Unconstrained distillation capacities of a pure-loss bosonic broadcast channel

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by  Masahiro Takeoka

Unconstrained distillation capacities of a pure-loss
bosonic broadcast channel
Masahiro Takeoka∗, Kaushik P. Seshadreesan†, and Mark M. Wilde‡
∗ Quantum ICT Laboratory, National Institute of Information and Communications Technology, Koganei, Tokyo 184-8795, Japan
† Max-Planck-Institut fu¨r die Physik des Lichts, 91058 Erlangen, Germany
‡ Hearne Institute for Theoretical Physics, Department of Physics and Astronomy,
Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA
6
1
0
2 Abstract—Bosonic channels are important in practice as they implementations of QKD to date are known to exhibit a rate-
formasimplemodelforfree-spaceorfiber-opticcommunication. loss tradeoff, in which the rate of secret key extraction drops
r
p Here we consider a single-sender multiple-receiver pure-loss withincreasingdistance[8].Inthecaseofthestandardoptical-
A bosonic broadcast channel and determine the unconstrained
fiber communication channel, the drop is exponential with
capacity region for the distillation of bipartite entanglement
7 and secret key between the sender and each receiver, whenever increasing distance.
they are allowed arbitrary public classical communication. We Some time after these limitations were observed, Refs. [9],

] show how the state merging protocol leads to achievable rates [10] provided a mathematical proof, using the notion of
h in this setting, giving an inner bound on the capacity region.
squashed entanglement [11], that the tradeoff is indeed a
p We also determine an outer bound on the region and find that
fundamental limitation even with unconstrained input energy.
- the outer bound matches the inner bound in the infinite-energy
nt limit, thereby establishing the unconstrained capacity region for One of the main results of [9], [10] is an upper bound on the
a such channels. Our result could provide a useful benchmark two-way LOCC assisted quantum and secret key agreement
u forimplementingabroadcastingofentanglementandsecretkey capacity of a pure-loss bosonic channel, which is solely a
q through such channels. An important open question relevant to
function of the channel transmittance η. If the transmitter
[ practice is to determine the capacity region in both this setting
can consume only a finite amount of energy (as is in some
andthesingle-sendersingle-receivercasewhenthereisanenergy
3 constraint on the transmitter. practical cases), then tighter bounds are available [9], [10].
v As a consequence, no yet-to-be-discovered protocol could
3 I. INTRODUCTION ever surpass the limitations established in [9], [10]. Ref. [12]
6
Quantum key distribution (QKD) [1], [2] and entanglement extendedthesquashedentanglementtechniquetoobtainupper
5
5 distillation (ED) [3] are two cornerstones of quantum com- bounds for a variety of phase-insensitive Gaussian channels.
0 munication. QKD enables two or more cooperating parties to Concurrently with [12], Ref. [13] improved the infinite-
1. distill and share unconditionally secure random bit sequences, energy bound from [9], [10] and conclusively established the
0 which could then be used for secure classical communication. unconstrained capacity of the pure-loss bosonic channel as
6 ED, on the other hand, allows them to distill pure maximal C(η)=−log (1−η).Itisstillanopenquestiontodetermine
2
1
entanglement from a quantum state shared via a noisy com- the constrained capacity (i.e., when the transmitter is limited
:
v munication channel, which could then be used to faithfully to consuming finite energy).
i transfer quantum states by using quantum teleportation [4]. In One of the long-term goals of quantum communication
X
both protocols, the parties are allowed to perform (in prin- is to establish a quantum internet [14]: a large collection
r
a ciple) an unlimited amount of local operations and classical of interconnected quantum networks between multiple users
communication (LOCC). that enables secure classical communication and distributed
Quantum communication technologies have matured quantum information processing. Apart from point-to-point
tremendously in recent years. In particular, QKD has been links, network architectures based on single-sender multiple-
available commercially for a number of years and has now receivers(modeledasbroadcastchannels)andviceversa(mul-
expanded to inter-city networks [5], [6]. Also, efforts are tiple access channels) are also important in this context. Even
currentlyunderwaytoaccomplishQKDinfreespacebetween though various network quantum communication scenarios
the earth and satellites [7]. havebeenexamined[15],[16],[17],[18],[19],[20],therehas
Quantumcommunication,however,facesanimportantchal- been limited work on the capacity of entanglement and secret
lenge. Like most other quantum technologies, its performance key distillation assisted by unlimited LOCC. Only recently in
is affected by noise. Loss is the main source of error in [21]wereouterboundsontheachievableratesestablishedfor
typical optical communication channels and severely limits multipartitesecretkeyagreementandentanglementgeneration
the rates and distances at which secret key or quantum between any subset of the users of a general single-sender m-
entanglement can be distilled using the channel. All practical receiver quantum broadcast channel (QBC) (for any m ≥ 1)
when assisted by unlimited LOCC between all the users. The A rate tuple (E , E , K , K ) is achievable if for
AB AC AB AC
main idea was to employ multipartite generalizations of the allε∈(0,1)andsufficientlylargen,thereexistsan(n,E ,
AB
squashedentanglement[22],[23]andthemethodsof[9],[10]. E ,K ,K ,ε)protocoloftheaboveform.Thecapacity
AC AB AC
In this paper, we consider a single-sender multiple-receiver region is the closure of the set of all achievable rates.
pure-loss bosonic QBC and establish the unconstrained ca- In the following, we concentrate on a specific channel: a
pacity region for the distillation of bipartite entanglement pure-loss bosonic QBC which we denote by L . For
A(cid:48)→BC
and secret key between the sender and each receiver assisted thischannel,theinputstateissplitintothreesystemsandone
by unlimited LOCC. To prove the statement, we establish system is sent to each of Bob, Charlie, and the environment
inner bounds on the achievable rate region by employing with transmittance η , η , and 1 − η − η , respectively,
B C B C
the quantum state merging protocol [24], [25]. The converse where η ,η ∈[0,1], η +η ≤1.
B C B C
part relies upon several tools. First, we utilize a teleportation Physically the signal splitting is modeled by a pair of two-
simulation argument originally introduced in [26, Section V] input two-output beam splitters in which the signal is mixed
and recently generalized in [13] to wider families of channels with the vacuum state. For example, one can construct such a
and continuous-variable systems. Next, it is known that the channelbyasequenceoftwobeamsplitterswithtransmittance
relative entropy of entanglement is an upper bound on the η + η and η /(η + η ), respectively, where the first
B C B B C
distillable key of a bipartite state [27], and the recent work beam splitter splits the signal and the environment, and the
in [13] stated how it is possible to combine the relative second one splits the signal into Bob’s and Charlie’s parts
entropy of entanglement upper bound with the teleportation (see Fig. 1(b)). Mathematically this is characterized by the
simulation argument to arrive at upper bounds on the secret- following input-output relation:
keyagreementcapacityofcertainsingle-sendersingle-receiver (cid:115)
√ η (1−η −η ) (cid:114) η
channels. We find that the outer bounds match the inner ˆb= η aˆ(cid:48)+ B B C fˆ+ C gˆ, (1)
bounds in the infinite-energy limit, thereby establishing the B ηB +ηC ηB +ηC
unconstrained capacity region. An important open question is (cid:115)
√ η (1−η −η ) (cid:114) η
to determine the constrained capacity region, i.e., when only cˆ=− η aˆ(cid:48)− C B C fˆ+ B gˆ, (2)
C η +η η +η
finite energy is available at the transmitter. B C B C
Thepaperisorganizedasfollows.InSectionII,wedescribe eˆ=−(cid:112)1−η −η aˆ(cid:48)+√η +η fˆ, (3)
B C B C
a general LOCC-assisted distillation protocol for a QBC and
where aˆ(cid:48), ˆb, cˆ, eˆ are annihilation operators for Alice’s input,
the mathematical and physical model of the pure-loss bosonic
Bob’s output, and Charlie’s output modes, respectively, and fˆ
QBC.TheunconstrainedcapacityregionisgiveninSectionIII
and gˆ are annihilation operators for vacuum inputs from the
along with a proof for the single-sender two-receiver case.
environment.
Section IV concludes the paper. The appendix generalizes the
Critical for our analysis is that the physical implementation
main theorem to the single-sender multiple-receiver case.
of L is not unique. One can model the same channel
A(cid:48)→BC
II. LOCC-ASSISTEDDISTILLATIONPROTOCOLANDTHE by a different concatenation of two other beam splitters: for
CHANNELMODEL example, we could have a first beam splitter split system
In the main text, we consider a single-sender two-receiver B from C and E, and then a second one split C and E.
QBC N (Fig. 1(a)) and an (n, E , E , K , Obviously, it is also possible to split C at the first beam
A(cid:48)→BC AB AC AB
K ,ε)protocoldescribedasfollows(theappendixconsiders splitter. These physical models are described in Fig. 1(c) and
AC
the generalization to multiple receivers). The sender, Alice, (d), respectively. In the next section, we will use these other
preparessomequantumsystemsinaninitialquantumstateand physical models to explicitly calculate the rate regions.
successivelysendssomeofthesesystemstothereceivers,Bob
III. UNCONSTRAINEDCAPACITYREGION
and Charlie, by interleaving n channel uses of the broadcast
Our main contribution is the following theorem:
channel with rounds of LOCC. The goal of the protocol is
Theorem 1: TheLOCC-assisted,unconstrainedcapacityre-
to distill bipartite maximally entangled states Φ and Φ
AB AC gion of the pure-loss bosonic QBC L is given by
andprivatestatesγ andγ (equivalentlysecretkeys[28]). A(cid:48)→BC
AB AC
After each channel use, they can perform an arbitrary number E +K ≤log ([1−η ]/[1−η −η ]), (4)
AB AB 2 C B C
of rounds of LOCC (in any direction with any number of
E +K ≤log ([1−η ]/[1−η −η ]), (5)
parties). The quantities E and E denote entanglement AC AC 2 B B C
AB AC
E +K +E +K ≤−log (1−η −η ). (6)
rates (i.e., the logarithm of the Schmidt rank of Φ and AB AB AC AC 2 B C
AB
ΦAC,respectively,normalizedbythenumberofchanneluses) See the appendix for a generalization of this theorem to the
and KAB and KAC are secret-key rates (i.e., the number of multiple-receiver model from [29].
secret-key bits in γ , and γ , respectively, normalized by
AB AC
A. Achievability part
the number of channel uses). The protocol considered here is
similar to the one described in [21], except that here we do Toachievetherateregion(4)–(6),weconsideradistillation
not consider the other possible rates E , K , E , and protocolwhichemploysquantumstatemerging.Statemerging
BC BC ABC
K . was introduced in [24], [25] and provides an operational
ABC
B
A’
C
E
(a) (b) Since one “ebit” of entanglement can generate one private bit
of secret key, the left-hand side of the above inequalities can
B be modified as E →E +K , E →E +K .
A’ AB AB AB AC AC AC
The right-hand side of these inequalities can be explicitly
C
calculated.RecallthatthemarginaloftheTMSVΨ (N )=
A(cid:48) S
Tr [|Ψ(N )(cid:105)(cid:104)Ψ(N )| ] is a thermal state with mean pho-
A S S AA(cid:48)
ton number N . Its entropy is equal to H(A(cid:48)) = g(N ),
S Ψ S
(c) (d)
where g(x) = (x+1)log (x+1)−xlog x. Also a pure-
2 2
loss channel with transmittance η maps a thermal state to
another thermal state with reduced average photon number.
Let UL be an isometric extension of L and let
A(cid:48)→BCE A(cid:48)→BC
|φ(cid:105) =UL |Ψ(N )(cid:105) , (11)
ABCE A(cid:48)→BCE S AA(cid:48)
be a purification of φ . By using |φ(cid:105) and observing
Fig. 1. (a) Single-sender two-receiver quantum broadcast channel. (b)-(d) ABC ABCE
the above facts, we have
Variousphysicalimplementationsofthepure-lossbosonicbroadcastchannel
withtransmittancesηB andηC.
−H(B|AC) =H(AC) −H(ABC)
φ φ φ
=H(BE) −H(E)
φ φ
meaningfortheconditionalquantumentropy.ForastateρAB, =g((1−ηC)NS)−g((1−ηB −ηC)NS).
its conditional quantum entropy is defined as H(A|B)ρ = (cid:16) (cid:17)
H(AB)ρ−H(B)ρ where H(AB)ρ and H(B)ρ are the quan- In the limit as NS → ∞, this converges to log2 1−1η−Bη−CηC .
tum entropies of ρ and its marginal ρ , respectively. For Similarly, we obtain
AB B
manycopiesofρ sharedbetweenAliceandBob,H(A|B) (cid:18) (cid:19)
AB ρ 1−η
is the optimal rate at which maximally entangled two-qubit −H(C|AB)φ →log2 1−η −Bη , (12)
statesneedtobeconsumedtotransferAlice’ssystemstoBob’s B C
(cid:18) (cid:19)
1
side via LOCC. If H(A|B)ρ is negative, the result is that −H(BC|A)φ →log2 1−η −η , (13)
after transferring Alice’s systems, they can gain (i.e., distill) B C
entanglement at rate −H(A|B)ρ. State merging also yields inthelimitofinfinitelylargeNS.Thus(4)–(6)areachievable
a quantum analog of the Slepian-Wolf theorem in classical when there is no energy constraint on the transmitter.
distributed compression problem and has been applied to the
QBC in [19], [20]. B. Converse part
Here we consider the following alternative state merging As stated at the end of Section I, the converse relies upon
based protocol. Alice first prepares n copies of a two-mode several tools and is given in terms of the relative entropy of
squeezed vacuum (TMSV) entanglement (REE) [31]. The REE for a quantum state ρ
AB
∞ is defined by
(cid:88) (cid:112)
|Ψ(N )(cid:105) = λ (N )|m(cid:105) |m(cid:105) , (7)
S AA(cid:48) m S A A(cid:48)
E (A;B) = inf D(ρ (cid:107)σ ) (14)
R ρ AB AB
m=0 σAB∈SEP
where |m(cid:105) is an m-photon state, λ (N ) =
m S whereD(ρ(cid:107)σ)=Tr[ρ(log ρ−log σ)]isthequantumrelative
Nm/(N +1)m+1, and N is the average photon number of 2 2
S S S entropy and SEP denotes the set of separable states. The orig-
the state per mode. She sends system A(cid:48) to Bob and Charlie
inal LOCC-assisted communication protocol can equivalently
through a pure-loss broadcast channel. After n uses of the
be rewritten by using a teleportation simulation argument [26,
channel, Alice, Bob, and Charlie share n copies of the state
Section V] suitably extended to continuous-variable bosonic
φ =L (|Ψ(N )(cid:105)(cid:104)Ψ(N )| ).
ABC A(cid:48)→BC S S AA(cid:48) channels [13]. Teleportation simulation in the case of a point-
Then by using φ⊗n , they perform state merging to estab-
ABC to-point channel can be understood as the possibility of
lish entanglement. More precisely, Bob and Charlie transfer
reducing a sequence of adaptive protocols involving two-way
their system back to Alice by LOCC (similar to reverse
LOCC into a sequence of non-adaptive protocols followed by
reconciliation in the point-to-point scenario). This could be
afinalLOCC.Forall‘teleportation-simulablechannels’(more
done by applying the point-to-point state merging protocol
precisely the channels arising from the action of teleportation
successively [24], [25] or alternatively, by applying the mul-
on a bipartite state) that allow for such a reduction, an
tiparty simultaneous decoding state merging [30]. Then we
upper bound on the entanglement and secret key agreement
obtain the achievable rate region for E and E as
AB AC capacity can be given by the REE [13], because the REE is
E ≤−H(B|AC) , (8) an upper bound on the distillable key of any bipartite state
AB φ
[27]. Furthermore, for pure-loss bosonic channels, one can
E ≤−H(C|AB) , (9)
AC φ use a concise formula for the REE given in [13]. With these
E +E ≤−H(BC|A) . (10)
AB AC φ techniques,anupperboundontheunconstrainedcapacityofa
1.0
point-to-pointpure-losschannelisequivalenttotheREEofthe
stateresulting fromsendingan infinite-energyTMSVthrough
0.8
thechannel,explicitlycalculatedtobeequalto−log (1−η).
2
Since the pure-loss bosonic QBC is covariant with re-
C0.6
spect to displacement operations (which are the teleportation KA
+
corrections for bosonic channels [32]), it is teleportation- EAC0.4
simulable. Then the original broadcasting protocol described
in the previous section can be replaced by the distillation of 0.2
n copies of φ =N (|Ψ(N )(cid:105)(cid:104)Ψ(N )| ) via the
ABC A(cid:48)→BC S S AA(cid:48)
final LOCC. This LOCC distills entanglement and secret key; 0.0
i.e., it generates a state ω which is ε-close to Φ˜ : 0.0 0.2 0.4 0.6 0.8 1.0
ABC ABC EAB+KAB
(cid:107)ω −Φ˜ (cid:107) ≤ε (15)
ABC ABC 1
Fig. 2. LOCC-assisted capacity region given by (4)–(6) for the pure-loss
with bosonicbroadcastchannel,where(ηB,ηC)=(0.2,0.3).
Φ˜ =Φ⊗nEAB ⊗Φ⊗nEAC ⊗γ⊗nKAB ⊗γ⊗nKAC, (16)
ABC A1B1 A2C1 A3B2 A4C2
where Ai, Bi, and Ci are subsystems of A, B, and C, Letφ(cid:48) betheTMSVinwhichonlythefirstbeamsplit-
ABC(cid:48)
respectively. Then by using several well known properties of
ter is applied. Observe that it is a pure state and its marginal
REE(monotonicityunderLOCC,continuity,andsubadditivity φ(cid:48) isathermalstatewithaveragephotonnumber(1−η )N .
C(cid:48) B S
for product states), we find that
Combining these two observations, we can conclude that the
n(E +K )≤E (B;AC) state has the following Schmidt decomposition:
AB AB R Φ˜
≤E (B;AC) +f(n,ε) ∞
R ω (cid:88) (cid:112)
|φ(cid:48)(cid:48)(cid:105) = λ ((1−η )N )|ϕ (cid:105) |m(cid:105) , (22)
≤nE (B;AC) +f(n,ε), (17) ABC(cid:48) m B S m AB C(cid:48)
R φ
m=0
wheref(n,ε)isafunctionsuchthatlim 1f(n,ε)=
ε→0,n→∞ n where {|ϕ (cid:105) } is some orthonormal basis. Since
0. Similar bounds can be obtained in terms of E (C;AB) m AB m
R φ
{|ϕ (cid:105) } is an orthonormal set, there exists a local unitary
and E (A;BC) , leading to the following outer bound for m AB
R φ
operation acting on systems A and B such that
the capacity region:
E +K ≤E (B;AC) (18) U : |ϕ (cid:105) →|m(cid:105) |aux(cid:105) , (23)
AB AB R φ AB m AB A B
E +K ≤E (C;AB) , (19)
AC AC R φ
where |aux(cid:105) is some constant auxiliary state. Then we have
E +K +E +K ≤E (A;BC) . (20)
AB AB AC AC R φ
U |φ(cid:48)(cid:48)(cid:105) =|aux(cid:105) |Ψ((1−η )N )(cid:105)
To calculate the right-hand side of each inequality, we use AB ABC(cid:48) B B S AC(cid:48)
a calculation from [13]: for a point-to-point pure-loss bosonic ≡|φ(cid:48)(cid:48)(cid:48)(cid:105)ABC(cid:48). (24)
channel with transmittance η, which we denote by Lη ,
A(cid:48)→B
the REE of LηA(cid:48)→B(|Ψ(NS)(cid:105)(cid:104)Ψ(NS)|AA(cid:48)) with NS → ∞ is Let φ˜ABC =LηC¯(cid:48)→C(|φ(cid:48)(cid:48)(cid:48)(cid:105)(cid:104)φ(cid:48)(cid:48)(cid:48)|ABC(cid:48)) where η¯=ηC/(1−
given by −log2(1−η). ηB). Note that φABC = LηC¯(cid:48)→C(|φ(cid:48)(cid:48)(cid:105)(cid:104)φ(cid:48)(cid:48)|ABC(cid:48)). Since the
For E (A;BC) , consider the physical implementation of localunitaryoperationU doesnotchangetheREEbetween
R φ AB
the channel in Fig. 1(b) and let φ(cid:48) be the state such that ABandC,wehaveE (C;AB) =E (C;AB) .Moreover,
AA(cid:48)(cid:48) R φ˜ R φ
onlythefirstbeamsplitterisapplied.Thesecondbeamsplitter E (C;AB) is the REE for the TMSV with (1 − η )N
R φ˜ B S
is a local unitary in the sense that it operates on B and C followedbyLη¯ .Thenbyusingtheresultin[13],wefind
C(cid:48)→C
whereas our partition is now between A and BC. Thus it
does not change the REE. Then we have ER(A;A(cid:48)(cid:48))φ(cid:48) = lim ER(C;AB)φ = lim ER(C;AB)φ˜
E (A;BC) . Also, since φ(cid:48) is a TMSV followed by a NS→∞ NS→∞
R φ AA(cid:48)(cid:48) =−log (1−η¯)
pure-loss channel with transmittance η +η , we get 2
B C
=log ([1−η ]/[1−η −η ]). (25)
lim E (A;BC) = lim E (A;A(cid:48)(cid:48)) 2 B B C
R φ R φ(cid:48)
NS→∞ NS→∞ Similarly, with the physical implementation picture in
=−log (1−η −η ). (21)
2 B C Fig. 1(d), we obtain
To calculate E (C;AB) , we employ the physical imple-
R φ
mentation of the channel in Fig. 1(c) in which the first beam lim ER(B;AC)φ =log2([1−ηC]/[1−ηB −ηC]), (26)
splitterwithtransmittance1−η separatesB fromtheothers. NS→∞
B
This beam splitter is followed by the second beam splitter which completes the proof of the converse part. Figure 2
with transmittance η /(1−η ) which separates C and the illustrates an example of the unconstrained capacity region
C B
environment E. given in (4)–(6).
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We acknowledge helpful discussions with Sam Braunstein,
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APPENDIX:THE1-TO-mBROADCASTCHANNEL
In this appendix, we generalize Theorem 1 to the 1-to-m
pure-loss broadcast channel for arbitrary positive integer m.
Consider the pure-loss broadcast channel L
A(cid:48)→B1···Bm
characterized by a set of transmittances {η ,··· ,η } with
(cid:80)m η ≤ 1 [29]. Let B = {B ,··· ,BB1 }, T ⊆BmB, and
i=1 Bi 1 m
T be a complement of set T. Then we have the following
theorem:
Theorem 2: The LOCC-assisted unconstrained capacity re-
gion of the pure-loss bosonic QBC L is given by
A(cid:48)→B1···Bm
(cid:88) (cid:18)1−η (cid:19)
E +K ≤log T , (27)
ABi ABi 2 1−η
B
Bi∈T
Fig.3. Implementationofthe1-to-mpure-lossbosonicbroadcastchannel.
for all non-empty T, where η = (cid:80)m η and η =
(cid:80) B i=1 Bi T
η .
Bi∈T Bi
the channel by a sequence of m beam splitters where the j-th
Proof:Thestrategyoftheproofisquitesimilartothatof
beam splitter’s transmittance is given by
Theorem 1. For the achievability, one can apply the point-to-
point state merging protocol successively, which leads to the 1−(cid:80)j η
following achievable rate region: η˜j = 1−(cid:80)kj=−11ηk. (30)
(cid:88) l=1 l
E ≤−H(T|AT) , (28)
ABi φ Now, for each given T involving t parties, consider the
Bi∈T following specific ordering. For ηi with 1 ≤ i ≤ t, assign
where η with B ∈ T, η = η , and for η with i > t+1,
Bj j t+1 E i
assign η with B ∈T. Then the transmittance of the t+1
φAB1···Bm =LA(cid:48)→B1···Bm(|Ψ(NS)(cid:105)(cid:104)Ψ(NS)|AA(cid:48)). (29) beam spBlijtter is ηj/(1−η ) where η = (cid:80) η (see
T T T Bi∈T Bi
The right-hand side of (28) is calculated to be Fig. 3). From the main text, we already know that the REE
for given partition T and AT is simply characterized by this
−H(T|AT) =H(AT) −H(ATT)
φ φ φ transmittance.Asaconsequence,theREEboundforN →∞
S
=H(TE)φ−H(E)φ turns out to be
=g((1−η )N )−g((1−η )N ), (cid:88)
T S B S (E +K )≤E (T;AT)
ABi ABi R φ
and taking NS → ∞, we get log2([1−ηT]/[1−ηB]). Since Bi∈T (cid:32) (cid:33)
one ebit of entanglement can generate one private bit of key, 1
wecanreplaceEABi withEABi+KABi whichcompletesthe =log2 1− ηT
achievability part. 1−ηT
Remark 3: Since the above rate region reflects the last =log (cid:18) 1−ηT (cid:19),
gain/consumption of entanglement after the sequential oper- 2 1−η −η
T T
ation of the point-to-point state mergings, it could be possible (cid:18)1−η (cid:19)
=log T , (31)
that the protocol is ‘catalytic,’ meaning that entanglement is 2 1−η
B
consumedatsomestatemergingwhichiscompensatedbythe
where η =η +η .
followingotherstatemergings.However,thisdoesnothappen B T T
in our case. We can check it by the following simple observa-
tion. Gain/consumption of entanglement at any state merging
isgivenby−H(S |AS ) whereS issomenonemptysubset
1 2 φ 1
of B and S is other subset (possibly empty) of B satisfying
2
S ∩S = ∅. Since −H(S |AS ) = H(S ) −H(S S )
1 2 1 2 φ 2 φ 1 2 φ
andS isnon-empty,thisquantityalwayshasapositivevalue
1
meaning that entanglement is generated at all steps of the
whole protocol.
For the converse, we need to configure the beam splitter
networkoftheQBCproperly.Notethatthechannelhasm+1
(cid:80)
transmittances η ,··· ,η , and η ≡1− η . We can
B1 Bm E i Bi
order these transmittances in some sequence and label it as
η ,η ,··· ,η ,η .Thenforanyordering,wecandescribe
1 2 m m+1