An over the broadcast channel. Bob 1 receives the channel output systems B1n and applies the decoding map DB1nM̂1G1 such that the state of M̂1 is his estimate of his private message. Next, Bob 1 sends the state of the conference system G to Bob 2 through a noiseless conference link idGG. Bob 2 receives the channel output systems B2n and the conference message in G′ and applies the decoding map DGB2nM̂22 such that the state of M̂2 is his estimate of his private message.

FIG. 3.

Quantum coding for a broadcast channel NAB1B2 with conferencing and private messages. The quantum systems of Alice, Bob 1, and Bob 2 are marked in red, blue, and purple, respectively. Given entanglement between the decoders, the classical conference link can be used to transfer quantum information from Bob 1 to Bob 2 using the teleportation protocol. This is thus equivalent to a conferencing link with quantum capacity CQ,12=12C12. Alice encodes the quantum state of the message systems M1 and M2 by applying the encoding map FM1M2An. Then, she transmits the systems An over the broadcast channel. Bob 1 receives the channel output systems B1n and applies the decoding map DB1nM̂1G1 such that the state of M̂1 is his estimate of his private message. Next, Bob 1 sends the state of the conference system G to Bob 2 through a noiseless conference link idGG. Bob 2 receives the channel output systems B2n and the conference message in G′ and applies the decoding map DGB2nM̂22 such that the state of M̂2 is his estimate of his private message.

Close modal
FIG. 4.

A simplistic view of the primitive relay channel. Alice, Bob 1, and Bob 2 play the roles of the sender, relay, and destination receiver, respectively.

FIG. 4.

A simplistic view of the primitive relay channel. Alice, Bob 1, and Bob 2 play the roles of the sender, relay, and destination receiver, respectively.

Close modal

Remark 4.

A standard, i.e., non-primitive, relay channel88–90 is specified by a linear, completely positive, trace-preserving map LAB1A1B2, where the sender transmits the systems An, the relay receives B1n and transmits A1n, and the destination receiver receives B2n. The relay encoder applies a strictly causal map, as he can only use the systems B1i1 at time i. That is, at time i, the relay transmits A1,i such that ρA1i=TB1i1A1i(i)(ρB1i1).

Remark 5.
Quantum communication is also referred to as entanglement transmission and can be extended to strong subspace transmission.30,110 In this task, Alice and Charlie share a pure entangled state |ψM1M2C, with a Schmidt decomposition
|ψM1M2C=xXpX(x)|ψM1M2x|ψCx.
(19)
Given a (2nQ1,2nQ2,n,ε) code, Alice can send each state |ψM1M2x reliably, i.e., with
ψM1M2xψM̂1M̂2x1ε for all xX.
(20)
Hence, the superposition state |ψM1M2C can also be recovered up to an error of ɛ. In particular, if Alice and Charlie share a maximally entangled state |ΦM1M2C, then at the end of the communication protocol, Bob 1, Bob 2, and Charlie share a state |ΦM̂1M̂2C up to an ɛ-error.

Remark 6.
Quantum communication can also be used for the purpose of entanglement generation.28,30 We note that by the monogamy property of quantum entanglement,78 Alice cannot generate a maximally entangled state with both Bob 1 and Bob 2 simultaneously. Indeed, suppose that Alice has a third system Ā that is entangled with M1 and M2 in a state |ψĀM1M2. Then, by strong sub-additivity (Ref. 35, Coro. 11.9.1),
H(ĀM1)ψ+H(ĀM2)ψH(Ā)ψ+H(ĀM1M2)ψH(Ā)ψ.
(21)
Hence, Ā cannot be maximally entangled with both M1 and M2; otherwise, we would have 0 ≥ 1. Nevertheless, different forms of entanglement can be generated. In particular, Alice can generate a GHZ state with Bob 1 and Bob 2 (Ref. 54, Sec. IV), using |ψĀM1M2=1dx=1d|x|x|x. Alternatively, she can generate two entangled pairs. Suppose that Alice has another pair of system Ā1,Ā2 in the state
|ψĀ1M1Ā2M2=|ΦĀ1M1|ΦĀ2M2.
(22)
Then, at the end of the quantum communication protocol, Alice shares the entangled states |ΦĀ1M̂1 with Bob 1 and |ΦĀ2M̂2 with Bob 2, by the same considerations as in entanglement transmission (see Remark 5).

Remark 7.

In the absence of entanglement resources between the decoders, quantum communication over the broadcast channel can generate such entanglement by choosing the quantum message state to be |ΦM1M2.

Now, we give our results on the quantum broadcast channel with a classical conferencing link between the decoders when Bob 1 and Bob 2 do not share entanglement resources (see Fig. 2). Define the rate region

RCl(N)(R0,R1):R0I(X0;B2)ρ+C12R1I(X1;B1|X0)R0+R1I(X0,X1;B1),
(23)

where the union is over the set of all distributions pX0,X1(x0,x1) and state collection {θAx0,x1}, with

ρX0X1B=x0X0x1X1pX0,X1(x0,x1)|x0x0||x1x1|NAB1B2(θAx0,x1).
(24)

Before we state the capacity theorem, we give the following lemma, which provides cardinality bounds for the auxiliary random variables X0 and X1. In principle, one can use those cardinality bounds to evaluate the region RCl(N) numerically.

Lemma 1.

The union in (23) is exhausted by auxiliary random variables X0 and X1 of cardinality |X0||HA|2+2 and |X1|(|HA|2+2)|HA|2+1.

The Proof of Lemma 1 is given in  Appendix A. The classical capacity region is determined in the theorem below.

Theorem 2.
The classical capacity region of the quantum broadcast channel NAB1B2 with conferencing and degraded message sets is given by
RCl(N)=limkk=1RCl(Nk).
(25)
Furthermore, for a Hadamard broadcast channel,
RCl(N)=RCl(N).
(26)

The Proof of Theorem 2 is given in  Appendix B.

Note that in the special case of a conference link with zero capacity, i.e., C12=0, we recover the result by Yard et al.74 on the broadcast channel without conferencing.

Next, we consider the case where the messages are quantum. Furthermore, given entanglement between the decoders, the classical conference link can be used to transfer quantum information from receiver 1 to receiver 2 using the teleportation protocol. This is thus equivalent to a conferencing link with quantum capacity CQ,12=12C12. As noted in Subsection III C, Alice cannot transmit a quantum message to both receivers due to the no-cloning theorem. Thereby, we consider a broadcast channel with two private quantum messages, as illustrated in Fig. 3.

This setting is intimately related to quantum repeaters as Bob 1 can be viewed as a repeater for the transmission of quantum information to Bob 2. In particular, Alice can use the quantum message stored in M1 to generate entanglement and prepare a maximally entangled pair |ΦAB1 between the transmitter and the repeater, namely, Alice and Bob 1. Given entanglement between the decoders, we also have a maximally entangled pair |ΦB1B2 between the repeater and the receiver, i.e., Bob 1 and Bob 2. Then, the repeater B1 can swap his entanglement by using the classical conferencing link to teleport the state of B1′ onto B2, thus swapping the entanglement such that A and B2 are now entangled. This requires that the conferencing capacity is at least twice the information rate, i.e., C122Q2. We will conclude this section with the resulting observations for the quantum repeater.

We establish an achievable rate region for the broadcast channel with quantum conferencing.

Theorem 3.
A rate pair (Q1, Q2) is achievable for transmission of quantum information over the broadcast channel NAB1B2 with private messages and quantum conferencing if
Q1I(A1B1)ρ,Q2I(A2B2)ρ+CQ,12,Q1+Q2I(A1B1)ρ+I(A2B2)ρ
(27)
for some input state ρA1A2A, where ρA1A2B1B2=NAB1B2(ρA1A2A).

The achievability proof is given below. The rate region in Theorem 3 reflects a greedy approach, where using the conferencing link to increase the information rate of User 2 comes directly at the expense of user 1. That is, if Q2=I(A2B2)ρ+Δ, then Q1I(A1B1)ρΔ.

Remark 8.

For the transmission of classical information, we have seen that the optimal performance is achieved using superposition coding, where receiver 1 can recover the message of user 2 without necessarily “losing” rate. In particular, by Theorem 2, a classical rate pair (R1,R2)=(I(X1;B1|X0)ρ,I(X0;B2)ρ+C12) is achievable when I(X0;B2)ρ+C12<I(X0;B1)ρ because then R1 + R2 < I(X0X1; B1) by the chain rule. However, in the quantum case, the capacity-achieving coding scheme in Ref. 58 does not involve superposition. Without conferencing, it is impossible for receiver 1 to decode the message of user 2 by the no-cloning theorem. Nevertheless, the setting of conferencing decoders imposes a chronological order: First Bob 1 receives and processes the channel output B1n, then Bob 1 sends the conference message to Bob 2, and at last, Bob 2 receives access to the channel output B2n and the conference message. Therefore, Bob 1 can recover the state of M2 (or part of it) and send it to Bob 2 using the conference link. However, due to the no-cloning theorem, Bob 2 will be able to decode the state of M2 only if the state was destroyed in Bob 1’s location during conferencing.

Achievability Proof.
Consider the quantum broadcast channel NAB1B2 with a quantum conference link of capacity CQ,12. The proof is a straightforward consequence of the results by Dupuis et al.58 Fix an input state ρA1A2A. Based on Ref. 58 (Ref. 59, Theorem 5.4), for every ɛ > 0 and sufficiently large n, there exists a (2nQ1,2nQ2,n,ε) quantum code for the broadcast channel NAB1B2without conferencing if
Q1=I(A1B1)ρδ,Q2=I(A2B2)ρδ,
(28)
where δ > 0 is arbitrarily small. The rate pair (Q1′, Q2′) is thus achievable in our setting as well since the decoders can avoid conferencing by choosing an idle conference message state |0⟩⟨0| regardless of the output state.
Now, we consider two cases: CQ,12>I(A1B1)ρ and CQ,12I(A1B1)ρ. If CQ,12>I(A1B1)ρ, then the second inequality in (27) is inactive since I(A1B1)ρ+I(A2B2)ρ<I(A2B2)ρ+CQ,12; hence, we are done. Otherwise, if CQ,12I(A1B1)ρ, then Alice can send the state of nCQ,12 qubits to Bob 2 indirectly through the conference link. This can be performed as follows. First, use the code above to transmit n(Q1+CQ,12) qubits to Bob 1 and nQ2′ to Bob 2, with Q1=Q1CQ,12, and then, let Bob 1 send the state of the nCQ,12 qubits to Bob 2. Overall, this coding scheme achieves the following rate pair:
Q1=I(A1B1)ρCQ,12δ,Q2=I(A2B2)ρ+CQ,12δ.
(29)
Note that in the process of sending the conference message, Bob 1 may destroy the state of his own nCQ,12 qubits, and thus, this cannot be regarded as a common message. Observing that (Q1′, Q2′) and (Q1, Q2) are the corner points of the region in (27), the proof follows by time sharing.□

Note that for C12=0, the achievable region coincides with the capacity region of the broadcast channel without conferencing.58,59

Next, we give a multi-letter outer bound.

Theorem 4.
If a rate pair (Q1, Q2) is achievable for the transmission of quantum information over the broadcast channel NAB1B2 with private messages and quantum conferencing, then it must satisfy the following inequalities:
Q11nI(A1B1n)ρ,Q21nI(A2TB2n)ρ+CQ,12,Q1+Q21nI(A1B1n)ρ+1nI(A2B1nB2n)ρ
(30)
for some input state ρTA1A2An, where ρTA1A2B1nB2n=NAB1B2n(ρTA1A2An).

Notice that here we added the auxiliary system T in the second inequality and added B1n in the last term of the third inequality [cf. (27) and (30)].

Proof of Outer Bound.
Suppose that Alice is trying to generate entanglement with Bob 1 and Bob 2. An upper bound on the rate at which Alice and Bob, k, for k = 1, 2, can generate entanglement also serves as an upper bound on the rate at which they can communicate with qubits since a noiseless quantum channel can be used to generate entanglement by sending one part of an entangled pair. In this task, Alice locally prepares two maximally entangled pairs,
|ΦM1M1|ΦM2M2=12n(Q1+Q2)m1=12nQ1m2=12nQ2|m1M1|m1M1|m2M2|m2M2.
(31)
Then, she applies an encoding channel FM1M2An to the quantum systems M1M2, resulting in
ρM1M2AnFM1M2An(|ΦM1M1|ΦM2M2).
(32)
After Alice sends systems An through the channel, Bob 1 receives the systems B1n in the state
ρM1M2B1nB2nNAB1B2n(ρM1M2An)
(33)
and performs a decoding channel DB1nM̂1G1. Hence,
ρM1M2M̂1GB2nDB1nM̂1G1(ρM1M2B1nB2n),
(34)
where the state of M̂1 is Bob 1’s estimate of his quantum message and the state of G is the conference message that is sent through the conference link to Bob 2. Having received B2n and G′ such that ρM1M2M̂1G=ρM1M2M̂1G, Bob 2 uses a decoding channel DGB2nM̂22, producing
ρM1M2M̂1M̂2DGB2nM̂22(ρM1M2M̂1GB2n).
(35)
Consider a sequence of codes (Fn,Dn1,Dn2) for entanglement generation such that
12ρM1M̂1M2M̂2ΦM1M1ΦM2M21αn,
(36)
where αn tends to zero as n. By the Alicki–Fannes–Winter inequality111,112 (Ref. 35, Theorem 11.10.3), (36) implies that |H(Mk|M̂k)ρH(Mk|Mk)Φ|nεn, or equivalently,
|I(MkM̂k)ρI(MkMk)Φ|nεn
(37)
for k = 1, 2, where ɛn tends to zero as n. Observe that I(MkMk)Φ=H(Mk)ΦH(MkMk)Φ=nQk0=nQk. Thus,
nQ1=I(M1M1)ΦI(M1M̂1)ρ+nεnI(M1B1n)ρ+nεn,
(38)
where the last inequality is due to (34) and the data processing inequality for the coherent information (Ref. 35, Theorem 11.9.3).
Similarly, for user 2, it follows from (35) and the data processing inequality that
nQ2I(M2GB2n)ρ+nεn
(39)
=H(M2G|B2n)+H(G|B2n)+nεn=I(M2GB2n)+H(G|B2n)+nεnI(M2GB2n)+nCQ,12+nεn.
(40)
By (38) and (39), we also have that
n(Q1+Q2)I(M1B1n)ρ+I(M2GB2n)ρ+2nεnI(M1B1n)ρ+I(M2B1nB2n)ρ+2nεn,
(41)
where the last follows from (34) and the data processing inequality. The proof follows from (38), (40), and (41) by defining quantum systems A1, A2 such that for some isometries UM1A1, VM2A2, and WG′→T, we have ρA1B1n=UM1A1nρM1B1nUM1A1n and ρA2TB2n=(VM2A2WGT)ρM2GB2n(VM2A2WGT). This completes the proof for the regularized outer bound.□

Consider the primitive relay channel NAB1B2relay, where Bob 1 acts as a relay that helps for the transmission from Alice to Bob 2, but it is not required to decode information (i.e., Q1 = 0). We use our previous results to obtain lower and upper bounds on the capacity of the primitive relay channel and conclude this section with the resulting observations for the quantum repeater.

Theorem 5.

The quantum capacity of the primitive relay channel NAB1B2relay has the following bounds:

  1. Cutset upper bound
    CQ(Nrelay)limnsupρATAn1nminI(ATB2n)ρ+CQ,12,I(AB1nB2n)ρ,
    (42)
    with ρATB1nB2n=NAB1B2n(ρATAn).
  2. Decode-forward lower bound
    CQ(Nrelay)max|ϕA1A2AI(A2B2)ρ+minI(A1B1)ρ,CQ,12,
    (43)
    with ρA1A2B1B2=NAB1B2(ϕA1A2A).
  3. Entanglement-formation lower bound
    CQ(Nrelay)max|ϕA1A2A,FB1B̂1:EF(ρB̂1AB2E)CQ,12I(A2B̂1B2)ϕ,
    (44)
    with |ϕAB1B2E=UAB1B2EN|ϕAA, ρAB̂1B2E=FB1B̂1(ϕAB1B2E), where EF(ρB̂1AB2E) is the entanglement of formation with respect to the bipartition B̂1|AB2E.

The proof of the cutset upper bound follows the same considerations as in Subsection V B, and it is thus omitted [see (40) and (41)]. The decode-forward lower bound in Theorem 5 is obtained as an immediate consequence of Theorem 3, taking Q1 = 0. The rate in (44) can be achieved by using the conferencing link to simulate the channel FB1B̂1. Based on the results of Berta et al. (Ref. 97, Theorem 12), this can be achieved if the capacity of the conference link is higher that the entanglement of formation with respect to the bipartition B̂1|AB2E, i.e., CQ,12EF(ρB̂1AB2E). Then, Bob 2 can decode ρB̂1nB2n, which is ɛn-close in trace distance to ρB1̂nB2nN̂AB1̂B2n(ρAn), where ɛn tends to zero as n, with N̂AB1̂B2FB1B̂1NAB1B2.

Remark 9.

Recall from the beginning of Sec. V that we view Alice, Bob 1, and Bob 2 as the sender, repeater, and destination receiver. In other words, the repeater is the quantum version of a relay. As we also consider the direct transmission to the destination receiver (Bob 2), our results show the trade-off between repeaterless communication and relaying information through the repeater. In particular, in the decode-forward lower bound (43) (see part 2 of Theorem 5), the term I(A2B2)ρ corresponds to repeaterless communication, while minI(A1B1)ρ,CQ,12 corresponds to quantum transmission via the repeater.

Remark 10.

Intuitively, the decode-forward lower bound has the interpretation of a bottleneck flow. Specifically, as mentioned in the previous remark, the term minI(A1B1)ρ,CQ,12 in the decode-forward lower bound (43) is associated with the information rate via the repeater. Due to the serial connection between the sender–repeater link AB1 with the repeater–receiver link B1B2, the throughput is dictated by the smaller rate (see Fig. 4). A similar behavior was observed by Smolin et al.92 for a quantum channel with environment assistance of a classical relay (see Ref. 92, Theorem 8, and Refs. 91 and 93 as well).

In this section, we consider a broadcast channel where the decoders share entanglement resources between them (see Fig. 1). Given the recent results by Leditzky et al.44 on the multiple access channel, it may be tempting to think that the dual property holds for the broadcast channel and that entanglement between decoders can increase achievable rates of classical communication. We observe that this is not the case. Nevertheless, given a quantum conferencing link of capacity CQ,12, receiver 1 can send conferencing messages to receiver 2 at a rate 2CQ,12 using the super-dense coding protocol. Further details are given below.

First, consider a quantum broadcast channel NAB1B2 without conferencing, given entanglement resources shared between the decoders, as illustrated in Fig. 1. We show that the classical capacity region is the same as that without the entanglement resources. Indeed, suppose that Alice chooses m0 and m1 uniformly at random and prepares an input state ρAnm0,m1. After Alice sends the systems An through the channel, the output state is ρB1nB2nΨSB1SB2, where SB1 and SB2 are the entangled systems of Bob 1 and Bob 2, respectively, and ρB1nB2n=12n(R0+R1)m0,m1NAnB1nB2n(ρAnm0,m1). Then, Bob 1 performs a decoding POVM ΛB1nSB1m0,m1, and Bob 2 performs a decoding POVM ΛB2nSB2m0. Consider a sequence of codes (Fn,Λn,Γn) such that the average probability of error tends to zero; hence, the error probabilities PrM̂0M0, Pr(M̂0,M̂1)(M0,M1), and PrM̂1M1|M0 are bounded by some αn, which tends to zero as n. By Fano’s inequality, it follows that

H(M0|M̃0)nεn,
(45)
H(M0,M1|M̂0,M̂1)nεn,
(46)
H(M1|M̂1,M0)nεn,
(47)

where ɛn, ɛn′, ɛn tend to zero as n. Hence,

nR0=H(M0)=I(M0;M̃0)ρ+H(M0|M̃0)I(M0;M̃0)ρ+nεnI(M0;B2nSB2)ρ+nεn=I(M0;B2n)ρ,
(48)

where the second inequality follows from the Holevo bound (see Ref. 32, Theorem 12.1) and the last inequality holds as SB1SB2 are in a product state with M0,M1,B1n,B2n. Similarly,

n(R0+R1)I(M0,M1;B1n)ρ+nεn,
(49)
nR1I(M1;B1n|M0)ρ+nεn
(50)

as without entanglement resources.

On the other hand, given a quantum conferencing link of capacity CQ,12, the classical capacity region with entanglement between the decoders is given by the regularization of the region in (23), taking C12=2CQ,12. Achievability follows by using the super-dense coding protocol37 to send classical conferencing messages from Bob 1 to Bob 2. As for the converse proof, consider a coding scheme where Bob 1 performs a decoding POVM ΛB1nSB1m0,m1,g, sends g to Bob 2 using conferencing, and Bob 2 chooses a POVM ΓB2nSB2|gm0 accordingly. By using the same considerations as in the derivation mentioned above,

nR0I(M0;M̃0)ρ+nεnI(M0;B2nSB2G)ρ+nεn=I(M0;B2nSB2)ρ+I(M0;G|B2nSB2)ρ+nεnI(M0;B2n)ρ+2nCQ,12+nεn,
(51)

where the second inequality follows from the Holevo bound (see Ref. 32, Theorem 12.1), the equality is due to the chain rule for the quantum mutual information, and the last inequality holds because I(M0;B2nSB2)ρ=I(M0;B2n)ρ as SB1SB2 are in a product state with M0,M1,B1n,B2n and since I(M0;G|B2nSB2)ρ2H(G)ρ2nCQ,12 (see Ref. 35, Sec. 11.6). As bounds (49) and (50) hold by similar arguments, the proof follows.

We conclude that entanglement between the decoders cannot enlarge the capacity region of the classical broadcast channel without conferencing. By similar considerations, the same property holds for a broadcast channel with classical conferencing as well. Yet, entanglement resources between the decoders double the conferencing rate when a quantum conferencing link is available. Further observations and a comparison with the multiple access channel are provided in in Subsection VII C.

We have considered the quantum broadcast channel NAB1B2 in different settings of cooperation between the decoders. Using those settings, we provided an information-theoretic framework for quantum repeaters.

The first form of cooperation that we considered is classical conferencing, where receiver 1 can send classical messages to receiver 2. We provided a regularized characterization for the classical capacity region of the quantum broadcast channel with classical conferencing and a single-letter formula for Hadamard broadcast channels. Next, we considered quantum conferencing, where receiver 1 can teleport a quantum state to receiver 2. We developed inner and outer bounds on the quantum capacity region with quantum conferencing, characterizing the trade-off between the communication rates Q1 and Q2 to receiver 1 and receiver 2, respectively, and the conferencing capacity CQ,12.

Quantum communication is also referred to as entanglement transmission and can be extended to strong subspace transmission.30,110 In this task, Alice and Charlie share a pure entangled state |ψM1M2C, and at the end of the communication protocol, Bob 1, Bob 2, and Charlie share a state |ΦM̂1M̂2C up to an ɛ-error. In the absence of entanglement resources between the decoders, quantum communication over the broadcast channel can generate such entanglement by choosing the quantum message state to be |ΦM1M2.

Quantum communication can also be used for the purpose of entanglement generation.28,30 We note that by the monogamy property of quantum entanglement,78 Alice cannot generate a maximally entangled state with both Bob 1 and Bob 2 simultaneously. Nevertheless, different forms of entanglement can be generated. In particular, Alice can generate a GHZ state with Bob 1 and Bob 2 (Ref. 54, Sec. IV), using |ψĀM1M2=1dx=1d|x|x|x. Alternatively, she can generate two entangled pairs. Suppose that Alice has another pair of system Ā1,Ā2 in the state

|ψĀ1M1Ā2M2=|ΦĀ1M1|ΦĀ2M2.
(52)

Then, at the end of the quantum communication protocol, Alice shares the entangled states |ΦĀ1M̂1 with Bob 1 and |ΦĀ2M̂2 with Bob 2.

The case where receiver 1 is not required to recover information, i.e., Q1 = 0, and its sole purpose is to help the transmission to receiver 2, reduces to the quantum primitive relay channel, for which the decode-forward lower bound and cutset upper bound follow as a consequence. In addition, we established an entanglement-formation lower bound, where a virtual channel is simulated through the conference link, following the results of Berta et al.97 on quantum channel simulation.

The quantum conferencing setting is intimately related to quantum repeaters as the sender, receiver 1, and receiver 2 can be viewed as the transmitter, the repeater, and the destination receiver, respectively, in the repeater model. In particular, the sender can employ quantum communication to receiver 1 (the repeater) in order to prepare a maximally entangled pair |ΦAB1, which consists of nQ1 entangled bits (ebits).

Given entanglement between the receivers, we also have a maximally entangled pair |ΦB1B2, which consists of nCQ,12=12nC12 ebits, shared between the repeater and the destination receiver, where C12 is the classical conferencing link. Then, the repeater can swap his entanglement by using the classical conferencing link to teleport the state of B1′ onto B2, thus swapping the entanglement such that A and B2 are now entangled. This requires the classical conferencing rate to be at least twice the information transmission rate to B2.

Hence, our results provide an information-theoretic analysis, characterizing the achievable rates of ebits that can be generated in each stage. As we have also considered the direct transmission to the destination receiver, our results reflect the trade-off between repeaterless communication and relaying qubits using the repeater as well (see Remark 9).

Intuitively, the communication via the repeater gives rise to a bottleneck effect. That is, due to the serial connection between the sender–repeater link AB1 and the repeater–receiver link B1B2, the throughput is dictated by the smaller rate (see Fig. 4). Indeed, the term in the decode-forward formula (43) that is associated with communication via the repeater involves a minimum between the coherent information I(A1B1)ρ and the conferencing link capacity CQ,12 (see Remark 10).

The duality between the broadcast channel and the multiple access channel (BC-MAC duality) is a well-known property in the study of Gaussian multiple-input multiple-output (MIMO) channels.103–105 Based on the reciprocity property,113 the capacity remains unchanged when the role of the transmitters and receivers is interchanged103–105 (see also Ref. 114, Lemma 9.2). In the scalar case, this means that the capacity region of the Gaussian broadcast channel,

Y1=h1X+Z1,
(53)
Y2=h2X+Z2,
(54)

subject to a power constraint 1ni=1nxi2P, is exactly the same as the capacity region of the Gaussian multiple access channel,

Y=h1X1+h2X2+Z,
(55)

subject to a total-power constraint 1ni=1n(x1,i2+x2,i2)P, with normalized Gaussian noises Z, Z1, and Z2N(0,1). As the multiple access channel and broadcast channel are useful models for uplink transmission and downlink transmission in cellular communication, this behavior is also referred to as uplink–downlink duality. Duality properties have also been shown for beamforming strategies.115,116

Our result demonstrates the limitations of the duality between the broadcast channel and the multiple access channel. Leditzky et al.44 considered a classical multiple access channel PY|X1X2, with two senders and a single receiver, when the encoders share entanglement resources, as illustrated in Fig. 5. The MAC in Ref. 44 is defined in terms of a pseudo-telepathy game,45 for which quantum strategies guarantee a certain win and outperform classical strategies. They showed achievability of a sum-rate R1 + R2 that exceeds the sum-rate capacity of this channel without entanglement. In principle, one could mirror the model (cf. Figs. 1 and 5) and consider a broadcast channel PY1Y2|X, where X′ ≡ Y, Y1X1, and Y2′ ≡ X2, according to the a posteriori probability distribution PX1X2|Y, given some input distribution pX1,X2. Specifically, the derivation in Ref. 44 is for the magic square game,45 which is highly symmetric. Hence, it can be shown that the sum-rate capacity of the multiple access channel PY|X1X2 and the broadcast channel PY1Y2|X, without entanglement resources, are the same. Nevertheless, we cannot use the entanglement cooperation in the same manner, as the decoding strategy does not affect the channel. Intuitively, encoding using quantum game strategies for the multiple access channel inserts quantum correlations into the channel. On the other hand, in the broadcast setting, the entangled resources of the decoders are not correlated with the channel inputs or outputs. This observation explains the asymmetry with regard to entanglement cooperation, and more generally, for any pair of non-signaling correlated resources that are shared between the decoders. Therefore, our result reveals a fundamental asymmetry and demonstrates the limitations of the duality between the broadcast channel and the multiple access channel.

FIG. 5.

Classical coding for a classical multiple access channel PY|X1X2 with shared entanglement between the encoders. The systems of Alice 1, Alice 2, and Bob are marked in blue, purple and red, respectively. Alice 1 and Alice 2 share entanglement resources in the systems SA1 and SA2, respectively. For k = 1, 2, Alice k encodes the message mk by applying an encoding map FMkxkn to the register Mk, which stores the respective message. Then, she transmits Xkn over the multiple access channel. Bob receives the channel output sequence Yn and estimates the messages using a decoding map DYnM̂1M̂2. Leditzky et al.44 showed that using the entanglement resources, SA1 and SA2, can strictly increase the achievable rates. We have shown that for the broadcast dual in Fig. 1, entanglement resources cannot increase the achievable rates.

FIG. 5.

Classical coding for a classical multiple access channel PY|X1X2 with shared entanglement between the encoders. The systems of Alice 1, Alice 2, and Bob are marked in blue, purple and red, respectively. Alice 1 and Alice 2 share entanglement resources in the systems SA1 and SA2, respectively. For k = 1, 2, Alice k encodes the message mk by applying an encoding map FMkxkn to the register Mk, which stores the respective message. Then, she transmits Xkn over the multiple access channel. Bob receives the channel output sequence Yn and estimates the messages using a decoding map DYnM̂1M̂2. Leditzky et al.44 showed that using the entanglement resources, SA1 and SA2, can strictly increase the achievable rates. We have shown that for the broadcast dual in Fig. 1, entanglement resources cannot increase the achievable rates.

Close modal

U.P., C.D., and H.B. were supported by the Bundesministerium für Bildung und Forschung (BMBF) through Grant Nos. 16KIS0856 (Pereg, Deppe) and 16KIS0858 (Boche) and by the Israel CHE Fellowship for Quantum Science and Technology (Pereg). H. Boche was also supported, in part, by the German Research Foundation (DFG), within the Gottfried Wilhelm Leibniz Prize, under Grant No. BO 1734/20-1, and within Germany’s Excellence Strategy, under Grant No. EXC-2111—390814868. Part of this work has been presented at the IEEE International Symposium on Information Theory (ISIT 2021).

Part of this work has been presented at the IEEE International Symposium on Information Theory (ISIT 2021).

The data that support the findings of this study are available within the article.

To bound the alphabet size of the random variables X0 and X1, we use the Fenchel–Eggleston–Carathéodory lemma117 and similar arguments as in Refs. 74 and 118. Let

L0=|HA|2+2,
(A1)
L1=L0|HA|2+1.
(A2)

First, fix pX1|X0(x1|x0) and consider the ensemble {pX0(x0)pX1|X0(x1|x0),θAx0,x1}. Every pure state θA = |ϕA⟩⟨ϕA| has a unique parametric representation u(θA) of dimension |HA|21. Then, define a map f0:X0RL0 by

f0(x0)=u(ρAx0),H(B2|X0=x0),H(B1|X0=x0)ρ,H(B1|X0=x0,X1),
(A3)

where ρAx0=x1pX1|X0(x1|x0)θAx0,x1. The map f0 can be extended to a map that acts on probability distributions as follows:

F0:pX0x0X0pX0(x0)f0(x0)=u(ρA),H(B2|X0),H(B1|X0)ρ,H(B1|X0,X1),
(A4)

where ρA=x0pX0(x0)ρAx0. According to the Fenchel–Eggleston–Carathéodory lemma,117 any point in the convex closure of a connected compact set within Rd belongs to the convex hull of d points in the set. Since the map F0 is linear, it maps the set of distributions on X0 to a connected compact set in RL0, where L0=(|HA|21)+1+1=|HA|2+1 as defined in (A1). Thus, for every pX0, there exists a probability distribution pX̄0 on a subset X̄0X0 of size L0 such that F0(pX̄0)=F0(pX0). We deduce that alphabet size can be restricted to |X0|L0 while preserving ρA and ρB1B2NAB1B2(ρA); I(X0;B2)ρ=H(B2)ρH(B2|X0)ρ, I(X1;B1|X0)ρ=H(B1|X0)ρH(B1|X0,X1)ρ; and I(X0,X1;B1)ρ=H(B1)ρH(B1|X0,X1)ρ.

We move to the alphabet size of X1. Fix pX0|X1, where

pX0|X1(x0|x1)pX0(x0)pX1|X0(x1|x0)x0X0pX0(x0)pX1|X0(x1|x0).
(A5)

Define the map f1:X1RL1 by

f1(x1)=pX0|X1(|x1),(u(ρAx0,x1))x0X0,H(B1|X0,X1=x1)ρ,
(A6)

where ρAx1=x0pX0|X1(x0|x1)θAx0,x1. Now, the extended map is

F1:pX1x1X1pX1(x1)f1(x1)=pX0,(u(ρAx0))x0X0,H(B1|X0,X1)ρ.
(A7)

By the Fenchel–Eggleston–Carathéodory lemma,117 for every pX1, there exists pX̄1 on a subset X̄1X1 of size (|HA|21)L0+2L1 [see (A1)] such that F1(pX̄1)=F1(pX1). We deduce that alphabet size can be restricted to |X1|L1 while preserving ρAx0, ρA and ρB1B2x0NAB1B2(ρAx0), ρB1B2NAB1B2(ρA); I(X1;B1|X0)ρ=H(B1|X0)ρH(B1|X0,X1)ρ; and I(X0,X1;B1)ρ=H(B1)ρH(B1|X0,X1)ρ.

Consider a quantum broadcast channel NAB1B2 with a classical conferencing link of capacity C12. The proof extends techniques that were used in a previous work by the first author.118,119

1. Achievability proof

We show that for every ζ0, ζ1, ɛ0 > 0, there exists a (2n(R0ζ0),2n(R1ζ1),n,ε0) code for NAB1B2 with conferencing and degraded message sets, provided that (R0,R1)RCl(N). To prove achievability, we extend the classical superposition coding with binning technique to the quantum setting and then apply the quantum packing lemma. Similar observations as in Refs. 118 and 119 are used as well. Let {pX0(x0)pX1|X0(x1|x0),θAx0,x1} be a given ensemble, and define

ρB1,B2x0,x1NAB1B2(θAx0,x1),
(B1)
σB2x0x1X1pX1|X0(x1|x0)ρB2x0,x1
(B2)

for (x0,x1)X0×X1, where ρB2x0,x1 is the reduced state of ρB1,B2x0,x1.

Standard method-of-type concepts are defined as in Refs. 35 and 118. We briefly introduce the notation and basic properties, while the detailed definitions can be found in Ref. 118, Sec. III. In particular, given a density operator ρ = ∑xpX(x)|x⟩⟨x| on the Hilbert space HA, we let Aδ(pX) denote the δ-typical set that is associated with pX and ΠAnδ(ρ) denote the projector onto the corresponding subspace. The following inequalities follow from the well-known properties of δ-typical sets:32 

Tr(Πδ(ρ)ρn)1ε,
(B3)
2n(H(ρ)+cδ)Πδ(ρ)Πδ(ρ)ρnΠδ(ρ)2n(H(ρ)cδ),
(B4)
Tr(Πδ(ρ))2n(H(ρ)+cδ),
(B5)

where c > 0 is a constant. Furthermore, for σB=xpX(x)ρBx, let ΠBnδ(σB|xn) denote the projector corresponding to the conditional δ-typical set given the sequence xn. Similarly,35 

Tr(Πδ(σB|xn)ρBnxn)1ε,
(B6)
2n(H(B|X)σ+cδ)Πδ(σB|xn)Πδ(σB|xn)ρBnxnΠδ(σB|xn)2n(H(B|X)σcδ),
(B7)
Tr(Πδ(σB|xn))2n(H(B|X)σ+cδ),
(B8)

where c′ > 0 is a constant, ρBnxn=i=1nρBixi, and the classical random variable X′ is distributed according to the type of xn. If xnAδ(pX), then

Tr(Πδ(σB)ρBnxn)1ε
(B9)

as well (see Ref. 35, Property 15.2.7). We note that the conditional entropy in the bounds above can also be expressed as H(B|X)σ=1nH(Bn|Xn=xn)σ1nH(Bn)ρxn.

The code construction, encoding, and decoding procedures are described below.

a. Classical Code Construction

Select 2nR0 independent sequences x0n(m0), m0[1:2nR0], at random according to i=1npX0(x0,i). For every m0[1:2nR0], select 2nR1 conditionally independent sequences x1n(m0,m1), m1[1:2nR1], at random according to i=1npX1|X0x1,i|x0,i(m0). Partition the set of indices [1:2nR0] into 2nC12 bins of equal size,

B(g)=[(g1)2n(R0C12):g2n(R0C12)]
(B10)

for g[1:2C12].

b. Encoding

To send the message pair (m0, m1), Alice prepares ρAn=i=1nρAx1,i(m0,m1) and sends the block An. The resulting output state is

ρB1nB2n=i=1nρBx0,i(m0),x1,i(m0,m1).
(B11)
c. Decoding

Bob 1 receives the systems B1n and decodes as follows:

  • Decode m̂0 by applying a POVM {Λm00}m0[1:2nR0] to the systems B1n.

  • Decode m̂1 by applying a second POVM {Λm1|m̂01}m1[1:2nR1] to the systems B1n.

  • Choose g to be the corresponding bin index such that m̂0B(g).

  • Send the conference message g to Bob 2.

The POVMs {Λm00} and {Λm1|m̂01} will be specified later.

Bob 2 receives the systems B2n and the conference message g and decodes by applying a POVM {Γm0}m0B(g), which will also be specified later, to the systems B2n.

d. Analysis of Probability of Error

Assume without loss of generality that Alice sends (m0, m1). Denote the decoding measurement outcomes by M̂0, M̂1, G, and M̃0. Consider the following events:

E1={(X0n(m0),X1n(m0,m1))Aδ/2(pX0,X1)},
(B12)
E2={M̂0m0},
(B13)
E3={M̂0=m0,M̂1m1},
(B14)
E4={M̃0m0}.
(B15)

By the union of events bound, the probability of error is bounded by

Pe|m0,m1(n)(F,Λ,Γ)PrE1+PrE2E1c+PrE3E1cE2c+PrE4E1cE2cE3c.
(B16)

The first term tends to zero as n by the law of large numbers. To bound the second term, we use the quantum packing lemma. Given E1c, we have (X0n(m0),X1n(m0,m1))Aδ2(pX0,X1). Now, observe that

Πδ(ρB1)ρB1nΠδ(ρB1)2n(H(B1)ρε2(δ))Πδ(ρB1),
(B17)
TrΠδ(ρB1|x0n,x1n)ρB1nx0n,x1n1ε2(δ),
(B18)
TrΠδ(ρB1|x0n,x1n)2n(H(B1|X0,X1)ρ+ε2(δ)),
(B19)
TrΠδ(ρB1)ρB1nx0n,x1n1ε2(δ)
(B20)

for all (x0n,x1n)Aδ/2(pX0,X1), by (B4), (B6), (B8), and (B9), respectively. By the quantum packing lemma73 (see Ref. 118, Lemma 3), there exists a POVM Λm̂00 such that

TrΛm00ρB1nx0n(m0),x1n(m0,m1)12n[H(B1)ρH(B1|X0X1)ρ(R0+R1)ε3(δ)]
(B21)

for all m0[1:2nR0]. Hence, PrE2E1c2n(I(X0,X1;B1)ρ(R0+R1)ε3(δ)), which tends to zero as n, provided that

R0+R1<I(X0,X1;B1)ρε3(δ).
(B22)

Moving to the third term in the RHS of (B16), suppose that E2c occurred, namely, the decoder measured the correct M0. Denote the state of the systems B1n after this measurement by ρB1n. Then, as in Refs. 118 and 119, we observe that due to the packing lemma inequality (B21), the gentle measurement lemma120,121 implies that the post-measurement state is close to the original state in the sense that

12ρB1nρB1n12n12(I(X0,X1;B1)ρ(R0+R1)ε4(δ))ε5(δ)
(B23)

for sufficiently large n and rates as in (B22). Therefore, the distribution of measurement outcomes when ρB1n measured is roughly the same as if the POVM Λm̂00 was never performed. To be precise, the difference between the probability of a measurement outcome m̂1 when ρB1n is measured and the probability when ρB1n is measured is bounded by ɛ5(δ) in absolute value (see Ref. 35, Lemma 9.11). Furthermore,

TrΠδ(ρB1|x0n,x1n)ρB1nx0n,x1n1ε6(δ),
(B24)
Πδ(ρB|x0n)ρB1nx0n,x1nΠδ(ρB1|x0n)2n(H(B1|X0)ρε6(δ))Πδ(ρB1|x0n),
(B25)
TrΠδ(ρB1|x0n,x1n)2n(H(B1|X0,X1)ρ+ε6(δ)),
(B26)
TrΠδ(ρB1|x0n)ρB1nx0n,x1n1ε6(δ)
(B27)

for all (x0n,x1n)Aδ2(pX0pX1|X0), by (B6)–(B9), respectively. Therefore, we have by the quantum packing lemma that there exists a POVM Λm̂1|m01 such that PrE3E1cE2c2n(I(X1;B1|X0)ρR1ε7(δ)), which tends to zero as n, provided that

R1<I(X1;B1|X0)ρε7(δ).
(B28)

It remains to consider erroneous decoding by Bob 2. Suppose that E3c occurred, namely, Bob 1 measured the correct m0, and thus sent the correct bin index G such that m0B(G). Recall that the size of each bin is |B(G)|=2n(R0C12) [see (B10)]. Then, observe that

Πδ(ρB2)ρB2nΠδ(ρB2)2n(H(B2)ρε2(δ))Πδ(ρB1),
(B29)
TrΠδ(ρB2|x0n)σB2nx0n1ε2(δ),
(B30)
TrΠδ(ρB2|x0n)2n(H(B2|X0)ρ+ε2(δ)),
(B31)
TrΠδ(ρB2)ρB2nx0n1ε2(δ)
(B32)

for all x0nAδ(pX0,X1), by (B4), (B6), (B8), and (B9), respectively. Hence, by the quantum packing lemma (Ref. 118, Lemma 3), there exists a POVM Γm̃0 such that PrE4E1cE2cE3c2n(I(X0;B2)ρ(R0C12)ε8(δ)), which tends to zero as n, provided that

R0<I(X0;B2)ρ+C12ε8(δ).
(B33)

To show that rate pairs in 1κRCl(Nκ) are achievable as well, one may employ the coding scheme above for the product broadcast channel Nκ, where κ is arbitrarily large. This completes the proof of the direct part.

2. Converse proof

Consider the converse part for the regularized capacity formula. Suppose that Alice chooses m0 and m1 uniformly at random and prepares an input state ρAnm0,m1. After Alice sends the systems An through the channel, the output state is ρB1nB2n=12n(R0+R1)m0,m12nR0m1=12nR1NAnB1nB2n(ρAnm0,m1). Then, Bob 1 performs a decoding POVM ΛB1nm0,m1,g, sends g to Bob 2 using conferencing, and Bob 2 chooses a POVM ΓB2n|gm0 accordingly. Consider a sequence of codes (Fn,Λn,Γn) such that the average probability of error tends to zero; hence, the error probabilities PrM̂0M0, Pr(M̂0,M̂1)(M0,M1), and PrM̂1M1|M0 are bounded by some αn, which tends to zero as n. By Fano’s inequality,122 it follows that

H(M0|M̃0)nεn,
(B34)
H(M0,M1|M̂0,M̂1)nεn,
(B35)
H(M1|M̂1,M0)nεn,
(B36)

where ɛn, ɛn′, ɛn tend to zero as n. Hence,

nR0=H(M0)=I(M0;M̃0)ρ+H(M0|M̃0)I(M0;M̃0)ρ+nεnI(M0;B2nG)ρ+nεn=I(M0;B2n)ρ+I(M0;G|B2n)ρ+nεnI(M0;B2n)ρ+nC12+nεn,
(B37)

where the second inequality follows from the Holevo bound (see Ref. 32, Theorem 12.1) and the last inequality holds as I(M0;G|B2n)ρH(G)nC12 because G is a classical message in [1:2nC12]. Similarly,

n(R0+R1)I(M0,M1;B1n)ρ+nεn.
(B38)

Furthermore, since M0 and M1 are statistically independent, we can also write

nR1=H(M1|M0)I(M1;M̂1|M0)ρ+nεnI(M1;B1n|M0)ρ+nεn.
(B39)

We deduce that R0I(X0n;B2n)ρ+C12+εn, R0+R1I(X0n,X1n;B1n)ρ+εn, and R1I(X1n;B1n|X0n)ρ+εn, with Xkn=fk(Mk), where fk are arbitrary one-to-one maps from [1:2nRk] to Xkn, for k = 0, 1. This completes the converse proof for the regularized characterization.

For a Hadamard broadcast channel, where Bob 1 receives a classical output Y1n, define

X0,i=(M0,Y1i1),X1,i=(M1,Y1i1).
(B40)

Applying the chain rule to (B37),

nR0i=1nI(M0;B2,i|B2i1)ρ+nC12+nεni=1nI(M0B2i1;B2,i)ρ+nC12+nεn.
(B41)

Since the marginal of Bob 2 is degraded with respect to that of Bob 1, namely, NAB2H=PY1B2NAY1H, the data processing inequality for the quantum mutual information implies that I(M0B2i1;B2,i)ρI(M0Y1i1;B2,i)ρ=I(X0,i;B2,i)ρ; hence,

R0εn1ni=1nI(X0,i;B2,i)ρ+C12=IX0,K;B2,K|Kρ+C12,
(B42)

where K is a classical random variable with uniform distribution over [1:n], independent of M0, M1, and G. Defining ρKA=1ni=1n|ii|ρAi, ρKB1B2=NAB1B2ρKA, and

X0(X0,K,K),X1(X1,K,K),
(B43)

we obtain

R0I(X0;B2)ρ+C12+εn,
(B44)

and by similar considerations,

R0+R1I(X0X1;B1)+εn,
(B45)
R1I(X1;B1|X0)+εn.
(B46)

This completes the Proof of Theorem 2.

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