An improved rate region for the classical-quantum broadcast channel

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by  Christoph Hirche

An improved rate region for the classical-quantum
broadcast channel
Christoph Hirche∗ and Ciara Morgan†
Institut fu¨r Theoretische Physik
Leibniz Universita¨t Hannover
Appelstraße 2, D-30167 Hannover, Germany
Email: ∗[email protected]†[email protected]
5
1
0
2 Abstract—We present a new achievable rate region for the Broadcastchannelshavealsobeengeneralizedtothesetting
two-userbinary-inputclassical-quantumbroadcast channel.The of classical-quantum communication [6], [7], [8]. To date
n resultisageneralizationoftheclassicalMarton-Gelfand-Pinsker
the best known rate region for the classical-quantum channel
a region and is provably larger than the best previously known
J has been established in [7], [8] and it is a generalization of
rate region for classical-quantumbroadcast channels. Theproof
9 of achievabilityisbased on therecentlyintroducedpolarcoding Marton’s region in the classical case.
2 scheme and its generalization to quantum network information Arikan [9] recently introduced the now celebrated polar
theory. coding scheme for classical channels. Indeed Arikan showed

h] Index Terms—Broadcast channel, polar codes, achievability that these codes can achieve the symmetric capacity of any
p classical single-sender single-receiver channel in the limit of
t- I. INTRODUCTION many channel uses and, remarkably, that this can be done
n with a complexity O(NlogN) for encoding and decoding,
a Oneofthefundamentaltasksinquantuminformationtheory where N is the number of channels used for communication.
u
is to determine the maximum possible rate at which informa- Moreover, polar codes make use of the effect of channel
q
[ tioncanbesentreliablyfromonepartytoanotheroveranoisy polarization, where a recursive construction is used to divide
communicationchannel.Indeedto showtheachievabilityofa the instances of a channel into a fraction that can be used for
1
certain rate, one needs to prove the existence of a code, that reliable communication and a fraction that is nearly useless.
v
is, an encoding and decoding scheme, that achieves this rate The crucial feature is that the fraction of good channels is
7
1 with vanishing error in the limit of many channel uses. approximatelyequaltothesymmetriccapacityofthechannel.
4 Networkinformationtheorycentersonthestudyandanaly- Polar codes have attracted great deal of attention and were
7 sisofcommunicationratesinthemulti-usersetting,generaliz- generalized to many additional communication settings, such
0
ingthesinglesenderandreceivercase.Thebroadcastchannel as the task of source coding [10], [11] and universal coding
.
1 is one the most fundamental channels in this field and, in for compound channels [12], [13].
0
the two-user case, it models the simultaneous communication Polar codes have also been generalized to the setting of
5
between a single sender and two receivers. In the classical sending classical information over quantum channels [14],
1
: setting, there exist several schemes to prove that certain rate [15], [16], in addition to sending quantum information [17],
v
regionsareachievableforthebroadcastchannel.Twoofthese [18], [19]. For the task of sending classical information, in
i
X schemes, which are of particular interest to us, are known additiontoasymmetricchannels[20],thequantumsettinghas
r as superposition coding [1], [2] and binning [3], also called also been generalized to certain multi-user channels, namely
a
multicoding. For the binning scheme, independent messages the multiple access and interference channels and compound
are sent simultaneously to both receivers and can be decoded multipleaccesschannel [21],[20].Recentlypolarcodeshave
by the respective receiver, according to an argument based alsobeenappliedtotheclassicalbroadcastchannel[22],[23].
on joint typicality. For the superposition scheme we exploit In [23] the authors show how polar codes can be used for
the fact that certain inputs are decodable by both receivers. the broadcast channel to achieve the Marton-Gelfand-Pinsker
Marton [3] combinedboth techniquesin orderto send private region with and without common messages.
messages over the two-user broadcast channel, achieving the In this work we will show that the approachof [23] can be
regionnowknownasMarton’sregion.Laterthisresultwasex- used to achieve the Marton-Gelfand-Pinsker region with and
tended by Gelfand and Pinsker [4] where a common message without common messages for classical-quantum broadcast
canbesenttobothreceivers,resultingintheso-calledMarton- channels, giving rise to the largest known rate region for the
Gelfand-Pinskerregionwith commonmessages. Interestingly, classical-quantum broadcast channel.
it can be shown, in the classical setting, that even when we The remainder of this work is organized as follows. In
set the common rate in the Marton-Gelfand-Pinskerregion to Section II we will state the necessary preliminaries and in
zero the resulting rate region is, in some cases, larger than Section III we show how to achieve the Marton-Gelfand-
Marton’s region [5]. Pinsker region for the broadcast channel using polar codes,
before we conclude in Section IV. We will now define the two-user classical-quantum broad-
cast channel. The broadcast channel can be modeled mathe-
II. NOTATION AND DEFINITIONS matically as the triple ,W, B1 B2 , with
X H ⊗H
(cid:0) (cid:1)
We begin by introducing certain notation which will be W :x ρB1B2. (6)
usedthroughoutthearticle,beforedefiningnecessaryentropic → x
quantities and measures. The information processing task for the two-user classical-
In the remaining work uN uN will denote a row vector quantumbroadcastchannelisdescribedasfollows.Thesender
(u ,...,u ) and correspo1nd≡ingly uj will denote, for 1 would like to communicatemessages to both receivers.These
i,j1 N, aNsubvector (u ,...,u ). Noite that if j <i then u≤j messagesareindependent,butcanalsocontainsomecommon
≤ i j i
isempty.Similarly,foravectoruN andasetA 1,...,N , part for both receivers. The model is such that the first
1 ⊂{ }
we write uA to denote the subvector (ui :i A). receiveronlyhasaccesstotheoutputsystemB1 andtherefore
A discrete classical-quantum channel W t∈akes realizations receives ρBx1 = TrB2ρBx1B2, similarly, the second receiver
xρBx∈, oXn aoffianitrea-nddiommenvsaiorinaablleHiXlbetrot sapaqcueanHtuBm, state, denoted fhoarseρaBxch2 =recTeirvBe1rρfrxBo1mB2a. TmheesssaegnedseertcMhooks=es{a1,m·e·s·s,a2gneRmk}k,
and encodes her messages with the resulting the codeword
W :x→ρBx, (1) xn(m1,m2) ∈ Xn. The receivers’ corresponding decoding
POVMs are denoted by Λ and Γ . The code is said
where each quantum state ρ is described by a positive { m1} { m2}
x to be a (n,R ,R ,ǫ)-code, if the average probabilityof error
semi-definite operator with unit trace. We will take the input 1 2
is bounded as follows
alphabet = 0,1 unless otherwise stated, and the tensor
X { } 1
product W⊗N of N channels is denoted by WN. p¯ = p (m ,m ) ǫ, (7)
e e 1 2
≤
To characterize the behavior of symmetric classical- |M1||M2|mX1,m2
quantumchannels,wewillmakeuseofthesymmetricHolevo
where the probability of error p (m ,m ) for a pair of
capacity, defined as follows: e 1 2
messages (m ,m ) is given by
1 2
I(W)≡I(X;B)ρ, (2) p (m ,m )=Tr (I Λ Γ )ρB1nB2n , (8)
where the quantum mutual information with respect to a e 1 2 n − m1 ⊗ m2 xn(m1,m2)o
classical-quantum state ρXB is given by with ρB1nB2n the state resulting when the sender transmits
thecodxenw(mo1rd,mx2n)(m ,m )throughninstancesofthechannel.
1 2
I(X;B) H(X) +H(B) H(XB) , (3)
ρ ρ ρ A rate pair (R ,R ) is said to be achievable for the two-user
≡ − 1 2
classical-quantum broadcast channel described above if there
withρXB = 1 0 0 ρB+1 1 1 ρB.Intheabove,thevon
2| ih |⊗ 0 2| ih |⊗ 1 exists an (n,R ,R ,ǫ)-code ǫ>0 and large enough n.
NeumannentropyH(ρ)isdefinedasH(ρ)≡−Tr{ρlog2ρ}. 1 2 ∀
We will also make use of the conditional entropy defined as
A. Polar codes for asymmetric channels
H(X B) =H(X) H(XB) andthequantumconditional
ρ ρ ρ
| − In this section we will review polar codes for achieving
mutual information defined for a tripartite state ρXYB as
the capacity of asymmetric channels, for more details we
I(X;B Y) H(XY) +H(YB) H(Y) H(XYB) .
ρ ρ ρ ρ ρ
| ≡ − − referto [20]. Essentially polar codesare describedby a linear
WecharacterizethereliabilityofachannelW asthefidelity
transformationgiven by xN =uNG , where uN is the input
between the output states N
sequence and
F(W) F(ρ0,ρ1), (4) GN =BNF⊗n (9)
≡
with F(ρ0,ρ1) ≡ k√ρ0√ρ1k21 and kAk1 ≡ Tr√A†A. Note with 1 0
that in the case of two commuting density matrices, ρ = F , (10)
p i i and σ = q i i the fidelity can be written ≡(cid:20)1 1(cid:21)
i i| ih | i 2i| ih |
Pas F(ρ,σ) = i√pPiqi . Note that, the Holevo capacity and BN is a permutation matrix known as a “bit reversal”
and the fidelity(cid:0)cPan be see(cid:1)n as quantumgeneralizationsof the operation[9].Thisiscalledchannelcombiningandtransforms
mutual information and the Bhattacharya parameter from the N single copies of a channel W to a channel W .
N
classical setting, respectively (see, e.g., [9]). In the second step, called channel splitting, W is used to
N
We will also use the quantity define W(i) as follows:
N
Z(X|B)ρ ≡2√p0p1F(ρ0,ρ1), (5) WN(i) :ui →ρ(Ui1)i−,u1iBN, (11)
introducedin [19], for a classical-quantum state ρ which can, where
again, be seen as quantum generalization of the Bhattacharya
parameter, for a classical variable, now with quantum side ρU1i−1BN = 1 ui−1 ui−1 1 ρBN. (12)
(i),ui 2i−1 | 1 ih 1 |⊗ 2N−i uN
information. uXi−1 uXN
1 i+1
This is equivalent to a decoder which estimates, by the i- III. BROADCAST CHANNEL
th measurement, the bit ui, with the following assumptions: A. Marton-Gelfand-Pinskerregion for private messages
the entire output is available to the decoder, the previous bits
In this section we will show how to achieve the Marton-
ui−1 are correctly decoded and the distribution over the bits
1 Gelfand-Pinsker region, initially without the use of common
uN is uniform. The assumptions that all previous bits are
i+1 messagesforclassical-quantumbroadcastchannelsusingpolar
correctly decoded is called “genie-aided” and can be ensured
codes. Indeed, we will use the technique of alignment as
by a limited amount of classical communication prior to the
explained in Section II-B to achieve the rate region
informationtransmission.Thedecoderdescribedaboveisthus
a “genie-aided” successive cancellation decoder. R1 I(V,V1;B1),
≤
These two steps give rise to the effect of channel po- R I(V,V ;B )
2 2 2
larization, which ensures that the fraction of channels W(i) ≤
which have the property I(W(i)) (1 δ,1] goes to tNhe R1+R2 ≤I(V,V1;B1)+I(V2;B2|V)−I(V1;V2|V),
symmetric Holevo informationNI(W∈) and−the fraction with R1+R2 ≤I(V,V2;B2)+I(V1;B1|V)−I(V1;V2|V),
(14)
I(W(i)) [0,1 δ) goes to 1 I(W) for any δ (0,1),
N ∈ − − ∈ for the classical-quantum two-user broadcast channel de-
as N goes to infinity through powers of two. This is one
scribed by a classical input X =ϕ(V,V ,V ) and a quantum
of the main insights of the work by Arikan [9] and the 1 2
generalization in [15] to the classical-quantum setting (see output ρxB1B2.
Let V,V ,V be auxiliary binary random variables with
[15] for a more detailedstatement). To achievethe symmetric 1 2
(V,V ,V ) p p p . Now, let X = ϕ(V,V ,V )
capacity we can now simply send information bits over the 1 2 ∼ V V2|V V1|V2V 1 2
channels I(W(i)) (1 δ,1] and send prearranged “frozen” be a deterministic function. Without loss of generality we
N ∈ − consider a broadcast channel such that I(V;B ) I(V;B ).
bits over the remaining channels. 1 ≤ 2
With G the usual polar coding transformation, set
It turns out that, the above approach essentially works for n
asymmetric channels as well, the crucial point to see this, Un =VnG , (15)
(0) n
is that the polar coding transform G is its own inverse
N Un =VnG , (16)
for binary inputs. We consider the reverse protocol of clas- (1) 1 n
sical lossless compression. Hence we can use a uniformly U(n2) =V2nGn. (17)
distributed input sequence and transform it to a distribution The variable Un carries the message of the first user, while
(1)
suitable to achieve the asymmetric capacity of the channel. Un andUn carrythemessageoftheseconduser.Toexploit
(0) (2)
DuetoapolarizationeffectwecanuseafractionofsizeH(X) the technique of superposition, we take Un to be decodable
0
forthefollowingchannelcoding.Notethatinthespecialcase by both receivers, while V carries information only for the
of a symmetric channel the uniformity of the required input secondreceiver.ThevariablesV ,V correspondtothebinning
1 2
distribution simply gives a fraction H(X)=1. It is shown in schemeandcanonlybedecodedbyonereceiver,respectively.
[20] that this approach achieves the asymmetric capacity To handle these additional auxiliary variables we use the
C(W)=maxI(X;B). (13) polarizationtechniqueusedtoachievetheasymmetriccapacity
p(x) of a channel introduced in Section II-A. Hence we introduce
This is a generalization of a result in [24] to the classical- setstodeterminethepolarizationoftheprobabilitydistribution
quantum setting. for the input and the channel independently. Define for l
∈
1,2 ,thefollowingsets,withinterpretationsprovidedbelow,
B. Alignment of polarized sets { }
From the definition of polar codes, it is clear that the set HV ={i∈[n]:Z(U(0),i |U(n0−)1)≥δn},
of channels which can be used for information transmission = i [n]:Z(U Un−1) δ ,
depend on the particular communication channel to be used. LV { ∈ (0),i | (0) ≤ n}
H = i [n]:Z(U Un−1Bn ) δ ,
Polarizing,inascenariowithmultiplepossiblechannels,such V|Bl { ∈ (0),i | (0) (l) ≥ n}
as the case of compound channels, where one must code at = i [n]:Z(U Un−1Bn ) δ ,
LV|Bl { ∈ (0),i | (0) (l) ≤ n}
rates which are achievable for all channels in a particular
H = i [n]:Z(U Un−1Un ) δ ,
known a set of channels, will hence, lead to the situation Vl|V { ∈ (l),i | (l) (0) ≥ n}
wheresomesynthesizedchannelsaregoodforonechannelbut LVl|V ={i∈[n]:Z(U(l),i |U(nl)−1U(n0))≤δn},
not for another, and vice versa. This problem can be solved H = i [n]:Z(U Un−1Un Bn ) δ ,
by the technique of alignment [12], which is described in Vl|V,Bl { ∈ (l),i | (l) (0) (l) ≥ n}
detail for the classical-quantum compound channel in [20]. LVl|V,Bl ={i∈[n]:Z(U(l),i |U(nl)−1U(n0)B(nl))≤δn},
The main idea is to combine the channels which are good in H = i [n]:Z(U Un−1Un Un ) δ ,
one case with the channels good in the other by additional V1|V,V2 { ∈ (1),i | (1) (0) (2) ≥ n}
= i [n]:Z(U Un−1Un Un ) δ ,
CNOT gates.Doingthisrecursively,wecan halvethe number LV1|V,V2 { ∈ (1),i | (1) (0) (2) ≤ n}
of incompatible indices in every step. With the number of which due to the polarization effect satisfy
channels uses approaching infinity, we can minimize the 1
number of incompatible indices. lim HV =H(V),
n→∞n| |
1
lim V =1 H(V), know that whenever I(V;B1) is not equal to I(V;B2) there
n→∞n|L | − will be unaligned indices remaining. Lets call this set (2).
lim 1 H =H(V B ), In the second step we choose a subset (1) of (1B) such
n→∞n| V|Bl| | l that (1) = (2) . We can then align thesBe two suIbsets and
1 |B | |B |
lim =1 H(V B ), therefore raise the number of indices from Un , which both
n→∞n|LV|Bl| − | l (0)
receivers can decode, to I(V;B ).
1 2
nl→im∞n|HVl|V|=H(Vl|V), In the third step we need to cope with the fact that the first
user cannot decode the informations in F(1). Again choose a
1
nl→im∞n|LVl|V|=1−H(Vl|V), subsetRbin ofI(1) suchthat|Rbin|=|F(1)|.WeuseRbin to
1 repeattheinformationforthefirstuserinF(1) ofthefollowing
nl→im∞n|HVl|V,Bl|=H(Vl|V,Bl), block.
1 In order to get the correct order for the successive cancel-
nl→im∞n|LVl|V,Bl|=1−H(Vl|V,Bl), lation decoder, we will encode U(n0) and U(n2) forward, while
1 Un will be decoded backwards. Moreover, the first receiver
nl→im∞n|HV1|V,V2|=H(V1|V,V2), de(c1)odes U(n0) and U(n1) forwards, while the second receiver
1 decodes Un and Un backwards.
lim =1 H(V V,V ). (0) (2)
n→∞n|LV1|V,V2| − 1| 2 Now if we let the number of blocks approach infinity, we
can calculate the rate for the first receiver as follows
Note that the polarization of the classical quantities follows
1
from polar codes for classical source coding [10], while the R = ( (1) (1) )
polarization of the quantities with quantum side information 1 n |I |−|B |−|Rbin|
is shown in [19]. =I(V1;B1 V) I(V1;V2 V) (I(V;B2) I(V;B2))
| − | − −
Intuitively H and describe the polarization of the =I(V,V ;B ) I(V ;V V) I(V;B ).
V V 1 1 1 2 2
L − | −
random variable V and correspond to whether or not the ith
We can also calculate the rate for the second receiver
bit is nearly completely deterministic given the previous bits,
1
respectively. Similarly H and determine whether R = ( (2) + (2) )
V|Bl LV|Bl 2 n |Isup| |Ibin|
the lth receiver can decode bits knowing the previous inputs
=I(V;B )+I(V ;B V)
saanmdealilntoeruptpreuttast.ioHn,Vwl|Vith, LthVel|aVd,diHtioVnl|aVl,Bsild,eLiVnlf|oVr,mBlathioanvefrothme =I(V,V22;B2). 2 2 |
first decoding V. HV1|V,V2 and LV1|V,V2 handle the indices Finallynotethatifweswaptheroleofthetworeceivers,the
decoded by the first user, with assumed knowledge of V and set (2) will beemptyduetothe assumptionthatI(V;B )
1
V2, while in our case that user does not have access to the I(VB;B2), therefore we can achieve the rates ≤
latter.
Recall that Un can be decoded by both users but only R1 =I(V,V1;B1) (18)
(0)
contains information for the second one. Define s(u2)p = R2 =I(V2;B2 |V)−I(V1;V2 |V). (19)
HV ∩LV|B2 to contain positions decodable for theIsecond For the classical case it is known [23] that these two rate
user and (1) = H those decodable for the first pairs coincide with the Marton-Gelfand-Pinsker rate region.
Iv V ∩LV|B1
user. Again, Un can only be decoded by the second user The proof can be directly translated to the setting of classical
(2)
and also only contains information for this receiver. Indeed quantum communication and therefore our scheme achieves
by (2) =H we denote the set of indices which the rate region stated in Equation 14.
Ibin V2|V ∩LV|B2
can be decoded by the second receiver. Un only needs to be
(1) B. Marton-Gelfand-Pinskerregion with common messages
decodedbythefirstuserandalsoonlycontainsinformationfor
We can simply extend our coding scheme in Section III-A
that user. The set (1) = H denotes the indices
I V1|V ∩LV|B1 to include the transmission of a common message for both
which the first receiver can decode reliably. We also have to
receivers, by noting that the information sent via Un can
take into account that the first user cannot decode Un there- (0)
(2) be reliably decoded by both users. Therefore we can use
foretheindicesinthesetF(1) = H H
LV1|V,V2∩ V1|V∩ V1|V,B1 these indices to send an amount of information equal to
are critical.
min I(V;B ),I(V;B ) tobothusers.Thisleadstotherate
1 2
We now use, in total, three different steps of alignment { }
region
construction as described in Section II-B and illustrated in
Figure III-A. First we handle Un . By definition these vari- R0 min I(V;B1),I(V;B2) ,
(0) ≤ { }
ables should be decoded by both receivers and contain infor- R +R I(V,V ;B ),
0 1 1 1
≤
mation only for the second one. We simply use the alignment R +R I(V,V ;B )
0 2 2 2
techniquefor classical-quantumchannelsto send the message ≤
R +R +R I(V,V ;B )+I(V ;B V) I(V ;V V),
assigned for the second receiver to both of them. By the 0 1 2 ≤ 1 1 2 2| − 1 2|
R +R +R I(V,V ;B )+I(V ;B V) I(V ;V V).
assumption that I(V;B ) I(V;B ) we conclude that we 0 1 2 2 2 1 1 1 2
1 2 ≤ | − |
≤ (20)
can reliably send I(V;B ) of information to both users. We
1
Is(2u)p Is(2u)p Is(2u)p
B(2) B(2) B(2)
U(0)
Iv(1) Iv(1) Iv(1)
F(1) F(1) F(1)
U(1)
I(1) B(1) Rbin I(1) B(1) Rbin I(1) B(1) Rbin
U(2)
I(2) I(2) I(2)
bin bin bin
Fig.1. Codingforthebroadcast channel. Indices indotted subsetsareconsidered tobegoodforaspecific receiver denoted bytheassociated set.Arrows
indicate thealignment process.Foracolored versionofthisfiguresee[20].
IV. CONCLUSION [9] E.Arikan, “Channel polarization: A method forconstructing capacity-
achievingcodesforsymmetricbinary-inputmemorylesschannels,”IEEE
We have shown that, using polar coding, it is possible to TransactionsonInformationTheory,vol.55,no.7,pp.3051–3073,July
achieve a new rate region for classical-quantum broadcast 2009,arXiv:0807.3917.
[10] ——,“Sourcepolarization,” Jan.2010,arXiv:1001.3087.
channel, which coincides with the classical Marton-Gelfand-
[11] ——, “Polar coding forthe Slepian-Wolf problem based onmonotone
Pinsker region and is larger than the previously known rate chainrules,”Proceedingsofthe2012IEEEInternationalSymposiumon
regions for this channel. Hence this work gives an example Information Theory,pp.566–570,July2012.
[12] S. H. Hassani and R. Urbanke, “Universal polar codes,” July 2013,
where polar coding can be used to prove the achievability of
arXiv:1307.7223.
new rate regions. [13] H. Mahdavifar, M. El-Khamy, J. Lee, and I. Kang, “Compound polar
codes,”February2013,arXiv:1302.0265.
ACKNOWLEDGMENTS [14] J. M. Renes, F. Dupuis, and R. Renner, “Efficient polar coding of
quantum information,” Physical Review Letters, vol. 109, no. 5, p.
We would like to thank Mark M. Wilde for enjoyable and 050504,August2012,arXiv:1109.3195.
[15] M.M.WildeandS.Guha,“Polarcodesforclassical-quantumchannels,”
fruitfuldiscussions.ThisworkwassupportedbytheEUgrants
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SIQS and QFTCMPS and by the cluster of excellence EXC 1187,February2013,arXiv:1109.2591.
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