Subadditivity of the minimum output entropy and superactivation of the classical capacity of quantum multiple access channels

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by  Lukasz Czekaj

Subadditivity of the minimum output entropy and superactivation of the classical
capacity of quantum multiple access channels
Ł. Czekaj1,2
1Faculty of Applied Physics and Mathematics,
Gdańsk University of Technology, 80-952 Gdańsk, Poland
2National Quantum Information Center of Gdańsk, 81-824 Sopot, Poland
[email protected]
Westudysubadditivityoftheminimumoutputentropy(Hmin)ofquantummultipleaccesschan-
nels (MACs). We provide an example of violation of the additivity theorem for Hmin known in
1 classical information theory. Ourresult isbasedon afundamentalpropertyof MACsi.e. indepen-
1 denceofeachsender. Thechannelsusedintheexamplecanbeconstructedexplicitly. Onthebasis
0 ofsubadditivityofHmin wealso providean exampleof extremalsuperadditivity(superactivation)
2 of the classical capacity region of MACs.

n
PACSnumbers: 03.67.Hk,89.70.Kn
a
J

5
Using quantum resources like quantum entangle- the explicit construction of the channels we present.

] ment [1] in quantum information theory [2] leads to It should allow a better understanding of the mech-
h a new class of effects, known as quantum activation, anism behind the subadditivity effect. The subaddi-
p
which are impossible in classical information theory tivity of H leads us to the quantum activation of
- min
t [3]. Some examples of quantum activation are: (i) . The example provided here exhibits superadditiv-
n R
superadditivity of the classical capacity in the fun- ity of the total rate R . We construct two sequences
a T
C
u damental case of 1-to-1 quantum channels [7] where of channels Γ˜(δ) , Γ˜(δ) and study its parallel setup
q transmission of entangled states leads to capacities Γ˜(δ) Γ˜({δ) A. W} i{thBou}t using entanglementin com-
[ larger than using product states, (ii) nonlocality ef- { A }⊗{ B }
munication, R 0 as δ 0. On the other hand,
2 fect for classicalcapacity region of quantum multi- using entangleTd s→tates allow→s to achieve R = 1 for
R T
v ple accesschannels [4, 5] where entanglementusedby
each δ. This can be view as a superactivation effect,
4 first sender increases maximal rate of an other (R )
2 since entanglement strongly activates channels with
6
without increasing the maximal total rate R +R .
2 1 2 almost 0 capacities.
The effect can be quite strong as shown in [6], (iii)
0 The superadditivity of in entanglement breaking
1. superactivation, i.e. extremal superadditivity, of the MACs suggests qualitativeCdifferences between bipar-
quantum capacity for 1-to-1 channels where two
0 Q tite and multipartite communication since it cannot
1 quantumchannelswith0quantumcapacitiesworking occur for entanglement breaking 1-to-1 channels. It
1 together allow for transmission of qubits [8].
was first pointed out in Ref. [6]. Superadditivity was
:
v shown for the entanglement breaking MACs cooper-
Thispaperaddressessubadditivityoftheminimum
i atingwith anidentity channel(whichis notentangle-
X output entropy H and quantum activation of the
min
ment breaking). Here we move one step further and
r classicalcapacityregion oftheentanglementbreak-
a R show that the very strong superadditivity takes place
ing [24] multiple access channels (MACs) and it con-
also if we use only entanglement breaking MACs.
tinues the research started in Ref. [4]. Effect of the
The paper is organized as follows: first we provide
subadditivity of H is manifested when transmis-
min
definitions and theorems used in the main part of the
sion of entangled states produces lower entropy than
paper. We stress on the explanation of the idea of a
transmission of any product states. The question of
randomness extractor which is of paramount impor-
subadditivityofH ofquantum1-to-1channelsap-
min
tance to further considerations. Then we present the
pearstohavebeenfirstconsideredinprintinRef.[9].
main results, i.e. subadditivity of H and superac-
In Ret. [17] the equivalence between additivity of min
tivation of the classical capacity of the MACs.
H and classical capacity χ was stated. Finally, an
min
example of subadditivity of H for quantum 1-to-1
min
channels was first provided in Ref. [7]. and explored
BACKGROUND
further in [20]. Hastings’ channels seems to be very
hard to explicitly construct since this task requires a
search through the set of unitary matrices. Here we A quantum channel Γ is a linear, completely posi-
studythesubadditivityofH intherealmofMACs. tive and trace preserving map from density matrices
min
ThesetupwepresentisintrinsicallyMACandcannot to density matrices ρ Γ(ρ) [2] and it models trans-
7→
be reduced to the setup of 1-to-1 channels, i.e. to the missionofquantumstatesinthepresenceofnoise. An
case studied by Hastings. The advantage of our ap- entanglement breaking channel is a quantum channel
proachistheexistenceofeffectivealgorithmsallowing whichcannotbeusedtocreateentanglementbetween
2
parts participating in communication [24]. It can be upper bound for the capacity region which can be
presentedinthe formofa measurementfollowedby a achieved due to quantum entanglement.
state preparation. In quantum multiple access chan- ForaMACΓwithnsenderswedefinetheminimum
nelsthereareatleasttwosenderstransmittingtoone output entropy H (Γ) as:
min
receiver. Eachsendersendshisstateindependentlyof
the other, i.e. their inputs are uncorrelated. For the Hmin(Γ)= min H(Γ(ρ1 ... ρn)) (5)
ρ1,...,ρn ⊗ ⊗
case of two senders, a MAC acts as a map:
where ρ belongs to the input space of the sender S .
i i
ρ1 ρ2 Γ(ρ1 ρ2) (1) Minimizationrunsoverallstatesfromtheinputspace
⊗ 7→ ⊗
of each sender. Due to the concavity of H(ρ), it is
where state ρ (ρ ) is sent by the sender S (S ).
1 2 1 2 sufficient to minimize only over pure states.
We will denote as Γ Γ a parallel setupof chan-
A B Intheclassicalsetupsenderstransmitonlyproduct
⊗
nels Γ ,Γ . It means that each sender has access to
A B statesfromorthogonalbasesoftheinputspacesofΓ
A
oneinputofthechannelΓ andoneinputofthechan-
A andΓ . BythepropertiesofthevonNeumanentropy,
B
nel Γ . They can transmit any states through their
B we can state for MACs the additivity theorem:
inputs where the first part of transmitted states goes
throughΓA andthesecondthroughΓB. Channelsare Hmin(ΓA)+Hmin(ΓB)=Hmin(ΓA ΓB) (6)
used synchronouslyandthe receiverhas access to the ⊗
outputs of both channels. Existenceofentangledstatesintheinputspaceofthe
A quantumchannelcanbe usedfortransmissionof Γ Γ extends the set we minimize H over and
A B min
⊗
either classical [22] or quantum information [21]. In makes the additivity theorem invalid in the quantum
the transmission of classical information, senders en- setup. Subadditivity of H occurs if transmission
min
code classicalmessages i , j intocode statestrans- ofentanglementstatethroughΓ Γ producelower
A B
{ } { } ⊗
mittedthroughthechanneli ρ(i),j ρ(j). Senders entropythanthesumofHminofeachchannelworking
7→ 1 7→ 2
and receiver know the ensemble of code states (i.e. separately.
the set of code states and the probabilities the states The additivity theorem for MACs can be stated
are transmitted with) p(i),ρ(i) , p(j),ρ(j) but one analogically:
{ 1 1 } { 2 2 }
sender does not know which state is transmitted by
(Γ )+ (Γ )= (Γ Γ ) (7)
the other sender at a given time. The receiver per- R A R B R A⊗ B
forms measurementon the output state and basedon
Here we use the geometrical sum of the sets in Euk-
itsresulttriestoinferswhichmessage(i,j)wastrans-
lides space. In the parallel setup of quantum MACs
mitted.
we can use entangled states as code words. Super-
The amount of classical information which can be
additivity of the classical capacity regions takes
reliabllytransmittedthroughtheMACisgivenbythe R
place if there exists a protocol using entangled code
pair of rates (R ,R ). The set of all achievable rates
1 2 states with classical capacity region (Γ Γ )
form the Holevo like classical capacity region (Γ). Rent A ⊗ B
R such that for each protocol using only product code
For a given ensemble of code states one can define
stateswith (Γ Γ )= (Γ )+ (Γ )occurs
the state ρ = p(i)p(j)e(i) e(j) Γ(ρ(i) ρ(j)) Rprod A⊗ B R A R B
i,j 1 2 1 ⊗ 2 ⊗ 1 ⊗ 2 prod ( ent. Superactivationdescribesthesituation
where e(i) ,( eP(j) ) are projectors on the standard wRhen R is huge in comparison with .
{ 1 } { 2 } Rent Rprod
basis of the Hilbert space of the input controlled by Inwhatfollows,we shallusegeneralizedBellstates
S1 (S2). The capacity region (Γ) is obtained as a [14] in the form:
R
convex closure of all rates (R ,R ) such that there
1 2
exists ρ for which the set of inequalities is fulfilled: 1 D−1 2πi
ψ = exp αl l l+β (8)
α,β
| i √D (cid:18) D (cid:19)| i| i
R1 I(S1 :RS2) (2) Xl=0
≤ |
R I(S :RS ) (3)
2 2 1 α,β 0,D 1 are indices. The states belong to
≤ |
∈ { − }
R =R +R I(S ,S :R) (4) the space CD CD. For the state µ = µ i
T 1 2 ≤ 1 2 ⊗ | i i i| i
where i is the standardbasis,wewillwritePµ∗ =
where I(S1,S2 :R)=H(ρS1)+H(ρS2)−H(ρR) and iµ∗i|{ii| ai}nd D =2d. | i
I(S1 : R|S1) = jpjI(S1 : R|S2 = j). H(ρ) = PFor two random variables X,Y with equal support
tr[ρlogρ] is thePvon Neuman entropy. RT denotes the statistical distance is defined as:
−
the total rate and is defined as R = R . We
shall denote "single shot" formulaT (1)(PΓ)i =i (Γ) dist(X,Y)= 1 p (e) p (e) (9)
andR(n)(Γ)= n1R(Γ⊗n) forthe situRationwhereRcode 2e∈sXup(X)| x − y |
states can be n-particle entangled states. The in-
teresting case is that of the regularized capacity re- We denoteby F am-bitrandomvariablewithaflat
m
gion (∞)(Γ)=lim 1 (Γ⊗n), whichexpressthe distribution over its support.
R n→∞ nR
3
Proposition 1 For a binary random variable X if
dist(X,F ) = ǫ then 1 H(X) = (2/ln2)ǫ2 +O(ǫ4)
2
−
and for ǫ (0,0.5) we have 1 H(X) 4ǫ2.
∈ − ≤ X1 MI mI
Proof: This follows directly from a Taylor series ex-
pansion.
A classical multiple source randomness extractor is X
2 pt
a function which distills entropy from independent mII fo Y=|fopt(mI,mII)〉
"week random sources" into random variables with
X
almost flat distribution. We put the word "classical" 3 MII
to distinguishfromsituationwhererandomnessisob-
tained on the base of quantum effects. X
4
Usability of a random source in a randomness ex-
traction process is characterized by the min-entropy
H∞ [11, 12] defined as: FIG.1: ThegeneralschemaofthechannelsΓAandΓB. Xi
H (X)= min logp(x) (10) are d-qubits input lines. MI and MII are measurements.
∞
x∈supX− Its result is denoted by mI and mII respectively. fopt is
aclassicalrandomnessextractorwithpropertiesdescribed
Definition 1 Multiple source randomness extractor
by theThm. 1.
[10]: A function f : 0,1 n×l 0,1 m which sat-
ext
{ } 7→{ }
isfies:
XA
1
dist(f (X ,...,X ),F ) ǫ (11)
ext 1 l m ≤ XA Γ
2
for every independent n-bit source X1,...,Xl with XA A
H∞(X ) k is called an l-source extractor with k 3
i
≥ XA
min-entropy requirement, n-bit input, m-bit output 4
and ǫ-statistical distance.
Theorem 1 Extractor existence [10]: Let m<k <n XB
1
be integers and let ǫ > 0. If k > logn + 2m + XB
Γ
2log(1/ǫ)+1holds then thereexists a2-sourceextrac- 2 B
tor fopt : 0,1 n×2 0,1 m with k-entropy require- XB3
ment and{dista}nce ǫ→. T{he }extractor can be computed XB
in time proportional to 25n222k. 4
FIG. 2: The parallel setup of the channels ΓA and ΓB.
Dashed lines depict entanglement of the inputs in case of
SUBADDITIVITY OF Hmin |Ψ0,0i transmission through ΓA⊗ΓB.
Here we provide two families of MACs Γ(δ) and
Γ(δ) indexed by δ which exhibit subad{ditAiv}ity of fopt(mI,mII), the channel produces the output state
{ B } 0 or 1 .
Hmin for δ <1/2. | iWe |wiill show that for any δ > 0, H (Γ(δ)) and
(δ) (δ) min A
The channels Γ ,Γ consist of 4 independent d-
A B H (Γ(δ)) cannot be lower than 1 δ. On the other
qubit inputs XA,...,XA (XB,...,XB) and 1-qubit min B −
1 4 1 4 handwewillshowthatifeachsendertransmits Ψ ,
output YA (YB). The inputs XA and XB are con- | 0,0i
i i the output entropy of the Γ(δ) Γ(δ) is equal to 1.
trolledbythesenderSi. Thesizeoftheinputsdepend A ⊗ B
on δ as follows: d=⌈2log(1/δ)+12⌉. Both channels SincethisistheupperboundforHmin(ΓA(δ)⊗ΓB(δ))we
are based on the same scheme (see FIG. 1) so we will willprovethatforδ <1/2,H (Γ(δ))+H (Γ(δ))>
min A min B
describethechannelΓAandpointoutwherethechan- H (Γ(δ) Γ(δ)).
nelsdiffer. Inthefirststep,thechannelperformmea- min A ⊗ B
We start by proving that H (Γ(δ)) =
surements M and M . M is a joint measurement min A
I II I H (Γ(δ)) 1 δ but first we give a proposi-
on inputs X1,X2 and MII is a joint measurement on min B ≥ −
tion which will be useful in what follows.
inputs X ,X . In the channel Γ , measurements M
3 4 A I
and M are performedin the basis Ψ while in
II α,β Proposition 2 For the random variables associated
{| i}
the channel Γ in the basis Ψ∗ . The result of
B {| α,βi} with the outputs of the measurements performed by
the measurement MI (MII) is denoted by mI (mII). the channels Γ(δ) and Γ(δ), the following holds:
m and m provide the 2d-bits inputs to the ran- A B
I II H∞(MA) = H∞(MA) = H∞(MB) = H∞(MB) =
domness extractor f , which produces a 1 bit out- I II I II
opt d. Here d denotes the input size of the channels.
put. Existence of the extractor f with proper fea-
opt
tures will be proven later. Depending on the value of Proof: We will only prove that H∞(MA) = d
I
4
since the other cases can be proved analogously. Let XA
the projector measurement MA be performed on the 1
I XA
productstate µ ν ,where µ = D−1µ j , ν = 2 Γ
PS1kDa=−n01dνSk|2kireas|preeic⊗dt-i|qveuilbyi.tWsteatwe|sililpsehrtPoawijn=tinh0gattjot|hsieenp|drioebrs- XXA43A A
ability p(α,β) = ψ µ ν 2 of getting the pair
m,n
| | i|
(α,β) asthe result(cid:10)ofmea(cid:12)(cid:12)su(cid:11)rementsatisfiesp(α,β)≤ FIG. 3: Construction of the channel Γ˜. The solid lines
D1. H∞(MIA)≥d is simple consequence of this fact. represent the qubits lines and the dashed represent the
bit lines. The bit lines controls CNOTs performed on the
Observe that:
output of thechannel.
p(α,β) = ψ µ ν (12)
α,β
| i
(cid:10) D(cid:12)−(cid:11)1
1 (cid:12) 2πi
= exp αl l l+β (13)
√D Xl=0 (cid:18) D (cid:19)h |h | p(αA,βA,αB,βB) (19)
1 2
D−1 D−1 = ψ∗ UαA UαB† ψ (20)
µj j νk k D2 (cid:12)h 0,0| βA ⊗ βB | 0,0i(cid:12)
| i | i (cid:12) (cid:12)
Xj=0 kX=0 1 (cid:12) 1 D−1 2π (cid:12)
1 D−1 2πi = D2 (cid:12)(cid:12)D hk|hk|exp(cid:20)iD(αA−αB)l(cid:21) (21)
= exp αl (14) (cid:12) kX,l=0
√D j=0,Xk=0,l=0 (cid:18) D (cid:19) l+(cid:12)(cid:12)(cid:12)βA l+βB 2 (22)
| i| i|
µjνk l j l+β k (15) D−1 2
1 2π
= √1D(cid:10)DXl=−(cid:12)(cid:12)01(cid:11)e(cid:10)xp(cid:18)2(cid:12)(cid:12)Dπ(cid:11)iαl(cid:19)µlνl+β (16) == D14δ(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Xl exδp(cid:20)iD(αA−αB)l(cid:21)δβA,βB(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ((2234))
= 1 µ∗ Uα ν (17) D2 αA,αB βA,βB
√Dh | β| i
(18) whereEq.(20)isobtainedinthesamewayasEq.(17).
This result can be viewed as generalized entangle-
ment swapping [14–16]. Entanglement between uses
whereUα = D−1 l+β l exp 2πiαl isanunitary. of channels ΓA and ΓB is swapped by the measure-
Finally bβy thPe pl=ro0pe|rty oifht|he sc(cid:0)alDarpr(cid:1)oduct we have ment MIA into entanglement between the inputs of
p(α,β)= 1 µ∗ Uα ν 2 1. 2 the channel ΓB belonging to senders S1 and S2. The
D|h | β| i| ≤ argumentpresentedisalsovalidforthemeasurements
Taking into account proposition 2, and noting that MA and MB.
II II
d/2 > logd is true for d > 4, we obtain that d =
As we have shown, in the case of entangled state
2log1/δ+12 fulfills the requirements of Thm. 1
w⌈ith ǫ = √δ/⌉2. Since f exists and has statisti- transmission, the outputs of fext for channels ΓA and
opt Γ areequal. SincetheoutputofthechannelΓ Γ
B A B
cal distance ǫ, by proposition 1 for each channel we ⊗
can be written in the form p00 00 +(1 p)11 11
have H 1 ǫ. | ih | − | ih |
min ≥ − for which the entropy is upper bounded by 1.
NowconsideroutputentropyoftheΓ(δ) Γ(δ) ifall
A ⊗ B
senders transmit ψ . The firstpartofthe 2d-qubit
0,0
| i
state is transmitted through the channel ΓA(δ) and the SUPERACTIVATION OF R
(δ)
secondthroughΓ (seeFIG.2). Wewillshowthatin
B
this case the output entropy of the Γ(δ) Γ(δ) cannot We now turn to on the superactivation of the clas-
A ⊗ B
exceed 1. sicalcapacityregionsofthe channelsΓ˜ ,Γ˜ . Namely
A B
we will show that if the senders can transmit only
The randomness extractor f is a determinis-
opt product states, classical capacity region =
tic function of the outcome of the measurements Rprod
MI,MII. Its output controlswhichof the pure states R(1)(Γ˜A(δ))+R(1)(Γ˜B(δ)) is bounded by the inequalities
|s0uil,ts|1iofwmilelabseurtehmeeonutstpMutAofantdheMchBan(nMelA. aIfndthMe rBe)- cRoSm=paPreit∈hSisRwi ≤ith2δa fporrotaoncyolsuubssinetgoefntsaenndgleerds Sst.atWese.
are identical, then so willIthe outpuIts of tIhe channIels Rent =R(1)(Γ˜A(δ)⊗Γ˜B(δ))achievedinthis caseis given
too. Let us focus on the measurements MIA and MIB. by inequality R1+R2+R3+R4 ≤1.
We will show that p(mA,mB)= p(α ,β ,α ,β ) Let us present the channel Γ˜ (see FIG. 3). It is a
I I A A B B ∝ A
δ δ . 4-to-1channel. Input of eachsender consists d-qubits
αA,αB βA,βB
5
line and 1-bit line. The channel acts as: ACKNOWLEDGMENTS
Γ˜A(δ)(ρ1⊗e(1i)⊗...⊗ρ4⊗e(4l)) (25) The author thanks P. Horodecki and R.W. Chha-
= CNOT ... CNOT (Γ(δ)(ρ ... ρ )) jlany for discussions and valuable comments on
i◦ ◦ l A 1⊗ ⊗ 4 manuscript. This work was supported by Ministry of
Since andHigherEducationGrantNo. NN202231937
where ρ is transmitted through qubit inputs and e(.)
and by the QESSENCE project.
throughbit inputs. CNOT (ρ)=ρ andCNOT (ρ)=
0 1
XρX†. In the same way we construct Γ˜(δ).
B
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≤ | ≤ −
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user rates R , however the superadditivity of the reg-
i
ularizedclassicalcapacity (∞) of1-to-1channelsand
C
superadditivity of R of (∞) of MACs still remains
T
R
an open questions.