Classical Information Capacity of the Bosonic Broadcast Channel
Classical Information Capacity of the Bosonic
Broadcast Channel
Saikat Guha
Research Laboratory of Electronics
Massachusetts Institute of Technology
Cambridge, MA 02139
saikat@MIT.edu
Jeffrey H. Shapiro
Research Laboratory of Electronics
Massachusetts Institute of Technology
Cambridge, MA 02139
jhs@MIT.edu
Abstract— We show that when coherent-state encoding is
employed in conjunction with coherent detection, the Bosonic
broadcast channel is equivalent to a classical degraded Gaussian
broadcast channel whose capacity region is dual to that of the
classical Gaussian multiple-access channel. We further show that
if a minimum output-entropy conjecture holds true, then the
ultimate classical information capacity of the Bosonic broadcast
channel can be achieved by a coherent-state encoding. We provide
some evidence in support of the conjecture.
I. INTRODUCTION
The past decade has seen several advances in evaluating
classical information capacities of several important quantum
communication channels [1]–[5]. Despite the theoretical ad-
vances that have resulted [1], exact capacity results are not
known for many important and practical quantum communi-
cation channels. Here we extend the line of research aimed
at evaluating capacities of Bosonic communication channels,
which began with the capacity derivation for the input photon-
number constrained lossless Bosonic channel [2], [3]. The
capacity of the lossy Bosonic channel was found in [4], where
it was shown that a modulation scheme using classical light
(coherent states) suffices to achieve ultimate communication
rates over this channel. Subsequent attempts to evaluate the
capacity of the noisy Bosonic channel with additive Gaussian
noise [5] led to a crucial conjecture on the minimum output
entropy of a class of Bosonic channels [6]. Proving that
conjecture would complete the capacity proof for the Bosonic
channel with additive Gaussian noise, and it would show that
this channel’s capacity is achievable with classical-light mod-
ulation. More recent work that addressed Bosonic multiple-
access communication channels [7] revealed that modulation
of information using non-classical states of light is necessary
to achieve ultimate single-user rates. In the present work,
we study the classical information capacity of the Bosonic
broadcast channel. A broadcast channel is the congregation of
communication media connecting a single transmitter to two or
more receivers. In general, the transmitter encodes and sends
out independent information to each receiver in a way that
each receiver can reliably decode its respective information.
In Sec. II, we describe some recent work on the capacity
region of the degraded quantum broadcast channel [8]. In
Sec. III, we introduce the noiseless Bosonic broadcast chan-
nel model, and derive its capacity region subject to a new
minimum output entropy conjecture. In Sec. IV we show that
a recent duality result between capacity regions of classical
multiple-input, multiple-output Gaussian multiple-access and
broadcast channels [9] does not hold for Bosonic channels.
II. QUANTUM DEGRADED BROADCAST CHANNEL
A quantum channel NA−B from Alice to Bob is a trace-
preserving completely positive map that maps Alice’s single-
use density operators ρ̂A to Bob’s, ρ̂B = NA−B(ρ̂A). The two-
user quantum broadcast channel NA−BC is a quantum channel
from sender Alice (A) to two independent receivers Bob (B)
and Charlie (C). The quantum channel from Alice to Bob is
obtained by tracing out C from the channel map, i.e.,NA−B ≡
TrC (NA−BC), with a similar definition for NA−C . We say
that a broadcast channel NA−BC is degraded if there exists a
degrading channel N degB−C from B to C satisfying NA−C =
N degB−C ◦ NA−B . The degraded broadcast channel describes
a physical scenario in which for each successive n uses of
NA−BC Alice communicates a randomly generated classical
message (m, k) ∈ (WB ,WC) to Bob and Charlie, where the
message-sets WB and WC are sets of classical indices of
sizes 2nRB and 2nRC respectively. The messages (m, k) are
assumed to be uniformly distributed over (WB ,WC). Because
of the degraded nature of the channel, Bob receives the entire
message (m, k) whereas Charlie only receives the index k.
To convey these message (m, k), Alice prepares n-channel
use states that after transmission through the channel, result
in bipartite conditional density matrices
, ∀(m, k) ∈
(WB ,WC). The quantum states received by Bob and Charlie,{
respectively, can be found by tracing
out the other receiver, viz., ρ̂B
m,k ≡ TrCn
, etc. A
(2nRB , 2nRC , n, �) code for this channel consists of an encoder
xn : (WB ,WC)→ An, (1)
a positive operator-valued measure (POVM) {Λmk} on Bn
and a POVM {Λ′k} on C
n which satisfy1
ρ̂xn(m,k)(Λmk ⊗ Λ′k)
≥ 1− � (2)
1An, Bn, and Cn are the n channel use alphabets of Alice, Bob, and
Charlie, with respective sizes |An|, |Bn|, and |Cn|.
for every (m, k) ∈ (WB ,WC). A rate-pair (RB , RC) is
achievable if there exists a sequence of (2nRB , 2nRC , n, �n)
codes with �n → 0. The classical capacity region of the
broadcast channel is defined as the convex hull of the closure
of all achievable rate pairs (RB , RC). The classical capacity
region of the two-user degraded quantum broadcast channel
NA−BC was recently derived by Yard et. al. [8], and can be
expressed in terms of the Holevo information [10],
χ(pj , σ̂j) ≡ S
pj σ̂j
pjS(σ̂j), (3)
where {pj} is a probability distribution associated with the
density operators σ̂j , and S(ρ̂) ≡ −Tr(ρ̂ log ρ̂) is the von
Neumann entropy of the quantum state ρ̂. Because χ may
not be additive, the rate region (RB , RC) of the degraded
broadcast channel must be computed by maximizing over
successive uses of the channel, i.e., for n uses
pj|i,N⊗nA−B(ρ̂
pj|iρ̂
pipj|iS
) , and (4)
RC ≤ χ
pi,∑
pj|iN⊗nA−C(ρ̂
 /n,
pipj|iρ̂
pj|iρ̂
 , (5)
where j ≡ (m, k) is a collective index and the states
live in the Hilbert space H⊗n of n successive uses of the
broadcast channel. The probabilities {pi} form a distribution
over an auxiliary classical alphabet T , of size |T |, satisfying
|T | ≤ min
|An|, |Bn|2 + |Cn|2 + 1
. The ultimate rate-
region is computed by maximizing the region specified by
Eqs. (4) and (5), over {pi},
, and n, subject to
the cardinality constraint on |T |. Fig. 1 illustrates the setup of
the two-user degraded quantum channel.
III. NOISELESS BOSONIC BROADCAST CHANNEL
The two-user noiseless Bosonic broadcast channel NA−BC
consists of a collection of spatial and temporal Bosonic
modes at the transmitter (Alice), that interact with a minimal-
quantum-noise environment and split into two sets of spatio-
temporal modes en route to two independent receivers (Bob
and Charlie). The multi-mode two-user Bosonic broadcast
channelNA−BC is given by
sNAs−BsCs , whereNAs−BsCs
Fig. 1. Schematic diagram of the degraded single-mode Bosonic broadcast
channel. The transmitter Alice (A) encodes her messages to Bob (B) and
Charlie (C) in a classical index j, and over n successive uses of the channel,
prepares a bipartite state ρ̂B
j for them.
is the broadcast-channel map for the sth mode, which can be
obtained from the Heisenberg evolutions
b̂s =
ηs âs +
1− ηs ês, and (6)
ĉs =
1− ηs âs −
ηs ês, (7)
where {âs} are Alice’s modal annihilation operators, and {b̂s},
{ĉs} are the corresponding modal annihilation operators for
Bob and Charlie, respectively. The modal transmissivities {ηs}
satisfy 0 ≤ ηs ≤ 1, ∀s, and the environment modes {ês} are
in their vacuum states. We will limit our treatment here to the
single-mode Bosonic broadcast channel, as the capacity of the
multi-mode channel can in principle be obtained by summing
up capacities of all spatio-temporal modes and maximizing the
sum capacity region subject to an overall input-power budget
using Lagrange multipliers, cf. [5], where this was done for the
capacity of the multi-mode single-user lossy Bosonic channel.
The principal result we have for the single-mode degraded
Bosonic broadcast channel depends on a minimum output
entropy conjecture (the strong form of Conjecture 2, see
Appendix). Assuming this conjecture to be true, we have
that the ultimate capacity region of the single-mode noiseless
Bosonic broadcast channel (see Fig. 2) with a mean input
photon-number constraint 〈â†â〉 ≤ N̄ is
RB ≤ g(ηβN̄), and (8)
RC ≤ g((1− η)N̄)− g((1− η)βN̄), (9)
for 0 ≤ β ≤ 1, where g(x) = (1 + x) log(1 + x) − x log(x).
This rate region is additive and achievable with single channel
use coherent-state encoding with the distributions
pT (t) =
, and (10)
pA|T (α|t) =
1− β t− α|2
. (11)
Proof — It is straightforward to show that if η > 1/2, the
Bosonic broadcast channel is a degraded quantum broadcast
channel, in which Bob’s is the less-noisy receiver and Charlie’s
is the more-noisy receiver. Yard et al.’s capacity region in
Eqs. (4) and (5) requires finite-dimensional Hilbert spaces.
Fig. 2. A single-mode noiseless Bosonic broadcast channel can be envisioned
as a beam splitter with transmissivity η. With η > 1/2, the Bosonic broadcast
channel reduces to a ‘degraded’ quantum broadcast channel, where Bob (B) is
the less-noisy receiver and Charlie (C) is the more noisy (degraded) receiver.
Nevertheless, we will use their result for the Bosonic broadcast
channel, which has an infinite-dimensional state space, by
extending it to infinite-dimensional state spaces through a
limiting argument.2 The n = 1 rate-region for the Bosonic
broadcast channel using a coherent-state encoding is thus:
pT (t)S
pA|T (α|t)|
η α〉〈
η α| dα
dt (12)
RC ≤ S
pT (t)pA|T (α|t)|
1− η α〉〈
1− η α| dα dt
pT (t)S
pA|T (α|t)
1− η α〉〈
1− η α| dα
dt, (13)
where we need to maximize the bounds for RB and RC over
all joint distributions pT (t)pA|T (α|t) subject to 〈|α|2〉 ≤ N̄ .
Note that A and T are complex-valued random variables, and
the second term in the RB bound (4) vanishes, because the von
Neumann entropy of a pure state is zero. Substituting Eqs. (10)
and (11) into Eqs. (12) and (13), shows that the rate-region
Eqs. (8) and (9) is achievable using single-use coherent state
encoding.
For the converse, assume that the rate pair (RB , RC) is
achievable. Let {xn(m, k)}, and POVMs {Λmk} and {Λ′k}
comprise any (2nRB , 2nRC , n, �) code in the achieving se-
quence. Suppose that Bob and Charlie store their decoded
messages in the classical registers ŴB and ŴC respectively.
Let us use pWB ,WC (m, k) = pWB (m)pWC (k) to denote the
joint probability mass function of the independent message
registers WB and WC . As (RB , RC) is an achievable rate-
2When |T | and |A| are finite, and we are using coherent states, there will be
a finite number of possible transmitted states, which leads to a finite number
of possible states received by Bob and Charlie. Suppose we limit the auxiliary-
input alphabet (T )—and hence the input (A) and the output alphabets (B and
C)—to truncated coherent states within the finite-dimensional Hilbert space
spanned by the Fock states {|0〉, |1〉, . . . , |K〉}, where K � N̄ . Applying
Yard et al.’s theorem to the Hilbert space spanned by these truncated coherent
states then gives us a broadcast channel capacity region that must be strictly an
inner-bound of the rate-region given by unconditional equations (12) and (13).
For made K sufficiently large, while maintaining the cardinality condition,
the rate-region expressions given by Yard et. al.’s theorem will converge to
Eqs (12) and (13).
pair, there must exist �′n → 0, such that
nRC = H(WC)
≤ I(WC ; ŴC) + n�′n
≤ χ(pWC (k), ρ̂
k ) + n�
n, (14)
where I(WC ; ŴC) ≡ H(ŴC) − H(ŴC |WC) is the Shan-
non mutual information, and ρ̂C
m pWB (m)ρ̂
m,k. The
second line follows from Fano’s inequality and the third line
follows from Holevo’s bound3. Similarly, for an �′′n → 0, we
can bound nRB as
nRB = H(WB)
≤ I(WB ; ŴB) + n�′′n
≤ χ(pWB (m), ρ̂
m ) + n�
pWC (k)χ(pWB (m), ρ̂
m,k) + n�
n, (15)
where the three lines above follow from Fano’s inequality,
Holevo’s bound and the concavity of Holevo information. In
order to prove the converse, we now need to show that there
exists a number β ∈ [0, 1], such that∑
pWC (k)χ(pWB (m), ρ̂
m,k) ≤ ng(ηβN̄),
and χ(pWC (k), ρ̂
k ) ≤ ng((1− η)N̄)− ng((1− η)βN̄).
From the non-negativity of the von Neumann entropy
, it follows that
k pWC (k)χ(pWB (m), ρ̂
m,k) ≤∑
k pWC (k)S
m pWB (m)ρ̂
, as the second term of
the Holevo information above is non-negative. Because the
maximum von Neumann entropy of a single-mode Bosonic
state with 〈â†â〉 ≤ N̄ is given by g(N̄), we have that
0 ≤ S
ηN̄kj
, (16)
where, N̄k ≡
N̄kj , and N̄kj is the mean photon
number of the jth symbol ρ̂
k of the n-symbol codeword
k , for j ∈ {1, . . . , n}. Therefore, ∃βk ∈ [0, 1], ∀k ∈ WC ,
such that
ηβkN̄k
. (17)
Because of the degraded nature of the channel, Charlie’s state
can be obtained as the output of a beam splitter whose input
states are Bob’s state (coupling coefficient η′ = (1 − η)/η
to Charlie) and a vacuum state (coupling coefficient 1 − η′
to Charlie). It follows, from assuming the truth of Strong
conjecture 2 (see Appendix), that
(1− η)βkN̄k
. (18)
N̄ is the average number of photons per-use at the transmitter
(Alice) averaged over the entire codebook. Thus, the mean
3Holevo’s bound [10]: Let X be the input alphabet for a channel, {pi, ρ̂i}
the priors and modulating states, {Πj} be a POVM, and Y the resulting
output (classical) alphabet. The Shannon mutual information I(X;Y ) is upper
bounded by the Holevo information χ(pi, ρ̂i)
photon-number of the n-use average codeword at Bob, ρ̂B
k pWC (k)ρ̂
k , is ηN̄ . Hence,
pWC (k)S
≤ S(ρ̂B
) ≤ ng
, (19)
where the second inequality follows from the convexity of
von Neumann entropy. The monotonicity of g(x) then implies
that there is a β ∈ [0, 1], such that
k pWC (k)S
ng(ηβN̄). Hence we have,∑
pWC (k)χ(pWB (m), ρ̂
m,k) ≤ ng(ηβN̄). (20)
for some β ∈ [0, 1]. Equation (17), and the uniform distribu-
tion pWC (k) = 1/2
nRC imply that∑
ηβkN̄k
. (21)
Using (21), the convexity of g(x), and η > 1/2, we have
shown (proof omitted) that∑
(1− η)βkN̄k
(1− η)βN̄
. (22)
From Eq. (22), and Eq. (18) summed over k, we then obtain∑
pWC (k)S
≥ ng((1− η)βN̄). (23)
Finally, writing Charlie’s Holevo information as
χ(pWC (k), ρ̂
k ) = S
pWC (k)ρ̂
pWC (k)S
≤ ng((1− η)N̄)
pWC (k)S
, (24)
we can use Eq. (23) to get
χ(pWC (k), ρ̂
k ) ≤ ng((1− η)N̄)− ng((1− η)βN̄), (25)
which completes the proof.
IV. DISCUSSION AND CONCLUSION
Recently, Vishwanath et. al. [9] established a duality be-
tween the dirty paper achievable region (recently proved to be
the ultimate capacity region [11]) for the classical multiple-
input, multiple-output (MIMO) Gaussian broadcast channel
and the capacity region of the MIMO Gaussian multiple-access
channel (MAC). The duality result states that if we evaluate
the capacity regions of the MIMO Gaussian MAC—with fixed
total received power P and channel-gain values—over all
possible power-allocations between the users, the corners of
those capacity regions trace out the capacity region of the
MIMO Gaussian broadcast channel with transmitter power
P and the same channel-gain values. Unlike this classical
result, the capacity region of the Bosonic broadcast channel
using coherent-state inputs is not equal to of the envelope of
Fig. 3. Comparison of Bosonic broadcast and multiple-access channel
capacity regions, in bits per channel use, for η = 0.8, and N̄ = 15. The red
line is the conjectured ultimate broadcast capacity region, which lies below
the green line—the envelope of the MAC capacity regions.
the MAC capacity regions using coherent-state inputs. The
capacity region of the Bosonic MAC using coherent-state
inputs was first computed by Yen [7]. In Fig. 3 we compare
the envelope of coherent-state MAC capacities to the capacity
region of the coherent-state broadcast channel. This figure
shows that with a fixed beam splitter and identical average
photon number budgets, more collective classical information
can be sent when the beam splitter is used as a multiple-access
channel as opposed to when it is used as a broadcast channel.
The broadcast channel capacity region that we have
derived—modulo Strong conjecture 2—exceeds what can be
accomplished with conventional optical receivers, as shown in
Fig. 4. In this figure we compare the capacity regions attained
by a coherent-state input alphabet using homodyne detection,
heterodyne detection, and optimum reception. As is known for
single-user Bosonic communications, homodyne detection per-
forms better than heterodyne detection when the transmitters
are starved for photons, because it has lower noise. Conversely,
heterodyne detection outperforms homodyne detection when
the transmitters are photon rich, because it has a factor-of-two
bandwidth advantage. To bridge the gap between the coherent-
detection capacity regions and the ultimate capacity region,
one must use joint detection over long codewords. Future
investigation will need to be done to realize better broadcast
communication rates over the Bosonic broadcast channel.
ACKNOWLEDGMENT
This research was supported by the Defense Advanced
Research Projects Agency. The authors thank Baris Erkmen for
helpful discussions and for proving the Gaussian-state version
of Strong conjecture 2.
APPENDIX: MINIMUM OUTPUT ENTROPY CONJECTURES
Let â and b̂ denote the two input modes of a lossless
beam splitter of transmissivity η, to produce output modes
η â +
1− η b̂ and d̂ =
1− η â −
η b̂. In [6], we
proposed the following minimum output entropy conjecture:
Fig. 4. Comparison of Bosonic broadcast channel capacity regions, in bits per
channel use, achieved by coherent-state encoding with homodyne detection
[red, circles], heterodyne detection [blue, dashed], and optimum reception
[black, solid], for η = 0.8, and N̄ = 1, 5, and 15.
Conjecture 1 — Let the input b̂ be in a zero-mean thermal
state with von Neumann entropy S(ρ̂B) = g(K). Then the von
Neumann entropy of output ĉ is minimized when â is in the
vacuum state, and the minimum output entropy is g((1−η)K).
In this paper, we propose a new output entropy conjecture:
Conjecture 2 — Let the input â be in its vacuum state,
input b̂ in a zero-mean state with von Neumann entropy
S(ρ̂B) = g(K). Then the von Neumann entropy of output ĉ
is minimized when b̂ is in a thermal state with average photon
number K, and the minimum output entropy is g((1− η)K).
For the capacity proof of the Bosonic broadcast channel,
we use Strong conjecture 2, which we now describe. Let the
input modes {âi : 1 ≤ i ≤ n} be in a product state of n
vacuum states, and let the von Neumann entropy of the joint
state of the inputs {b̂i : 1 ≤ i ≤ n} be ng(K). Then, putting
{b̂i : 1 ≤ i ≤ n} in a product state of mean-photon-number K
thermal states minimizes the output von Neumann entropy of
the joint state of {ĉi : 1 ≤ i ≤ n}. Moreover, this minimum
output entropy is ng((1− η)K).
Previous work has provided considerable evidence in sup-
port of Conjecture 1 [6], [12]. In particular, we know that
Conjecture 1 is true: when the state of â is Gaussian; when
Wehrl entropy4 is considered instead of von Neumann entropy;
and when Rényi entropy of integer order n ≥ 2 is considered
instead of von Neumann entropy. Strong conjecture 1, i.e., the
n-use version, has been proven: when the joint state of the
{âi} is Gaussian [13]; and when Wehrl entropy is considered
instead of von Neumann entropy. Other evidence in support
of Conjecture 1 has been developed from entropy bounds [6],
which show that the conjecture is asymptotically correct in the
limit of weak and strong noise, and from simulated annealing
starting with randomly selected initial states.
In unpublished work, we have shown that Conjecture 2 is
true: when the state of b̂ is Gaussian; when Wehrl entropy is
considered instead of von Neumann entropy; and when the
state of b̂ is mixed and diagonal in the Fock basis with a
probability distribution that is either Poisson, Binomial, or
Bose-Einstein. For Strong conjecture 2 we have shown that
it is true: when the {b̂i} are in a Gaussian state; and when
Wehrl entropy is considered instead of von Neumann entropy.
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4The Wehrl entropy of a state with density operator ρ̂ is the differential
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	Introduction
	Quantum Degraded Broadcast Channel
	Noiseless Bosonic Broadcast Channel
	Discussion and Conclusion
	References
