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.

## I. INTRODUCTION

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 Internet^{2} 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, $|\Phi AP1\u3009$ between the sender and the repeater and $|\Phi P2B\u3009$ between the repeater and the receiver. At the next stage, the repeater teleports the quantum state of *P*_{1} 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 jammer^{20} 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 Holevo^{22} 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 coding^{37} 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 game^{45} where quantum strategies guarantee a certain win and outperform classical strategies, extending ideas by Nötzel^{46} 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 phenomenon^{19} 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 Winter^{93} 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 Servetto^{94} (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 *Q*_{1} and *Q*_{2} 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 $|\Phi AP1\u3009$, which consists of *nQ*_{1} entangled bits (ebits). Given entanglement between the receivers, we also have a maximally entangled pair $|\Phi P2B\u3009$, 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 *P*_{1} 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.

## II. DEFINITIONS

### A. Notation, states, and information measures

We use the following notation conventions. Script letters $X,Y,Z,\u2026$ 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) *p*_{X}(*x*) over a finite set $X$. We use *x*^{j} = (*x*_{1}, *x*_{2}, …, *x*_{j}) to denote a sequence of letters from $X$. A random sequence *X*^{n} 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 $|\psi \u3009\u2208HA$, where ⟨*ψ*| is the Hermitian conjugate of |*ψ*⟩. In general, a density operator has a spectral decomposition,

where $X={1,2,\u2026,|HA|}$, *p*_{X}(*x*) is a probability distribution over $X$, and ${|\psi x\u3009}x\u2208X$ forms an orthonormal basis of the Hilbert space $HA$. A measurement of a quantum system is any set of operators {Λ_{j}} that forms a positive operator-valued measure (POVM), i.e., the operators are positive semi-definite and $\u2211j\Lambda j=1$, 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 *p*_{A}(*j*) = Tr(Λ_{j}*ρ*). The trace distance between two density operators *ρ* and *σ* is $\rho \u2212\sigma 1$, where $F1=Tr(F\u2020F)$.

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 $HA\u2297HB$ of the corresponding Hilbert spaces. Given a bipartite state *σ*_{AB}, define the quantum mutual information as

Furthermore, conditional quantum entropy and mutual information are defined by *H*(*A*|*B*)_{σ} = *H*(*σ*_{AB}) − *H*(*σ*_{B}) and *I*(*A*;*B*|*C*)_{σ} = *H*(*A*|*C*)_{σ} + *H*(*B*|*C*)_{σ} − *H*(*A*,*B*|*C*)_{σ}, respectively. The coherent information is then defined as

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 $|\Phi AB\u3009=1D\u2211j=0D\u22121|j\u3009A\u2297|j\u3009B$, where ${|j\u3009A}j=0D\u22121$ and ${|j\u3009B}j=0D\u22121$ are respective orthonormal bases. Note that *I*(*A*;*B*)_{|Φ⟩⟨Φ|} = 2 log(*D*) and *I*(*A*⟩*B*)_{|Φ⟩⟨Φ|} = log(*D*).

The entanglement of formation of a joint state *ρ*_{AB} is defined as^{97,106}

where the infimum is over all pure state decompositions $\rho AB=\u2211xpX(x)|\psi ABx\u3009\u3008\psi ABx|$.

### B. Quantum broadcast channel

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 $NA\u2192B1B2$ corresponding to a quantum physical evolution. We assume that the channel is memoryless. That is, if the systems *A*^{n} = (*A*_{1}, …, *A*_{n}) are sent through *n* channel uses, then the input state $\rho An$ undergoes the tensor product mapping $NAn\u2192B1nB2n\u2261NA\u2192B1B2\u2297n$. The marginal channel $NA\u2192B1(1)$ is defined by

for receiver 1, and similarly, $NA\u2192B2(2)$ for receiver 2. One may say that $NA\u2192B1B2$ is an extension of $NA\u2192B1(1)$ and $NA\u2192B2(2)$. 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 *CC*_{G→G′}, 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

for some set of operators *N*_{j} such that $\u2211jNj\u2020Nj=1$.

*X*and

*Y*

_{1},

*Y*

_{2}, respectively, while the Kraus operators are $Nx,y1,y2=PY1Y2|X(y1,y2|x)|y1,y2\u3009\u3008x|$ for some probability kernel $PY1Y2|X$ and orthonormal bases {|

*x*⟩}, {|

*y*

_{1},

*y*

_{2}⟩}. Therefore, given an input $x\u2208X$, the output state of a classical broadcast channel is

### C. Degraded broadcast channel, Hadamard broadcast channel, and degradable marginals

We will also be interested in the following special cases.

^{57}). A quantum broadcast channel $NA\u2192B1B2$ is called

*degraded*if there exists a degrading channel $PB1\u2192B2$ such that the marginals satisfy the following relation:

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 $NA\u2192Y1B2H$ can be viewed as a measure-and-prepare channel where the marginal channel $NA\u2192Y1(1)$ acts as a measurement device, while the degrading channel $PY1\u2192B2$ corresponds to state preparation. In this case, the marginal quantum channel $NA\u2192B2(2)$ 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 $MA\u2192B$ has an isometric extension $UA\u2192BEM(\rho A)=U\rho AU\u2020$, also called a Stinespring dilation, where the operator *U* is an isometry, i.e., $U\u2020U=1$ (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 $M\u0302A\u2192E(\rho A)=TrB(U\rho AU\u2020)$ is called the complementary channel for $MA\u2192B$.

(Degradable marginals). A point-to-point quantum channel $MA\u2192B$ is called *degradable* if there exists an isometric extension such that the complementary channel $M\u0302A\u2192E$ is degraded with respect to $MA\u2192B$. 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 $NA\u2192B1B2$ has degradable marginals if both marginals $NA\u2192B1(1)$ and $NA\u2192B2(2)$ are degradable.

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

## III. CODING FOR THE BROADCAST CHANNEL

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.

### A. Classical coding with entangled decoders

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.

A $(2nR0,2nR1,n)$ classical code for the quantum broadcast channel $NA\u2192B1B2$ 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 $FM0M1\u2192An$, where

*M*_{0}and*M*_{1}are classical registers that store the common and private messages, respectively.A pure entangled state $\Psi SB1,SB2$.

Two decoding POVMs: ${\Lambda SB1B1nm0,m1}$ for receiver 1 and ${\Gamma SB2B2nm0}$ for receiver 2, where the measurement outcome

*m*_{k}is an index in $[1:2nRk]$ for*k*= 0, 1.

We denote the code by $(F,\Psi ,\Lambda ,\Gamma )$.

The communication scheme is depicted in Fig. 1. The sender Alice has the systems *A*^{n}, and the receivers Bob 1 and Bob 2 have the systems $B1n,SB1$ and $B2n,SB2$, respectively. Alice chooses a common message $m0\u2208[1:2nR0]$ that is intended for both users and a private message $m1\u2208[1:2nR1]$ for Bob 1 and stores them in the classical registers *M*_{0} and *M*_{1}, respectively. She encodes the messages by applying the encoding map $FM0M1\u2192An$, which results in an input state $\rho Anm0,m1=FM0M1\u2192An(m0,m1)$, and transmits the systems *A*^{n} over *n* channel uses of $NA\u2192B1B2$. Hence, the output state is

Bob 1 receives the channel output systems $B1n$, combines them with his entangled system $SB1$, and applies the POVM ${\Lambda SB1B1nm0,m1}$. Bob 1 then obtains from the measurement outcome an estimate of the message pair $(m\u03020,m\u03021)\u2208[1:2nR0]\xd7[1:2nR1]$. Similarly, Bob 2 finds an estimate of the common message $m\u03030\u2208[1:2nR0]$ by performing a measurement using ${\Gamma SB2B2nm0}$ on the output systems $B2n$ and his entangled system $SB2$. The conditional probability of error of the code, given that the message pair (*m*_{0}, *m*_{1}) was sent, is given by

A $(2nR0,2nR1,n,\epsilon )$ classical code satisfies $Pe|m0,m1(n)(F,\Psi ,\Lambda ,\Gamma )\u2264\epsilon $ for all $(m0,m1)\u2208[1:2nR0]\xd7[1:2nR1]$. A rate pair (*R*_{0}, *R*_{1}) is called achievable with entangled decoders if for every *ɛ* > 0 and sufficiently large *n*, there exists a $(2nR0,2nR1,n,\epsilon )$ code. The classical capacity region is defined as the set of achievable pairs (*R*_{0}, *R*_{1}) with entangled decoders.

One may also consider the broadcast channel with independent messages, i.e., when the common message *m*_{0} is replaced by a private message *m*_{2} 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.

### B. Classical coding with conferencing

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 $C12$.

A $(2nR0,2nR1,n)$ classical code for the quantum broadcast channel $NA\u2192B1B2$ 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 $FM0M1\u2192An$, where

*M*_{0}and*M*_{1}are classical registers that store the common and private messages, respectively.A decoding POVM ${\Lambda B1nm0,m1,g}$ for receiver 1, where the measurement outcome is a triplet of indices (

*m*_{0},*m*_{1},*g*) in $[1:2nR0]\xd7[1:2nR1]\xd7[1:2C12]$.A collection of decoding POVMs ${\Gamma B2n|gm0}$, $g\u2208[1:2C12]$, for receiver 2, where the measurement outcome is an index in $[1:2nR0]$.

We denote the code by $(F,\Lambda ,\Gamma )$.

The communication scheme is depicted in Fig. 2. The sender Alice has the systems *A*^{n}, and the receivers Bob 1 and Bob 2 have the systems $B1n$ and $B2n$, respectively. Alice chooses a common message $m0\u2208[1:2nR0]$ and a private message $m1\u2208[1:2nR1]$ for Bob 1 and stores them in the classical registers *M*_{0} and *M*_{1}, respectively. She encodes the messages by applying the encoding map $FM0M1\u2192An$, which results in an input state

and transmits the systems *A*^{n} over *n* channel uses of $NA\u2192B1B2$. Hence, the output state is

Bob 1 receives the channel output systems $B1n$ and applies the POVM ${\Lambda B1nm0,m1,g}$. Bob 1 then obtains from the measurement outcome an estimate of the message pair $(m\u03020,m\u03021)\u2208[1:2nR0]\xd7[1:2nR1]$ and a conference message $g\u2208[1:2nC12]$. Next, Bob 1 sends the conference message *g* to Bob 2. Given the conference message *g*, Bob 2 chooses a POVM ${\Gamma B2n|gm0}$ to perform on the channel output systems $B2n$, producing an estimate of the common message $m\u03030\u2208[1:2nR0]$ as the measurement outcome. The conditional probability of error of the code, given that the message pair (*m*_{0}, *m*_{1}) was sent, is given by

A $(2nR0,2nR1,n,\epsilon )$ classical code satisfies $Pe|m0,m1(n)(F,\Lambda ,\Gamma )\u2264\epsilon $ for all $(m0,m1)\u2208[1:2nR0]\xd7[1:2nR1]$. A rate pair (*R*_{0}, *R*_{1}) is called achievable with conferencing if for every *ɛ* > 0 and sufficiently large *n*, there exists a $(2nR0,2nR1,n,\epsilon )$ code. The classical capacity region $RCl(N)$ is defined as the set of achievable pairs (*R*_{0}, *R*_{1}) with conferencing.

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 (*B*_{1}) 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 channel^{109} consists of a single user and a helper (see Definition 6).

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 $C12$.

### C. Quantum coding with conferencing, entanglement generation, and entanglement transmission

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.

A $(2nQ1,2nQ2,n)$ quantum code for the quantum broadcast channel $NA\u2192B1B2$ with independent messages and conferencing consists of the following:

A quantum message state $\rho M1M2$, where

*M*_{1}and*M*_{2}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 $FM1M2\u2192An$.

A decoding map $DB1n\u2192M\u03021G1$ 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 $DG\u2032B2n\u2192M\u030302$ 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 *M*_{1}, *M*_{2}, and *A*^{n}; Bob 1 has the systems $B1n$, *G*, and $M\u03021$; and Bob 2 has the systems $B2n$, *G*′, and $M\u03022$. Alice encodes the quantum state of the message systems *M*_{1} and *M*_{2} by applying the encoding map $FM1M2\u2192An$, which results in the input state

and transmits the systems *A*^{n} over *n* channel uses of $NA\u2192B1B2$. Hence, the output state is

Bob 1 receives the channel output systems $B1n$ and applies the decoding map $DB1n\u2192M\u03021G1$, which results in

The reduced state of $M\u03021$ is Bob 1’s estimate of the original state of his private message system *M*_{1}. Next, Bob 1 sends the conference message from *G* to *G*′ using the noiseless conference link id_{G→G′}; hence, $\rho M\u03021G\u2032B2n=idG\u2192G\u2032(\rho M\u03021GB2n)=\rho M\u03021GB2n$. Bob 2 receives the channel output systems $B2n$ and the conference message in *G*′ and applies the decoding map $DG\u2032B2n\u2192M\u030222$ such that $M\u03022$ is his estimate of his private message. The estimated state is then given by

and the estimation error is given by

A $(2nQ1,2nQ2,n,\epsilon )$ quantum code satisfies $e(n)(F,D1,D2,\rho M0M1)\u2264\epsilon $ for all $\rho M0,M1$. A rate pair (*Q*_{1}, *Q*_{2}) is called achievable with conferencing if for every *ɛ* > 0 and sufficiently large *n*, there exists a $(2nQ1,2nQ2,n,\epsilon )$ code. The quantum capacity region $RQ(N)$ is defined as the set of achievable pairs (*Q*_{1}, *Q*_{2}) with conferencing.

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

A primitive relay channel $NA\u2192B1B2relay$ is a broadcast channel with conferencing when user 1 does not send information, i.e., *Q*_{1} = 0. Alice, Bob 1, and Bob 2 are then called the source, relay, and destination receiver, respectively. A quantum rate *Q*_{2} > 0 is called achievable for the primitive relay channel if (0, *Q*_{2}) 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 channel^{109} where information is received and encoded at the relay in a strictly causal manner.

A standard, i.e., non-primitive, relay channel^{88–90} is specified by a linear, completely positive, trace-preserving map $LA\u2192B1A1B2$, where the sender transmits the systems *A*^{n}, 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 $B1i\u22121$ at time *i*. That is, at time *i*, the relay transmits *A*_{1,i} such that $\rho A1i=TB1i\u22121\u2192A1i(i)(\rho B1i\u22121)$.

^{30,110}In this task, Alice and Charlie share a pure entangled state $|\psi M1M2C\u3009$, with a Schmidt decomposition

*ɛ*. In particular, if Alice and Charlie share a maximally entangled state $|\Phi M1M2C\u3009$, then at the end of the communication protocol, Bob 1, Bob 2, and Charlie share a state $\u2248|\Phi M\u03021M\u03022C\u3009$ up to an

*ɛ*-error.

^{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 $A\u0304$ that is entangled with

*M*

_{1}and

*M*

_{2}in a state $|\psi A\u0304M1M2\u3009$. Then, by strong sub-additivity (Ref. 35, Coro. 11.9.1),

*M*

_{1}and

*M*

_{2}; 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 $|\psi A\u0304M1M2\u3009=1d\u2211x=1d|x\u3009\u2297|x\u3009\u2297|x\u3009$. Alternatively, she can generate two entangled pairs. Suppose that Alice has another pair of system $A\u03041,A\u03042$ in the state

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 $|\Phi M1M2\u3009$.

## IV. MAIN RESULTS—CLASSICAL CONFERENCING

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

where the union is over the set of all distributions $pX0,X1(x0,x1)$ and state collection ${\theta Ax0,x1}$, with

Before we state the capacity theorem, we give the following lemma, which provides cardinality bounds for the auxiliary random variables *X*_{0} and *X*_{1}. In principle, one can use those cardinality bounds to evaluate the region $RCl(N)$ numerically.

The union in (23) is exhausted by auxiliary random variables *X*_{0} and *X*_{1} of cardinality $|X0|\u2264|HA|2+2$ and $|X1|\u2264(|HA|2+2)|HA|2+1$.

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

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.

## V. MAIN RESULTS—QUANTUM 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 *M*_{1} to generate entanglement and prepare a maximally entangled pair $|\Phi AB1\u3009$ between the transmitter and the repeater, namely, Alice and Bob 1. Given entanglement between the decoders, we also have a maximally entangled pair $|\Phi B1\u2032B2\u2032\u3009$ between the repeater and the receiver, i.e., Bob 1 and Bob 2. Then, the repeater *B*_{1} can swap his entanglement by using the classical conferencing link to teleport the state of *B*_{1}′ onto *B*_{2}, thus swapping the entanglement such that *A* and *B*_{2} are now entangled. This requires that the conferencing capacity is at least twice the information rate, i.e., $C12\u22652Q2$. We will conclude this section with the resulting observations for the quantum repeater.

### A. Achievable region

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

*Q*

_{1},

*Q*

_{2}) is achievable for transmission of quantum information over the broadcast channel $NA\u2192B1B2$ with private messages and quantum conferencing if

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(A2\u3009B2)\rho +\Delta $, then $Q1\u2264I(A1\u3009B1)\rho \u2212\Delta $.

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 $(R1,R2)=(I(X1;B1|X0)\rho ,I(X0;B2)\rho +C12)$ is achievable when $I(X0;B2)\rho +C12<I(X0;B1)\rho $ because then *R*_{1} + *R*_{2} < *I*(*X*_{0}*X*_{1}; *B*_{1}) 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 *M*_{2} (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 *M*_{2} only if the state was destroyed in Bob 1’s location during conferencing.

*et al.*

^{58}Fix an input state $\rho A1A2A\u2032$. Based on Ref. 58 (Ref. 59, Theorem 5.4), for every

*ɛ*> 0 and sufficiently large

*n*, there exists a $(2nQ1\u2032,2nQ2\u2032,n,\epsilon )$ quantum code for the broadcast channel $NA\u2192B1B2$

*without*conferencing if

*δ*> 0 is arbitrarily small. The rate pair (

*Q*

_{1}′,

*Q*

_{2}′) 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.

*nQ*

_{2}′ to Bob 2, with $Q1\u2033=Q1\u2032\u2212CQ,12$, 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:

*Q*

_{1}′,

*Q*

_{2}′) and (

*Q*″

_{1},

*Q*″

_{2}) 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}

### B. Outer bound

Next, we give a multi-letter outer bound.

*Q*

_{1},

*Q*

_{2}) is achievable for the transmission of quantum information over the broadcast channel $NA\u2192B1B2$ with private messages and quantum conferencing, then it must satisfy the following inequalities:

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)].

*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,

*A*′

^{n}through the channel, Bob 1 receives the systems $B1n$ in the state

*G*is the conference message that is sent through the conference link to Bob 2. Having received $B2n$ and

*G*′ such that $\rho M1M2M\u03021G\u2032=\rho M1M2M\u03021G$, Bob 2 uses a decoding channel $DG\u2032B2n\u2192M\u030222$, producing

*α*

_{n}tends to zero as

*n*→

*∞*. By the Alicki–Fannes–Winter inequality

^{111,112}(Ref. 35, Theorem 11.10.3), (36) implies that $|H(Mk|M\u0302k)\rho \u2212H(Mk|Mk\u2032)\Phi |\u2264n\epsilon n$, or equivalently,

*k*= 1, 2, where

*ɛ*

_{n}tends to zero as

*n*→

*∞*. Observe that $I(Mk\u3009Mk\u2032)\Phi =H(Mk)\Phi \u2212H(MkMk\u2032)\Phi =nQk\u22120=nQk$. Thus,

*A*

_{1},

*A*

_{2}such that for some isometries $UM1\u2192A1$, $VM2\u2192A2$, and

*W*

_{G′→T}, we have $\rho A1B1n=UM1\u2192A1n\rho M1B1nUM1\u2192A1n\u2020$ and $\rho A2TB2n=(VM2\u2192A2\u2297WG\u2032\u2192T)\rho M2G\u2032B2n(VM2\u2192A2\u2297WG\u2032\u2192T)\u2020$. This completes the proof for the regularized outer bound.□

### C. Primitive relay channel

Consider the primitive relay channel $NA\u2192B1B2relay$, 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., *Q*_{1} = 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.

The quantum capacity of the primitive relay channel $NA\u2192B1B2relay$ has the following bounds:

- Cutset upper boundwith $\rho ATB1nB2n=NA\u2032\u2192B1B2\u2297n(\rho ATA\u2032n)$.(42)$CQ(Nrelay)\u2264limn\u2192\u221esup\rho ATA\u2032n1nminI(AT\u3009B2n)\rho +CQ,12,I(A\u3009B1nB2n)\rho ,$
- Decode-forward lower boundwith $\rho A1A2B1B2=NA\u2032\u2192B1B2(\varphi A1A2A\u2032)$.(43)$CQ(Nrelay)\u2265max|\varphi A1A2A\u2032\u3009I(A2\u3009B2)\rho +minI(A1\u3009B1)\rho ,CQ,12,$
- Entanglement-formation lower boundwith $|\varphi AB1B2E\u3009=UA\u2032\u2192B1B2EN|\varphi AA\u2032\u3009$, $\rho AB\u03021B2E=FB1\u2192B\u03021(\varphi AB1B2E)$, where $EF(\rho B\u03021AB2E)$ is the entanglement of formation with respect to the bipartition $B\u03021|AB2E$.(44)$CQ(Nrelay)\u2265max|\varphi A1A2A\u2032\u3009,FB1\u2192B\u03021:EF(\rho B\u03021AB2E)\u2264CQ,12I(A2\u3009B\u03021B2)\varphi ,$

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 *Q*_{1} = 0. The rate in (44) can be achieved by using the conferencing link to simulate the channel $FB1\u2192B\u03021$. 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 $B\u03021|AB2E$, i.e., $CQ,12\u2265EF(\rho B\u03021AB2E)$. Then, Bob 2 can decode $\rho B\u03021nB2n$, which is *ɛ*_{n}-close in trace distance to $\rho B1\u0302nB2n\u2261N\u0302A\u2192B1\u0302B2\u2297n(\rho A\u2032n)$, where *ɛ*_{n} tends to zero as *n* → *∞*, with $N\u0302A\u2192B1\u0302B2\u225cFB1\u2192B\u03021\u25e6NA\u2192B1B2$.

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(A2\u3009B2)\rho $ corresponds to repeaterless communication, while $minI(A1\u3009B1)\rho ,CQ,12$ corresponds to quantum transmission via the repeater.

Intuitively, the decode-forward lower bound has the interpretation of a bottleneck flow. Specifically, as mentioned in the previous remark, the term $minI(A1\u3009B1)\rho ,CQ,12$ 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 *A* → *B*_{1} with the repeater–receiver link *B*_{1} → *B*_{2}, 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).

## VI. ENTANGLED DECODERS

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 $2\u22c5CQ,12$ using the super-dense coding protocol. Further details are given below.

First, consider a quantum broadcast channel $NA\u2192B1B2$ 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 *m*_{0} and *m*_{1} uniformly at random and prepares an input state $\rho Anm0,m1$. After Alice sends the systems *A*^{n} through the channel, the output state is $\rho B1nB2n\u2297\Psi SB1SB2$, where $SB1$ and $SB2$ are the entangled systems of Bob 1 and Bob 2, respectively, and $\rho B1nB2n=12n(R0+R1)\u2211m0,m1NAn\u2192B1nB2n(\rho Anm0,m1)$. Then, Bob 1 performs a decoding POVM $\Lambda B1nSB1m0,m1$, and Bob 2 performs a decoding POVM $\Lambda B2nSB2m0$. Consider a sequence of codes $(Fn,\Lambda n,\Gamma n)$ such that the average probability of error tends to zero; hence, the error probabilities $PrM\u03020\u2260M0$, $Pr(M\u03020,M\u03021)\u2260(M0,M1)$, and $PrM\u03021\u2260M1|M0$ are bounded by some *α*_{n}, which tends to zero as *n* → *∞*. By Fano’s inequality, it follows that

where *ɛ*_{n}, *ɛ*_{n}′, *ɛ*″_{n} tend to zero as *n* → *∞*. Hence,

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,

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 $C12=2\u22c5CQ,12$. Achievability follows by using the super-dense coding protocol^{37} 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 $\Lambda B1nSB1m0,m1,g$, sends *g* to Bob 2 using conferencing, and Bob 2 chooses a POVM $\Gamma B2nSB2|gm0$ accordingly. By using the same considerations as in the derivation mentioned above,

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)\rho =I(M0;B2n)\rho $ as $SB1SB2$ are in a product state with $M0,M1,B1n,B2n$ and since $I(M0;G\u2032|B2nSB2)\rho \u22642H(G\u2032)\rho \u22642\u22c5nCQ,12$ (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.

## VII. SUMMARY AND DISCUSSION

We have considered the quantum broadcast channel $NA\u2192B1B2$ in different settings of cooperation between the decoders. Using those settings, we provided an information-theoretic framework for quantum repeaters.

### A. Conferencing

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 *Q*_{1} and *Q*_{2} 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 $|\psi M1M2C\u3009$, and at the end of the communication protocol, Bob 1, Bob 2, and Charlie share a state $\u2248|\Phi M\u03021M\u03022C\u3009$ 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 $|\Phi M1M2\u3009$.

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 $|\psi A\u0304M1M2\u3009=1d\u2211x=1d|x\u3009\u2297|x\u3009\u2297|x\u3009$. Alternatively, she can generate two entangled pairs. Suppose that Alice has another pair of system $A\u03041,A\u03042$ in the state

Then, at the end of the quantum communication protocol, Alice shares the entangled states $\u2248|\Phi A\u03041M\u03021\u3009$ with Bob 1 and $\u2248|\Phi A\u03042M\u03022\u3009$ with Bob 2.

The case where receiver 1 is not required to recover information, i.e., *Q*_{1} = 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.

### B. Quantum repeaters

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 $|\Phi AB1\u3009$, which consists of *nQ*_{1} entangled bits (ebits).

Given entanglement between the receivers, we also have a maximally entangled pair $|\Phi B1\u2032B2\u2032\u3009$, 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 *B*_{1}′ onto *B*_{2}, thus swapping the entanglement such that *A* and *B*_{2} are now entangled. This requires the classical conferencing rate to be at least twice the information transmission rate to *B*_{2}.

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 *A* → *B*_{1} and the repeater–receiver link *B*_{1} → *B*_{2}, 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(A1\u3009B1)\rho $ and the conferencing link capacity $CQ,12$ (see Remark 10).

### C. BC-MAC duality

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 interchanged^{103–105} (see also Ref. 114, Lemma 9.2). In the scalar case, this means that the capacity region of the Gaussian broadcast channel,

subject to a power constraint $1n\u2211i=1nxi2\u2264P$, is exactly the same as the capacity region of the Gaussian multiple access channel,

subject to a total-power constraint $1n\u2211i=1n(x1,i2+x2,i2)\u2264P$, with normalized Gaussian noises *Z*, *Z*_{1}, and $Z2\u223cN(0,1)$. 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 *R*_{1} + *R*_{2} 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 $PY1\u2032Y2\u2032|X\u2032$, where *X*′ ≡ *Y*, *Y*_{1} ≡ *X*_{1}, and *Y*_{2}′ ≡ *X*_{2}, 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 $PY1\u2032Y2\u2032|X\u2032$, 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.

## ACKNOWLEDGMENTS

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).

## ACKNOWLEDGMENTS

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

## DATA AVAILABILITY

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

### APPENDIX A: PROOF OF LEMMA 1

To bound the alphabet size of the random variables *X*_{0} and *X*_{1}, we use the Fenchel–Eggleston–Carathéodory lemma^{117} and similar arguments as in Refs. 74 and 118. Let

First, fix $pX1|X0(x1|x0)$ and consider the ensemble ${pX0(x0)pX1|X0(x1|x0),\theta Ax0,x1}$. Every pure state *θ*_{A} = |*ϕ*_{A}⟩⟨*ϕ*_{A}| has a unique parametric representation *u*(*θ*_{A}) of dimension $|HA|2\u22121$. Then, define a map $f0:X0\u2192RL0$ by

where $\rho Ax0=\u2211x1pX1|X0(x1|x0)\theta Ax0,x1$. The map *f*_{0} can be extended to a map that acts on probability distributions as follows:

where $\rho A=\u2211x0pX0(x0)\rho Ax0$. 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 *F*_{0} is linear, it maps the set of distributions on $X0$ to a connected compact set in $RL0$, where $L0=(|HA|2\u22121)+1+1=|HA|2+1$ as defined in (A1). Thus, for every $pX0$, there exists a probability distribution $pX\u03040$ on a subset $X\u03040\u2286X0$ of size *L*_{0} such that $F0(pX\u03040)=F0(pX0)$. We deduce that alphabet size can be restricted to $|X0|\u2264L0$ while preserving *ρ*_{A} and $\rho B1B2\u2261NA\u2192B1B2(\rho A)$; $I(X0;B2)\rho =H(B2)\rho \u2212H(B2|X0)\rho $, $I(X1;B1|X0)\rho =H(B1|X0)\rho \u2212H(B1|X0,X1)\rho $; and $I(X0,X1;B1)\rho =H(B1)\rho \u2212H(B1|X0,X1)\rho $.

We move to the alphabet size of *X*_{1}. Fix $pX0|X1$, where

Define the map $f1:X1\u2192RL1$ by

where $\rho Ax1=\u2211x0pX0|X1(x0|x1)\theta Ax0,x1$. Now, the extended map is

By the Fenchel–Eggleston–Carathéodory lemma,^{117} for every $pX1$, there exists $pX\u03041$ on a subset $X\u03041\u2286X1$ of size $(|HA|2\u22121)L0+2\u2264L1$ [see (A1)] such that $F1(pX\u03041)=F1(pX1)$. We deduce that alphabet size can be restricted to $|X1|\u2264L1$ while preserving $\rho Ax0$, *ρ*_{A} and $\rho B1B2x0\u2261NA\u2192B1B2(\rho Ax0)$, $\rho B1B2\u2261NA\u2192B1B2(\rho A)$; $I(X1;B1|X0)\rho =H(B1|X0)\rho \u2212H(B1|X0,X1)\rho $; and $I(X0,X1;B1)\rho =H(B1)\rho \u2212H(B1|X0,X1)\rho $.

### APPENDIX B: PROOF OF THEOREM 2

Consider a quantum broadcast channel $NA\u2192B1B2$ 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 *ζ*_{0}, *ζ*_{1}, *ɛ*_{0} > 0, there exists a $(2n(R0\u2212\zeta 0),2n(R1\u2212\zeta 1),n,\epsilon 0)$ code for $NA\u2192B1B2$ with conferencing and degraded message sets, provided that $(R0,R1)\u2208RCl(N)$. 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 ${pX0(x0)pX1|X0(x1|x0),\theta Ax0,x1}$ be a given ensemble, and define

for $(x0,x1)\u2208X0\xd7X1$, where $\rho B2x0,x1$ is the reduced state of $\rho B1,B2x0,x1$.

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* = ∑_{x}*p*_{X}(*x*)|*x*⟩⟨*x*| on the Hilbert space $HA$, we let $A\delta (pX)$ denote the *δ*-typical set that is associated with *p*_{X} and $\Pi An\delta (\rho )$ denote the projector onto the corresponding subspace. The following inequalities follow from the well-known properties of *δ*-typical sets:^{32}

where *c* > 0 is a constant. Furthermore, for $\sigma B=\u2211xpX(x)\rho Bx$, let $\Pi Bn\delta (\sigma B|xn)$ denote the projector corresponding to the conditional *δ*-typical set given the sequence *x*^{n}. Similarly,^{35}

where *c*′ > 0 is a constant, $\rho Bnxn=\u2297i=1n\rho Bixi$, and the classical random variable *X*′ is distributed according to the type of *x*^{n}. If $xn\u2208A\delta (pX)$, then

as well (see Ref. 35, Property 15.2.7). We note that the conditional entropy in the bounds above can also be expressed as $H(B|X\u2032)\sigma =1nH(Bn|Xn=xn)\sigma \u22611nH(Bn)\rho xn$.

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

##### a. *Classical Code Construction*

Select $2nR0$ independent sequences $x0n(m0)$, $m0\u2208[1:2nR0]$, at random according to $\u220fi=1npX0(x0,i)$. For every $m0\u2208[1:2nR0]$, select $2nR1$ conditionally independent sequences $x1n(m0,m1)$, $m1\u2208[1:2nR1]$, at random according to $\u220fi=1npX1|X0x1,i|x0,i(m0)$. Partition the set of indices $[1:2nR0]$ into $2nC12$ bins of equal size,

for $g\u2208[1:2C12]$.

##### b. *Encoding*

To send the message pair (*m*_{0}, *m*_{1}), Alice prepares $\rho An=\u2297i=1n\rho Ax1,i(m0,m1)$ and sends the block *A*^{n}. The resulting output state is

##### c. *Decoding*

Bob 1 receives the systems $B1n$ and decodes as follows:

Decode $m\u03020$ by applying a POVM ${\Lambda m00}m0\u2208[1:2nR0]$ to the systems $B1n$.

Decode $m\u03021$ by applying a second POVM ${\Lambda m1|m\u030201}m1\u2208[1:2nR1]$ to the systems $B1n$.

Choose

*g*to be the corresponding bin index such that $m\u03020\u2208B(g)$.Send the conference message

*g*to Bob 2.

The POVMs ${\Lambda m00}$ and ${\Lambda m1|m\u030201}$ will be specified later.

Bob 2 receives the systems $B2n$ and the conference message *g* and decodes by applying a POVM ${\Gamma m0}m0\u2208B(g)$, which will also be specified later, to the systems $B2n$.

##### d. *Analysis of Probability of Error*

Assume without loss of generality that Alice sends (*m*_{0}, *m*_{1}). Denote the decoding measurement outcomes by $M\u03020$, $M\u03021$, *G*, and $M\u03030$. Consider the following events:

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

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 $(X0n(m0),X1n(m0,m1))\u2208A\delta 2(pX0,X1)$. Now, observe that

for all $(x0n,x1n)\u2208A\delta /2(pX0,X1)$, by (B4), (B6), (B8), and (B9), respectively. By the quantum packing lemma^{73} (see Ref. 118, Lemma 3), there exists a POVM $\Lambda m\u030200$ such that

for all $m0\u2208[1:2nR0]$. Hence, $PrE2\u2223E1c\u22642\u2212n(I(X0,X1;B1)\rho \u2212(R0+R1)\u2212\epsilon 3(\delta ))$, which tends to zero as *n* → *∞*, provided that

Moving to the third term in the RHS of (B16), suppose that $E2c$ occurred, namely, the decoder measured the correct *M*_{0}. Denote the state of the systems $B1n$ after this measurement by $\rho B1n\u2032$. Then, as in Refs. 118 and 119, we observe that due to the packing lemma inequality (B21), the gentle measurement lemma^{120,121} implies that the post-measurement state is close to the original state in the sense that

for sufficiently large *n* and rates as in (B22). Therefore, the distribution of measurement outcomes when $\rho B1n\u2032$ measured is roughly the same as if the POVM $\Lambda m\u030200$ was never performed. To be precise, the difference between the probability of a measurement outcome $m\u03021$ when $\rho B1n\u2032$ is measured and the probability when $\rho B1n$ is measured is bounded by *ɛ*_{5}(*δ*) in absolute value (see Ref. 35, Lemma 9.11). Furthermore,

for all $(x0n,x1n)\u2208A\delta 2(pX0pX1|X0)$, by (B6)–(B9), respectively. Therefore, we have by the quantum packing lemma that there exists a POVM $\Lambda m\u03021|m01$ such that $PrE3\u2223E1c\u2229E2c\u22642\u2212n(I(X1;B1|X0)\rho \u2212R1\u2212\epsilon 7(\delta ))$, which tends to zero as *n* → *∞*, provided that

It remains to consider erroneous decoding by Bob 2. Suppose that $E3c$ occurred, namely, Bob 1 measured the correct *m*_{0}, and thus sent the correct bin index *G* such that $m0\u2208B(G)$. Recall that the size of each bin is $|B(G)|=2n(R0\u2212C12)$ [see (B10)]. Then, observe that

for all $x0n\u2208A\delta (pX0,X1)$, by (B4), (B6), (B8), and (B9), respectively. Hence, by the quantum packing lemma (Ref. 118, Lemma 3), there exists a POVM $\Gamma m\u03030$ such that $PrE4\u2223E1c\u2229E2c\u2229E3c\u22642\u2212n(I(X0;B2)\rho \u2212(R0\u2212C12)\u2212\epsilon 8(\delta ))$, which tends to zero as *n* → *∞*, provided that

To show that rate pairs in $1\kappa RCl(N\u2297\kappa )$ are achievable as well, one may employ the coding scheme above for the product broadcast channel $N\u2297\kappa $, 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 *m*_{0} and *m*_{1} uniformly at random and prepares an input state $\rho Anm0,m1$. After Alice sends the systems *A*^{n} through the channel, the output state is $\rho B1nB2n=12n(R0+R1)\u2211m0,m12nR0\u2211m1=12nR1NAn\u2192B1nB2n(\rho Anm0,m1)$. Then, Bob 1 performs a decoding POVM $\Lambda B1nm0,m1,g$, sends *g* to Bob 2 using conferencing, and Bob 2 chooses a POVM $\Gamma B2n|gm0$ accordingly. Consider a sequence of codes $(Fn,\Lambda n,\Gamma n)$ such that the average probability of error tends to zero; hence, the error probabilities $PrM\u03020\u2260M0$, $Pr(M\u03020,M\u03021)\u2260(M0,M1)$, and $PrM\u03021\u2260M1|M0$ are bounded by some *α*_{n}, which tends to zero as *n* → *∞*. By Fano’s inequality,^{122} it follows that

where *ɛ*_{n}, *ɛ*_{n}′, *ɛ*″_{n} tend to zero as *n* → *∞*. Hence,

where the second inequality follows from the Holevo bound (see Ref. 32, Theorem 12.1) and the last inequality holds as $I(M0;G|B2n)\rho \u2264H(G)\u2264nC12$ because *G* is a *classical* message in $[1:2nC12]$. Similarly,

Furthermore, since *M*_{0} and *M*_{1} are statistically independent, we can also write

We deduce that $R0\u2264I(X0n;B2n)\rho +C12+\epsilon n$, $R0+R1\u2264I(X0n,X1n;B1n)\rho +\epsilon n\u2032$, and $R1\u2264I(X1n;B1n|X0n)\rho +\epsilon n\u2033$, with $Xkn=fk(Mk)$, where *f*_{k} 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

Applying the chain rule to (B37),

Since the marginal of Bob 2 is degraded with respect to that of Bob 1, namely, $NA\u2192B2H=PY1\u2192B2\u25e6NA\u2192Y1H$, the data processing inequality for the quantum mutual information implies that $I(M0B2i\u22121;B2,i)\rho \u2264I(M0Y1i\u22121;B2,i)\rho =I(X0,i;B2,i)\rho $; hence,

where *K* is a classical random variable with uniform distribution over [1:*n*], independent of *M*_{0}, *M*_{1}, and *G*. Defining $\rho KA=1n\u2211i=1n|i\u3009\u3008i|\u2297\rho Ai$, $\rho KB1B2=NA\u2192B1B2\rho KA$, and

we obtain

and by similar considerations,

This completes the Proof of Theorem 2.