Broadcast Classical-Quantum Capacity Region of Two-Phase Bidirectional Relaying Channel

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by  Holger Boche

Noname manuscript No.
(will be inserted by the editor)
The broadcast classical-quantum capacity region of a
two-phase bidirectional relaying channel
Holger Boche · Minglai Cai · Christian
Deppe
thedateofreceiptandacceptanceshouldbeinsertedlater
5 Abstract We studied a three-node quantum network that enables bidirec-
1 tional communication between two nodes with a half-duplex relay node for
0
transmittingclassicalmessages.Adecode-and-forwardprotocolisusedtoper-
2
formthecommunicationintwophases.Inthefirstphase,themessagesoftwo

p
nodes are transmitted to the relay node. The capacity of the first phase is
e
well-knownby previous works. In thesecond phase, the relay node broadcasts
S
a re-encoded composition to the two nodes. We determine the capacity region
8
of the broadcast phase. To the best of our knowledge, this is the first paper
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analyzing quantum bidirectional relay networks.

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T Keywords Quantum information theory; Quantum network; Quantum relay
I channel; Quantum broadcast channel
.
s
c
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1 Introduction

4
v Thestudyofquantumchannelnetworkshasbecomemoreandmoreimportant
2
in the last few years.
8
Some of the first applications of this will be secret key transmission/gene-
6
3 ration and transmitting of secure messages over quantum networks. The ca-
. pacities for secret key transmission/generation and the secrecy capacities for
1
message transmission have been determined in [14] and [11].
0
4 For these applications, the transmitters have to solve two main problems.
1 First, the message (a secret key or a secure message) has to be encoded in
v: such a way that it can be decoded correctly by the legal receiver. Second,
i the message has to be encoded such that the wiretapper’s knowledge of the
X
transmitted classical message can be kept arbitrarily small.
r
a
LehrstuhlfürTheoretischeInformationstechnik,
TechnischeUniversitätMünchen,
Munich,Germany
E-mail:{boche,minglai.cai,christian.deppe}@tum.de
2 HolgerBocheetal.
Transmission of secret keys and secure messages over long distances is
an essential requirement for those applications. Thus, in our paper, we con-
sider the problem how to ensure that the legal receivers are able to reproduce
the original messages. This is a necessary condition for reliable secret key
transmissions/generations and transmitting of secure messages over quantum
networks.
Theproblemoflong-distancetransmissionsofquantuminformationisone
of the biggest problems in the realization of quantum networks. The sending
of photons in optical fibers is presently limited by 200 km because of losses
in the optical-fiber channel due to absorption on the way. One solution to
solve this problem is the development of quantum repeaters (cf. [10] and [1]).
Unfortunately, the practical realization of this component is not given until
now. The researchers already built quantum repeaters in their laboratories,
but until now it is not possible to extend the limit of 200 km.
Inthispaper,weusearelayinsteadofaquantumrepeater.Theadvantage
of this protocol is that it is realizable and enables quantum communication
betweentwopartiesoverthedoublelengthofthedistanceforsendingphotons
in optical fibers. This protocol can also be used for free space optical commu-
nications between satellites. The communications in several classical practical
applications such as satellite communication and cellular communication are
modeled with channels with relay nodes. Channel networks with relay nodes
have been studied extensively in the context of classical information theory
(cf. [24] and [13]). The study of quantum channels with relay nodes has just
recently begun (cf. [36]).
We analyze a quantum channel network model which was introduced for
classical channel networks in [33]. It is called the two-phase bidirectional re-
laying channel (Figure 1). In this model we consider a three-node quantum
network with two message sets M and M , which is called a two-user bidi-
1 2
rectional quantum channel. The message m ∈ M is located at node 1, and
2 2
themessagem ∈M islocatedatnode2,respectively,whilearelaynodeen-
1 1
ables the bidirectional communication between these nodes. We assume that
the relay node cannot transmit and receive data at the same time. This is
usually called a half-duplex relay. This assumption is reasonable for practical
components in communication systems in general.
Our goal is that after the transmission the message m ∈M is known at
2 2
node2andthemessagem ∈M isknownatnode1,respectively.Wesimplify
1 1
theproblembyassuminganaprioriseparationofthecommunicationintotwo
phases.
Fig. 1: Two-phase bidirectional relaying channels
BidirectionalRelayingClassical-QuantumChannel 3
There exist several strategies which are usually classified by the process at
the relay node, namely the entanglement swapping-and-forward strategy (cf.
[23]) and the decode-and-forward strategy. We consider a two-phase decode-
and-forward protocol. The relay node’s task is to decode the messages that it
receives in the first phase and to forward the information to its destinations
in the second phase. This basically means that in the first phase the relay
measures and decodes and that in the second phase the relay prepares and
encodes. The disadvantage of this strategy is that the coherence is destroyed.
The advantage of this strategy is that in the second phase each receiver can
useitsownmessagefromthefirstphaseassideinformationtogatherahigher
capacity.
The goal of the communication is the transmission of classical signals, be-
tweentwopartners.Theproblemoftheoptimaltransmissionoftheseclassical
signals can be divided into two parts:
1. Quantummodulation(i.e.,choosinganoptimalsetofpossibleinputstates
which will be the input alphabet. This is equivalent to consider a special
classical-quantum channel, i.e., a quantum channel depending on the set
of chosen input states whose sender’s inputs are classical variables.)
2. Optimal coding for the classical-quantum channel.
We also consider this problem because the model of classical-quantum
channels is a very important tool for understanding the capacity formulas
for quantum channels. The capacity of classical-quantum channels has been
determinedin[20],[21],and[41].Itturnsoutthatclassical-quantumchannels
arenotonlyoftheoreticalinterest,butaswellasoftechnicalinterest,too;for
instance,acodeforaclassical-quantumchannelcanbeusedforentanglement
generation (cf. [14]). Furthermore, new phenomenons such as super-additivity
andsuper-activationappearforcarryingclassicalinformationthroughaquan-
tum channel (cf. [8]).
We would like to point out that the three-node bidirectional relay network
is an extremely advantageous tool in the relay network theory. The model
of classical three-node bidirectional relay network has been extended to more
complexmodelssuchasMIMOchannels([30],[15],[37],[32]),Gaussianchan-
nels([37]),polarcodes([5]),andcross-layerdesigns([27],[28],[29]).Inviewof
these previous works on the classical bidirectional relay network, our further
tasks will be to analyze these models for quantum networks (cf. [22], [17], and
[43]).
For classical models, the authors of [33] use results from coding theory for
degradedbroadcastchannel(cf.[6]).Oneofourmajorchallengesinthispaper
is that there are no equivalent tools in the quantum information theory yet.
Thus, we can only establish the capacity region with average errors, but not
the capacity region with maximal errors as in the classical case (cf. Remark
2). It still reminds open how the capacity region with maximal errors can be
established.
4 HolgerBocheetal.
Recent research has been done toward security for classical bidirectional
relay networks (cf. [39] and [5]). It is a promising task to find similar results
for quantum networks.
Another important basic feature for classical bidirectional relay networks
isthechanneluncertainty([38],[31]).Thecapacityforquantumchannelswith
uncertainty has been determined in [4], [7], and [8].
Thedesignofcommunicationprotocolsforquantumnetworksisachalleng-
ing task. For the design of efficient protocols, it is important to incorporate
sideinformationintothecodingschemes.Inthispaper,weconsideredthecase
where both receivers have side information about the messages. [9] considered
recently the case where the transmitter has side information about the chan-
nel state. This corresponds to the classical “writing on dirty paper” coding.
This coding strategy plays an important role in modern communication sy-
stems. It is an interesting research topic to develop on the basic of [9] and the
bidirectional relaying protocol of this paper for more complicated quantum
networks.
Thestudyofrelaychannelsinthequantumscenarioisnovel.Wehopeour
results may raise interests in further analysis.
2 Basic definitions
2.1 A two-phase protocol
By the separation of the communication, we have a multiple-access phase,
where node 1 and node 2 transmit messages m and m to the relay node,
2 1
and a broadcast phase, where the relay forwards the messages to node 2 and
node 1, respectively. We look at the two phases separately.
Fig. 2: The multiple access phase
In the multiple-access phase, we have a classical-quantum multiple-access
channel. The multiple-access channel is a channel such that two (or more)
senders send information to a common receiver via this channel. The optimal
coding strategies and capacity regions for classical multiple-access channels
BidirectionalRelayingClassical-QuantumChannel 5
have been given in [3] and [25]. The optimal coding strategies and capacity
regionsformultiple-accessquantumchannelshavebeengivenin[45]and[46].
Fig. 3: The broadcast phase
Inthebroadcastphase,wehaveabroadcastquantumchannel.Inabroad-
cast channel, one single sender sends information to two (or more) receivers.
The optimal coding strategies and capacity regions for classical broadcast
channels have been given in [26], [6], and [18]. An optimal coding strategy
andacapacityregionforbroadcastquantumchannelshavebeengivenin[35].
For the broadcast phase, we assume that the relay node has successfully
decoded the messages m and m in the multiple-access phase. Of course, the
1 2
message m is also known at node 1 and the message m is also known at
2 2
node 2.
The goal of the relay node is to broadcast a message to node 1 and node 2
which allows both nodes to recover the unknown source. This means that
node1wantstorecovermessagem andthatnode2wantstorecovermessage
1
m .
2
2.2 Notations and communication scenarios
For finite-dimensional complex Hilbert spaces G and G(cid:48), a quantum channel
N: S(G)→S(G(cid:48)), S(G)(cid:51)ρ→N(ρ)∈S(G(cid:48)) is represented by a completely
positive trace-preserving map that accepts input quantum states in S(G) and
produces output quantum states in S(G(cid:48)). Here, S(G) stands for the space of
density operators on the space G.
If the sender wants to transmit a classical message of a finite set A to the
receiver using a quantum channel N, his encoding procedure will include a
classical-to-quantum encoder to prepare a quantum message state ρ ∈ S(G)
suitable as an input for the channel. If the sender’s encoding is restricted to
transmitanindexedfinitesetofquantumstates{ρ :x∈A}⊂S(G),thenwe
x
can consider the choice of the signal quantum states ρ as a component of the
x
channel. Thus, we obtain a channel σ := N(ρ ) with classical inputs x ∈ A
x x
6 HolgerBocheetal.
and quantum outputs, which we call a classical-quantum channel. This is a
mapN:A→S(G(cid:48)),A(cid:51)x→N(x)∈S(G(cid:48))whichisrepresentedbythesetof
|A| possible output quantum states {σ =N(x):=N(ρ ):x∈A} ⊂ S(G(cid:48)),
x x
meaningthateachclassicalinputofx∈Aleadstoadistinctquantumoutput
σ ∈S(G(cid:48)). In view of this, we have the following definition.
x
Let H be a finite-dimensional complex Hilbert space. A classical-quantum
channel is a map N :A→S(H), A(cid:51)a→N(a)∈S(H).
For a probability distribution P on a finite set A and a positive constant
δ, we denote the set of typical sequences by
(cid:26) (cid:12) (cid:12) (cid:27)
Tn := xn ∈An :(cid:12)(cid:12)1 (cid:104)xn |a(cid:105)−P(a)(cid:12)(cid:12)≤ δ ,
P,δ (cid:12)n (cid:12) |A|
where (cid:104)xn |a(cid:105) is the number of occurrences of the symbol a in the sequence
xn.
Let n ∈ N, we define An := {(a ,...,a ) : a ∈ A ∀i ∈ {1,...,n}}. The
1 n i
spacewhichthevectors{v ⊗...⊗v :v ∈H ∀i∈{1,...,n}}spanisdefined
1 n i
by H⊗n. We also write an for the elements of An.
Associatedwithaclassicalquantumchannel,N:A→S(H)isthechannel
maponthen-blockN⊗n:An →S(H⊗n)suchthatforan =(a ,...,a )∈An.
1 n
We have N⊗n(an)=N(a )⊗...⊗N(a ).
1 n
For a quantum state ρ ∈ S(G), we denote the von Neumann entropy of ρ
by
S(ρ)=−tr(ρlogρ) .
Let V: A → S(G) be a classical-quantum channel. For P ∈ P(A) the con-
ditional entropy of the channel for V with input distribution P is denoted
by
(cid:88)
S(V|P):= P(x)S(V(x)) .
x∈A
Remark 1 The following definition is a more general definition of the con-
ditional entropy in quantum information theory. Let P and Q be quantum
systems. We denote the Hilbert space of P and Q by GP and GQ, respec-
tively. Let φPQ be a bipartite quantum state in S(GPQ). We denote S(P |
Q) :=S(φPQ)−S(φQ). Here φQ =tr (φPQ).
ρ P
Let Φ := {ρ : x ∈ A} be a set of quantum states labeled by elements of
x
A. For a probability distribution P on A the Holevo χ quantity is defined as
(cid:32) (cid:33)
(cid:88) (cid:88)
χ(P;Φ):=S P(X)ρ − P(X)S(ρ ) .
x x
x∈A x∈A
We denote the identity operator on a space G by id .
G
A collection of positive semi-definite operators {M : i} on G is called a
i
positiveoperatorvaluedmeasure,orPOVM,ifitisapartitionoftheidentity,
(cid:80)
i.e., M =id .
i i G
BidirectionalRelayingClassical-QuantumChannel 7
2.3 Code concepts
A two-user multiple-access quantum channel N : HBC → HA has two
BC−A
senders B and C, and a single receiver A. It is defined as a map N :HBC →
HA.
An (n,J(1),J(2)) code carrying classical information for a two-user quan-
n n
tum multiple access channel N : HBC → HA consists of a ensemble
BC−A
of quantum states {w(m ) : m = 1,...,J(1)} ⊂ S(HB⊗n), quantum states
1 1 n
(cid:26)
{v(m ) : m = 1,...,J(2)} ⊂ S(HC⊗n), and a POVM D : m ∈
2 2 n m1,m2 1
(cid:27)
{1,...,J(1)},m ∈{1,...,J(2)} on HA⊗n.
n 2 n
A pair of nonnegative numbers (R ,R ) is an achievable rate pair with
1 2
classical inputs for the quantum multiple-access channel N : HBC →
BC−A
HA with average error if for every positive ε, δ, and a sufficiently large n
there is an (n,J(1),J(2)) code carrying classical information ({w(m ) : m =
n n 1 1
1,...,J(1)}, {v(m ) : m = 1,...,J(2)},{D : m ∈ {1,...,J(1)}, m ∈
n 2 2 n m1,m2 1 n 2
{1,...,J(2)}}) such that 1 logJ(1) ≥R −δ, 1 logJ(2) ≥R −δ and
n n n 1 n n 2
J(1) J(2)
1 (cid:88)n (cid:88)n tr(cid:0)(id −D )N⊗n ((w(m ),v(m )))(cid:1)≤ε .
J(1)J(2) HA⊗n m1,m2 BC−A 1 2
n n m1=1m2=1
(1)
Thetwo-userbroadcastquantumchannelN :HA →HBC isaquan-
A−BC
tum channel from a single sender A to two independent receivers B and C.
The quantum channel W from A to B is obtained by tracing out C from
1
the channel map, i.e., W = N : HA → HB, which is the quantum
1 A−B
channel from A to B, is defined as W (σ) = tr (N (σ)). Furthermore,
1 C A−BC
W = N : HA → HB, which is the quantum channel from A to C, is
2 A−C
defined as W (σ)=tr (N (σ)).
2 B A−BC
An (n,J(1),J(2)) code carrying classical information for a two-user broad-
n n
castquantumchannelN :HA→HBC consistsofaensemble{w((m ,m )):
A−BC 1 2
(cid:110) (cid:111)
m =1,...,J(1),m =1,...,J(2)}⊂S(A⊗n),aPOVM D(1) :m ∈{1,...,J(1)}
1 n 2 n m1 1 n
(cid:110) (cid:111)
on HB⊗n, and a POVM D(2) :m ∈{1,...,J(2)} on HC⊗n.
m2 2 n
A pair of nonnegative numbers (R ,R ) is an achievable rate pair with a
1 2
classical input for the two-user broadcast quantum channel N : HA →
A−BC
HBC withaverageerrorifforeverypositiveε,δ,andasufficientlylargenthere
is an (n,J(1),J(2)) code carrying classical information ({w (j) : j},{D : j})
n n t j
such that 1 logJ(1) ≥R −δ, 1 logJ(2) ≥R −δ, for every m ∈M
n n 1 n n 2 2 2
J(1)
1 (cid:88)n (cid:16) (cid:17)
tr (id −D(1))W⊗n(w((m ,m ))) ≤ε , (2)
J(1) HB⊗n m1 1 1 2
n j=1
8 HolgerBocheetal.
and for every m ∈M
1 1
J(2)
1 (cid:88)n (cid:16) (cid:17)
tr (id −D(1))W⊗n(w((m ,m ))) ≤ε , (3)
J(2) HC⊗n m2 2 1 2
n j=1
where W =N and W =N .
1 A−B 2 A−C
The capacity regions of multiple-access quantum channels and broadcast
quantum channels are convex by the time-sharing principle: Let (R ,R ) and
1 2
(R(cid:48),R(cid:48))beratetuplesofmandnblockcodes,respectively,witherrorproba-
1 2
bilities (cid:15) and (cid:15) , respectively. We get an (m+n) block code with error prob-
1 2
(cid:16) (cid:17)
ability at most (cid:15) +(cid:15) and with rates m R + n R(cid:48), m R + n R(cid:48)
1 2 m+n 1 m+n 1 m+n 2 m+n 2
by concatenating the code words to (m+n) blocks and tensoring the corre-
sponding decoding observables.
3 The classical-quantum capacity region of the bidirectional
relaying quantum channel
For the multiple-access phase, the optimal coding strategy is well known
from [45], where the following lemma for the classical-quantum rate region
of multiple-access quantum channels was given.
Lemma 1 Let N be a two-user multiple-access quantum channel. Let
Y2Y1−X
HY1 be the Hilbert space whose unit vectors correspond to the pure states of
node 1’s quantum system, HY2 be the Hilbert space whose unit vectors corre-
spond to the pure states of node 2’s quantum system, and HX be the Hilbert
spacewhoseunitvectorscorrespondtothepurestatesoftherelaynode’squan-
tum system.
We assume node 1’s encoding is restricted to transmitting an indexed finite
set of orthogonal quantum states Y1 ⊂HY1.
We assume node 2’s encoding is restricted to transmitting an indexed finite
set of orthogonal quantum states Y2 ⊂HY2.
Theclassical-quantumcapacityregionofthemultiple-accessquantumchan-
nel N with average error is given by the set of all rate pairs (R ,R ),
Y2Y1−X 2 1
satisfying
R ≤χ(Q ;σX) , (4)
2 1
R ≤χ(Q ;σX) , (5)
1 2
and
R +R ≤χ(Q ;σX) (6)
2 1 1,2
for any joint probability distribution Q on Y ×Y . Here, Q is the marginal
1,2 1 2 1
probability distribution of Q on Y , Q is the marginal probability distribu-
1,2 1 2
tion of Q on Y , and σX is the resulting quantum state at the outcome of
1,2 2
the relay node.
BidirectionalRelayingClassical-QuantumChannel 9
Thus, if n is sufficiently large and if for M and M it holds
1 2
|M |≤(cid:98)2n(χ(Q1;σX)−(cid:15))(cid:99) ,
2
|M |≤(cid:98)2n(χ(Q2;σX)−(cid:15))(cid:99) ,
1
and
|M |+|M |≤(cid:98)2n(χ(Q1,2;σX)−(cid:15))(cid:99)
2 1
forsomepositive(cid:15),wecanassumethattherelaynodehassuccessfullydecoded
the messages m ∈M and m ∈M .
1 1 2 2
Note that the author of [45], using block codes and showing a weak con-
verse,isabletogive(4),(5),and(6)insingleletterformula.Pleasesee[9]for
more discussions on the value of multiletter formulas for quantum communi-
cation networks.
For the broadcast phase, since each node has perfect knowledge about the
message intended for the other, one can use this knowledge as a support for
the choice of the decoding strategy to decode the message intended for itself.
In view of these facts, we have the following Theorem 1.
Theorem 1 Let N be a two-user bidirectional quantum channel. Let HY1 be
the Hilbert space whose unit vectors correspond to the pure states of node 1’s
quantumsystem,HY2 betheHilbertspacewhoseunitvectorscorrespondtothe
pure states of node 2’s quantum system, and HX be the Hilbert space whose
unit vectors correspond to the pure states of the relay node’s quantum system.
Let N be the broadcast quantum channel in the broadcast phase.
X−Y1Y2
We assume that the relay node’s encoding is restricted to transmitting an
indexed finite set of orthogonal quantum states {φ :x∈X}⊂HX.
x
For all probability distribution P on X, the capacity region of the bidi-
rectional broadcast quantum channel N during the broadcast phase for
X−Y1Y2
transmitting classical information with average error is given by the set of all
rate pairs (R ,R ), satisfying
1 2
R ≤limsup 1χ(Pn;σY1⊗n) (7)
1 n
n→∞
and
R ≤limsup 1χ(Pn;σY2⊗n) . (8)
2 n
n→∞
Here, σY1 is the resulting quantum state at the outcome of node 1, while σY2
is the resulting quantum state at the outcome of node 2.
Proof Itiseasytoverifythateveryachievableratepaircannotexceed(7)and
(8). R1 cannot exceed limsupn→∞ n1χ(Pn;σY1⊗n), even if the relay node only
sendsamessagetonode1withoutsendinganymessagetonode2(cf.[21]).For
thesamereason,R2 cannotexceedlimsupn→∞ n1χ(Pn;σY2⊗n)either.Nowwe
will prove the achievability of the extremal point of the rate region given by
10 HolgerBocheetal.
(7) and (8), since then every rate pair in the rate region is achievable by the
time-sharing principle.
At first, we present some tools which that were used for our proof:
Let H be a Hilbert space. For ρ∈S(H) and α>0 there exists an orthog-
onal subspace projector Π commuting with ρ⊗n and satisfying
ρ,α
tr(cid:0)ρ⊗nΠ (cid:1)≥1− d , (9)
ρ,α 4nα2
√
tr(Π )≤2nS(ρ)+Kdα n , (10)
ρ,α
√
Π ·ρ⊗n·Π ≤2−nS(ρ)+Kdα nΠ , (11)
ρ,α ρ,α ρ,α
where d:=dimH and K is a positive constant (cf. [42]).
Let A be a finite set and let V : A → S(H) be a classical-quantum channel.
For a probability distribution P on A, α > 0, and xn ∈ Tn there exists an
P
orthogonal subspace projector Π (xn) commuting with V⊗n and satisfying
V,α xn
tr(cid:0)V⊗n(xn)Π (xn)(cid:1)≥1− ad , (12)
V,α 4nα2
√
tr(Π (xn))≤2nS(V|P)+Kadα n , (13)
V,α
Π (xn)·V⊗n(xn)·Π (xn)
V,α V,α
√
≤2−nS(V|P)+Kadα nΠ (xn) , (14)
V,α
where a:=#{A} and K is a positive constant (cf. [42]).
Let V : A → S(H) be a classical-quantum channel. Then, every probability
distribution P on A defines a quantum state PV on S(H), which is the re-
sulting quantum state at the output of V when the input is sent according to
P.Thus,forα(cid:48) >0wecandefineanorthogonalsubspaceprojectorΠ √
√PV,α(cid:48) a
which fulfills (9), (10), and (11) (here, we set ρ = PV and α = α(cid:48) a). Fur-
thermore, for Π √ , we have the following inequality
PV,α(cid:48) a
tr(cid:0)V⊗n(xn)·Π √ (cid:1)≥1− ad , (15)
PV,α(cid:48) a 4nα2
where K is a positive constant (cf. [42]).
Lemma 2 (Measurement on Approximately Close States, cf. [42])
Let σ and ρ be two quantum states, and let Π be a positive operator such that
Π ≤id; then,
tr(Πσ)≥tr(Πρ)−(cid:107)σ−ρ(cid:107) .
1
Lemma 3 (Tender Operator, cf. [44] and [34]) Letρbeaquantumstate.
Let X be a positive operator such that X ≤id and 1−tr(ρX)≤λ≤1; then,
√ √ √
(cid:107)ρ− Xρ X(cid:107)≤ 8λ . (16)