Relation between classical communication capacity and entanglement capability for two-qubit unitary operations

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by  Dominic W. Berry

Relation between classical communication capacity and entanglement capability for
two-qubit unitary operations
Dominic W. Berry and Barry C. Sanders
Department of Physics and Centre for Advanced Computing – Algorithms and Cryptography,
Macquarie University, Sydney, New South Wales 2109, Australia
(Dated: February 1, 2008)
Two-qubit operations may be characterized by their capacities for communication, both with
and without free entanglement, and their capacity for creating entanglement. We establish a set
of inequalities that give an ordering to the capacities of two-qubit unitary operations. Specifically,
we show that the capacities for entanglement creation and bidirectional communication without
entanglementassistance are at least as great ashalf thebidirectional communication capacity with
entanglement assistance. In addition, we show that the bidirectional communication that can be
performed using an ensemble may be increased via a two-qubit unitary operation by twice the
4
0 operation’s capacity for entanglement.
0
2 PACSnumbers: 03.67.Hk,03.65.Ud

n
a I. INTRODUCTION swap gate may create two ebits of entanglement or per-
J
formtwobitsofcommunicationinoneorbothdirections
9
Quantuminformationprocessingreliesonapplications simultaneously [12, 13, 15]. In both cases the communi-

6 ofsingle-qubitunitarytransformationsandnonlocaltwo- cation is achieved while consuming entanglement.
v qubit unitary transformations, where the term “nonlo- Therelationshipbetweencommunicationcapacityand
5 cal”referstojointoperationsonseparateHilbertspaces. entanglement capability has been addressed by Bennett,
6 Thesenonlocaloperationsarepowerfultoolsinquantum Harrow, Leung, and Smolin (BHLS) [16]. They define
0 information theory and can be characterized by the en- asymptotic entanglement capabilities and communica-
7 tanglement they can generate between the two Hilbert tion capacities for unitary operations and derive many
0
spaces as well as by the communication capacity they relations for these quantities.
2
0 can deliver. These two seemingly disparate measures of Ourgoalhereistoestablishrigorousrelationsbetween
/ gate strengths are useful for quite different applications entanglementcapabilityandcommunicationcapacityfor
h
of two-qubit nonlocal gates. However, we establish that two-qubit unitary operations. By investigating the en-
p
- these two capacities are closely related. tanglementgeneratedbyasuperpositionofclassicalmes-
t Entanglement capability has been investigated in de- sages, we recently showed that the entanglement capa-
n
a tail [1, 2, 3, 4, 5, 6, 7, 8, 9]. The entanglement increase bility of two-qubit unitary operations is at least as great
u for finite unitaries has been analyzed for initially unen- as half the capacity for error-free entanglement-assisted
q tangled states [5] and investigated by numerical meth- communication [17]. Here we provide an extended ver-
v: ods for initially entangled states [6]. An important fea- sionofthoseresults,withrigorousderivationsfornonzero
i ture of the entanglement capability is that it quantifies error rate.
X the “strength” of the gate, which is particularly useful In Sec. II we review the definitions of Ref. [16], with
r in considering the resource requirements for simulation minor modifications, and introduce the notation used
a
of specific gate operations or Hamiltonian dynamics by throughout the paper. We show that the communica-
other gates [10]. tion rates may be achieved with an upper bound on the
Thecommunicationcapacityofnonlocalunitarytrans- dimension of the ancilla in Sec. III. We then apply this
formationsisasimportant. Thecommunicationcapacity result to prove an inequality between the entanglement-
is relevant to performing quantum gates remotely using assisted communication capacity and the capacities for
entanglementandclassicalcommunication[1,11, 12,13, creating and destroying entanglement (Sec. IVA).
14]. Anearlyexampleoftheremarkablerelationbetween The results presented up to Sec. IVA may be applied
classical communication and entanglement was seen in to all bipartite unitary operations. We then restrict to
superdense coding [15]. two-qubit unitaries, and in Secs. IVB to VII establish
a sequence of inequalities between the various capacities
Itis knownthatthere isa relationshipbetweenentan-
for two-qubit unitaries. We discuss a possible method of
glementcapabilityandcommunicationcapacityforsome
simple gates. For example, the cnot gate can create provingequalities in Sec.VII, andconclude in Sec.VIII.
one ebit [the entanglementinherent in anideal Einstein-
Podolsky-Rosen(EPR),orBell,pair]ofentanglementbe-
tween two initially unentangled systems; the cnot gate II. CAPACITY DEFINITIONS
can thus be used to generate a Bell pair. In addition,
thecnotgatecanbeusedtocommunicateonebitofin- The asymptotic definitions of capacities for entangle-
formation in each direction simultaneously [12, 13]. The ment creation and communication were introduced by
2
BHLS. In this section we summarize the definitions of The communication scheme consists of t applications
BHLS,andmakesomeminorimprovements. Weconsider of U, interspersed with the intermediate local unitary
two parties, Alice and Bob, with the combined Hilbert operations V(k) V(k). The final state is therefore
space . We call a unitary operator U acting on A ⊗ B
A B
H ⊗H
local if it is of the form V V , where V
HA ⊗HB A ⊗ B A η =(V(t) V(t))U(V(t−1) V(t−1))U
andVB acton A and B, respectively. Operatorsthat | xyiAB A ⊗ B A ⊗ B ···
H H
are not of this form we will term nonlocal. (This is not U(V(0) V(0))x y ψ . (4)
meant to imply physical separation between the physi- ··· A ⊗ B | iA1| iB1| iA2B2
cal systems representedby and .) In general,the
A B
H H ForperfectcommunicationBob’smessageyistransferred
Hilbert spaces and will be of the form
A B
H H to A and Alice’s message x is transferred to B . For
1 1
= , = , (1) communication with fidelity at least 1 ǫ, we have
HA HAU ⊗HAanc HB HBU ⊗HBanc −
such that U acts only upon . The Hilbert
spaces HAanc and HBanc will bHeAcUal⊗ledHtBhUe ancillas. F (|yiA1|xiB1,TrA2B2|ηxyiABhηxy|)≥1−ǫ, (5)
The maximum increase in entanglement that may be
producedbyasingleimplementationoftheunitarytrans- where F(ψ ,ρ) ψ ρψ . This condition means that
| i ≡ h | | i
formation U is denoted as E . That is, we define the classical error rate is no greater than ǫ.
U
As in Ref. [16], we say that the pair (R ,R ) is
→ ←
E sup [E(U Ψ ) E(Ψ )]. (2)
U ≡ | i − | i achievable if, for all ǫ > 0, there exists a t such that it
|Ψi∈HA⊗HB is possible to perform communication of fidelity at least
The quantity E( ) is the entropy of entanglement 1 ǫ in this way with n tR and n tR .
a → b ←
··· − ≥ ≥
E(Ψ ) = S[TrA(Ψ Ψ)], where S(ρ) = Tr(ρlogρ). For communication that is not assisted by entangle-
| i | ih | −
Throughout we employ logarithms to base 2, so the en- ment, we take the initial ancilla state ψ to be
tanglement is expressed in units of ebits. The ancil- an unentangled state of arbitrary finite d|imieAn2sBi2on. For
las may be of arbitrarily large, but finite, dimension. entanglement-assisted communication, we allow an an-
Theentanglementcapabilityismaximizedforpurestates cilla state that is a tensor product of an unentangled
[5, 6, 16], and we therefore restrict attention to pure state and a maximally entangled state, both of which
statesinthispaper. Wealsodefinethemaximaldecrease may be of arbitrary finite dimension.
in entanglement
For the entanglement-assisted case, it is necessary to
E− sup [E(Ψ ) E(U Ψ )]. (3) allow an unentangled component to the ancilla because
U ≡|Ψi∈HA⊗HB | i − | i the communication scheme is composed entirely of uni-
tary operations. Some of the processes that communica-
BHLSshowthattheasymptoticentanglementcapability
tion schemes use require an unentangled subsystem, and
peroperationinthelimitofalargenumberofoperations
it is not possible to obtain an unentangled subsystem
is equal to E . Similarly, E− may be interpreted as the
U U using local unitary operations.
asymptotic entanglement capability of U† [16].
For example, consider the measurement of some state
Toquantifythecommunicationcapacityoftheunitary
ψ via a d-element positive operator-valued measure
operation U, we adopt the definitions of asymptotic ca- | i
Π . Inordertotreatthemeasurementasaunitarypro-
pacities introduced by BHLS. The bidirectional commu- { i}
cess,weaddaregisterofdimensiond,whichisinitialized
nication is quantified by the pair of numbers (R ,R ).
→ ← inan unentangledstate 0 . The measurementmay then
The pair (R→,R←) is said to be achievable if, for any be represented by a uni|tairy operation U such that
ǫ > 0, there exists t such that it is possible to commu- meas
U 0 ψ = i √Π ψ . The register stores the re-
nicate tR→ bits from Alice to Bob and tR← bits from sumlteaos|f tih|eimeasuir|emienti,|wihich is only possible because
Bob to Alice with fidelity 1 ǫ via t applications of U P
− it is initially unentangled.
interspersed with local unitary operations.
To explain this communicationscheme in more detail, Thus it is necessary to have an unentangled subsys-
Alice and Bob share the Hilbert space , where tem in order to perform the unitary equivalent of local
A B
= and = H. ⊗ThHis division measurements. In particular, because the communica-
HofAtheHHAil1be⊗rtHsAp2ace shHouBld nHotB1be⊗cHonBf2used with that tion scheme considered by Ref. [16] uses local measure-
in Eq. (1); here we do not specify the component of the ments,itrequiresanunentangledresource. Itistherefore
Hilbert space upon which U acts. Subsystems A and necessaryto allow an unentangled component to the an-
1
B1 are both of dimension 2max(na,nb). Subsystem A1 is cilla for our definition to be consistent with Ref. [16].
initially in one of 2na orthogonal states corresponding We denote the set of achievable rate pairs without en-
to Alice’s n -bit message x, and B is initially in a state tanglement assistance by S , and the set of achievable
a 1 U
correspondingtoBob’sn -bitmessagey. SubsystemsA rate pairs with entanglement assistance by SE. We will
b 2 U
and B are the ancillas for Alice and Bob, respectively, also use a superscript X to indicate both cases. The ex-
2
andareinitiallyinasharedstate ψ . Thecomplete plicit expressions for S and SE are given in Appendix
| iA2B2 U U
initial state may be expressed as x y ψ . A. The communication rates for the bidirectional and
| iA1| iB1| iA2B2
3
unidirectional cases are then defined by For Bob’s message y, Bob may take Alice’s final re-
duced density matrix in A to be
1
CX = sup (R +R ), (6)
+ → ←
(R→,R←)∈SUX 2na−1
C→X = sup R→, C←X = sup R←. (7) ρ(Ay1) =2−na TrA2B(|ηxyiABhηxy|). (10)
(R→,R←)∈SUX (R→,R←)∈SUX xX=0
Here we have averaged over x, as Bob does not have
HerewehaveusedtheX superscripttoindicatethatthe
this information. Using the concavity of the fidelity and
same definitions hold with entanglement assistance and
the fact that the fidelity is nondecreasing under partial
without entanglement assistance.
trace [18], the fidelity of ρ(y) with y is at least 1 ǫ.
A1 | iA1 −
Similarly, Alice may take Bob’s density matrix to be
III. LIMITING THE ANCILLA
2nb−1
ρ(x) =2−nb Tr (η η ). (11)
Intheabovedefinitions,thesizeoftheancillaisalways B1 AB2 | xyiABh xy|
y=0
finite, but has no upper limit, and can grow arbitrarily X
quickly with t. This unbounded nature of the dimension
This will have fidelity at least 1 ǫ with x .
of the ancilla causes difficulties in deriving results for − | iB1
When density matrix ρ is given with probability p ,
i i
the communication capacities. To avoid these problems, we denote this by the ensemble E= p ,ρ . From Refs.
i i
in this section we show how to put an upper bound on { }
[19,20],the communicationthatmaybe performedwith
the dimensionofthe ancilla; thatis, weshow thatfor all
such an ensemble is given by the Holevo information
achievableratepairs(thatarenotontheboundaryofthe
regionofachievableratepairs),thereexistsaconstantK
such that this rate pair may be achieved with an ancilla χ(E)=S p ρ p S(ρ ). (12)
i i i i
space of dimension no larger than 2Kt. i !− i
X X
In Ref. [16] similar results were derived for communi-
This communication may be obtained asymptotically by
cationwithout priorentanglementandfor one-waycom-
using error correcting codes over multiple states.
munication with entanglement. Here we give a general
For the communication scheme above, the ensemble
derivation that may be applied to these cases, and also
to bidirectional communication with entanglement. possessedby Alice is EA ={2−nb,ρ(Ay1)}. The elements of
The set of achievable rate pairs (R ,R ), SX, forms this ensemble have fidelity at least1 ǫ with the ensem-
a two-dimensional region with an up→per b←oundUary [16]. ble E0A = {2−nb,|yiA1hy|}. Using th−e concavity of the
The interior of SUX is an open set; therefore, for any rate fidelity, the average density matrix ρ¯A1 = 2−nb yρ(Ay1)
pair (R ,R ) in the interior of SX, there must be an hasfidelityatleast1 ǫwiththe averagedensity matrix
asfoocrhRiteh→′veab>e→nletRap→n←agirlaen(mRde→′nRt,←′Ras←′>si)stRcel←dos.oerrTuthoneUastsehseirsauttepedppecaraisrbeso;umthnaadytarbiyse,, ifsornItE.i0AsTgeoaiveseislntyimbsyhaotσ¯ewAtn1he=t−hHa2ot−lntehvbeoPiHnyof|oylerivmAo1ahitynio|f.onrfmoratEionPwfeormEa0Ay
b A
our derivation applies to both cases. use Fannes’ inequality [21]
It is possible to achieve the rate pair (R′ ,R′ ) with
→ ←
fidelity 1 ǫ, where S(ρ) S(σ) T(ρ,σ)logD+η(T(ρ,σ)), (13)
− | − |≤
∆R 2 ∆R 4 1 2 whereDisthedimensionoftheHilbertspaceandη(x)
ǫ min , , , (8) ≡
xlogx. T(ρ,σ) Trρ σ is the trace distance, and
≤ ((cid:18)32Rmax(cid:19) (cid:18) 16 (cid:19) (cid:18)2e(cid:19) ) w−e require that T(≡ρ,σ)| −1/e|.
≤
with Forρ=ρ(y)andσ = y y ,orρ=ρ¯ andσ =σ¯ ,
A1 | iA1h | A1 A1
thefidelityisatleast1 ǫ. Usingtheinequalitybetween
∆R=min{R→′ −R→,R←′ −R←}, (9) trace distance and fidel−ity given in Ref. [18], this implies
that T(ρ,σ) 2√ǫ. The fact that we have taken ǫ
and Rmax = max{R→′ ,R←′ ,1}. The reason for putting (1/2e)2 implie≤s that T(ρ,σ) 1/e is satisfied. ≤
this restriction on ǫ will become clear below. This com- ≤
Using these results, and the fact that η(x) √2x, the
municationmaybeachievedwithanancillaofsomefinite Holevo information for E must satisfy ≤
dimension, d, and τ applications of U. Without loss of A
generality we restrict the length of the messages com- χ(E ) n 4n√ǫ 4ǫ1/4, (14)
A b
municated,n andn ,tobenolongerthan2τR bits. ≥ − −
a b max
Anyexcessbitsareunnecessaryforachievingthecommu- where n = max n ,n . Using the inequalities n
a b
nication rate, and will be included as part of the ancilla. 2τR , n τR{′ , τ }1 and the restrictions put on≤ǫ
We may reduce the error rate by coding over M blocks in Emq.ax(8),bw≥e get← ≥
ofτ operations,foratotalnumberofoperationsequalto
t=Mτ. χ(E ) τ(R′ +R )/2. (15)
A ≥ ← ←
4
From Ref. [19], by coding over a number of ensem- IV. COMMUNICATION WITH
bles M no less than some minimum M , it is possi- ENTANGLEMENT
A
ble to communicate information per ensemble from Bob
to Alice arbitrarily close to χ(EA) with average fidelity The fact that it is possible to limit the dimension of
arbitrarily close to 1. For example, we may achieve the ancilla in this way considerably simplifies some of
communication Mτ(R←′ +3R←)/4 with average fidelity the analysis in Ref. [16], and allows additional relations
1 − 12ǫ0(R←′ − R←)/(R←′ + 3R←). The fact that the tobederived. Itispossibletoderiveinequalitiesbetween
result of Ref. [19] is given in terms of the average fi- the classical communication capacity and the entangle-
delity, rather than the minimum fidelity, is unimportant ment capability using a quantum superposition of clas-
because the codewords that result in low fidelity may be sical messages. However, there is the complication that
omitted. For this example, no more than a proportion theentanglementonlyconvergesifthereisanupperlimit
(R←′ −R←)/(R←′ +3R←) of the codewords may give fi- placed on the dimension.
dmeulintiycaletsiosnthoafnM1τ−Rǫ←0/2w.itOhmfiidtteilnitgytohfeasettlehaesntg1iv−esǫ0c/o2m.- C→EInwReerfe.p[1r6o]vethne. TinheequaanlcitilileassEwUer≥e bCo+unadneddEuUsin+gEtUe−ch≥-
Thesameconsiderationsholdforcommunicationfrom niques that are specific to these inequalities and cannot
Alice to Bob. For Bob’s ensemble we find be generalized. The bound that we have placed on the
dimension of the ancilla in Sec. III is sufficiently general
χ(E ) τ(R′ +R )/2. (16) tobeappliedtoshowtheseinequalities,andshowfurther
B ≥ → → inequalities. In Sec. IVA below, we show the additional
inequality E +E− CE. This inequality is for gen-
Similarly, by coding over a large number M M of U U ≥ +
B eralbipartite unitary operations,andis not restrictedto
≥
ensembles, it is possible to achieve communication from two-qubit unitaries. In Sec. IVB we show a simplified
Alice to Bob of MτR with fidelity of at least 1 ǫ /2.
→ 0 inequality that is restricted to two-qubit unitary opera-
−
If we take the number of ensembles M to be larger than
tions.
both M and M , then it is possible to perform the
A B
coding for communication in each direction on the same
block of M states. In this way, communication MτR← A. Arbitrary bipartite unitary operations
fromBobtoAliceandMτR fromAlicetoBobmaybe
→
performed, with the total fidelity at least 1−ǫ0. Let (R→,R←) be a pair of rates in the interior of SUE.
Thus we see that, by coding over M blocks of τ ap- FromSec.II,there existsa protocolthattransmitsn
a
plications of U, we may perform communication per op- tR bitsfromAlicetoBobandn tR bitsfromBo≥b
→ b ←
eration of R← from Bob to Alice and R→ from Alice toAlice,viatusesofU andwithfid≥elity1 ǫ. FromSec.
to Bob with arbitrarily high fidelity. M copies of the III, there exists a constant K such that ǫ−may be made
siniodnividd,ufaolraanctilolatasl|ψdiimA2eBn2siaorneorfeqduMir.ed,Ineaachddoiftiodnim, ewne- agrrbeaittrearrtilhyasnm2aKllt.wThihlee tfihnealdsimtaetnesiηon of thseatainsficielsla is no
xy AB
should take account of the M copies of subsystems A | i
1
and B1. These subsystems contain the codewords, and F(|yiA1|xiB1,TrA2B2|ηxyiABhηxy|)=1−ǫxy, (17)
the decoded messages will be contained in separate sub-
with ǫ ǫ for all messages x and y. Using Uhlmann’s
xy
systems. We therefore find that the dimension of the ≤
theorem, the state η may be expressed as [22]
xy AB
ancilla required for the total communication scheme is | i
nopoemraotrioentshaannd2KKt,=w(h2enre+tl=ogMd)τ/τi.s the total number of |ηxyiAB = 1−ǫxy|yiA1|xiB1|cxyiA2B2 +√ǫxy|exyi(A1B8,)
p
Thus we see that any achievable rate pair (R ,R ) where Tr e e has support orthogonal to
→ ← A2B2| xyiABh xy|
in the interior of SUX may be achieved with a bound 2Kt |yiA1|xiB1. The change in the entanglement is
on the dimension of the ancilla. The restriction that the
∆E =E(η ) E(ψ ). (19)
ratepairmustbeintheinteriorofSUX doesnotaffectthe xy | xyiAB − | iA2B2
results for the communication capacities, because these Now we add additional subsystems A and B that
3 3
capacities are defined via a supremum, and we may take contain copies of Alice’s message x and Bob’s message
the rate pair to be arbitrarily close to the upper bound- y. In addition, rather than selecting specific classical
ary. messages x and y, we have a coherent superposition of
Notealsothat,becausewehavetakenn andn forthe all possible messages. Then the input state is
a b
individual communication schemes (with τ applications
ofU)tobenolargerthan2τRmax,thetotalcommunica- 2−(na+nb)/2 |xiA1|xiA3|yiB1|yiB3|ψiA2B2, (20)
tion in either direction may be bounded by tK , where xy
n X
K =2R . Thisboundisrelativelytrivial,butwillbe
n max and the output state is
used later. Another application of these results is that
t may be taken to approach infinity in the limit ǫ 0, η =2−(na+nb)/2 x y η . (21)
whichisnotexplicitlygiveninthedefinitionofther→ates. | ǫi | iA3| iB3| xyiA12B12
xy
X
5
Here we use the notation of multiple subscripts to indi- Using this, and the fact that ∆E tE−, we find
xy ≥− U
cate the combined Hilbert space, for example, A in-
12 E(c ) E(ψ )
ednictairteesHHilbAe1rt⊗sHpaAc2e.s oWf AeluicseeaAndanBdobB, rteospiencdtiicvaetlye;tshoe, ≥|Ex(y|ηiAxy2BiA212−B12)−| √iAǫ2(B22n+Kt)−Q−E(|ψiA2B2)
for example, A is equivalent to A123. tE− √ǫ(2n+Kt) Q. (30)
In order to put bounds on the entanglement that may ≥− U − −
Substituting this into Eq. (28), and noting that ∆E
be created, we compare Eq. (21) with the state
≤
tE , gives
U
|ηi=2−(na+nb)/2 |yiA1|xiA3|xiB1|yiB3|cxyiA2B2. tEU ≥na+nb−tEU−−2√ǫ[3n+Kt]−2Q. (31)
xy
X Using the inequalities n tK (from Sec. III), n
(22) n a
≤ ≥
tR , n tR , and dividing on both sides by t, gives
We find that → b ←
≥
E +E− R +R 2√ǫ[3K +K] 2Q/t. (32)
hη|ηǫi=2−(na+nb) (A1hy|B1hx|A2B2hcxy|)|ηxyiA12B12 U U ≥ → ←− n −
In the limit ǫ 0, as K and K are fixed, the third
xy n
X →
term on the right-hand side goes to zero. The fourth
2−(na+nb) √1 ǫ=√1 ǫ, (23)
≥ − − term goes to zero in this limit, because t may be taken
Xxy to approach infinity (as mentioned in the preceding sec-
giving η ηǫ 2 1 ǫ. tion). WethereforeobtainEU+EU− ≥R→+R←,forany
|h | i| ≥ − achievable pair of rates (R ,R ) that is in the interior
From the continuity of the entropy of entanglement → ←
of SE. This implies that
[23],orFannes’inequality[21],wefindthatthedifference U
between the entanglement of the two states η and ηǫ E +E− CE. (33)
is bounded by | i | i U U ≥ +
Note that it is the upper limit K on the number of
E(η ) E(η ) T(η , η )(2n+Kt/2)+Q, (24) auxiliary qubits that may be used per operation that al-
ǫ ǫ
| | i − | i |≤ | i | i
lows us to take the limit ǫ 0. When the dimension of
where n = max(na,nb), Q = log(e)/e and T(a , b ) = the system is limited in th→is way, the entanglement has
| i | i
Tr a a b b isthetracedistance. Herewehaveused good continuity properties. This continuity means that
| ih |−| ih |
the fact that the dimension of the ancilla is bounded by weobtainthe same resultinthe limitǫ 0 aswe would
2K(cid:12)(cid:12)t. Using the(cid:12)(cid:12) inequality between trace distance and if the communication were exact (ǫ=0)→.
fidelity given in Ref. [18], we find that
E(η ) E(ηǫ ) 1 η ηǫ 2(4n+Kt)+Q B. Two-qubit unitary operations
| | i − | i |≤ −|h | i|
p√ǫ(4n+Kt)+Q. (25)
≤
For the case of two-qubit unitaries, we may simplify
Thereforethe increaseinthe entanglementfort appli-
this result. In this case the maximum increase in the
cations of U is
entanglement E is equal to the maximum decrease in
U
theentanglementE−,whichmaybeshowninthefollow-
∆E =E(η ) E(ψ ) U
| ǫi − | iA2B2 ing way. Any two-qubit interaction U is equivalent, up
≥E(|ηi)−√ǫ(4n+Kt)−Q−E(|ψiA2B2). (26) to local unitary operations, to an operation of the form
[3, 5, 24, 25]
Theentanglementof η maybefound,usingEq.(11.58)
of Ref. [18], to be | i U (α ,α ,α )=e−i(α1σ1⊗σ1+α2σ2⊗σ2+α3σ3⊗σ3), (34)
d 1 2 3
where σ , σ , and σ are the Pauli sigma matrices.
E(η )=n +n +2−(na+nb) E(c ). (27) 1 2 3
| i a b | xyiA2B2 As entanglement capability and classical communica-
xy
X tioncapacityareindependentoflocalunitaryoperations,
Substituting this into Eq. (26) gives wemayrestricttooperationsofthisform. Itisthensim-
ple to show that U∗ = U† = U−1. As discussed in Ref.
d d d
∆E na+nb √ǫ(4n+Kt) Q [5], for any measure of entanglement, E(Ψ )=E(Ψ∗ ).
≥ − − | i | i
+2−(na+nb) [E(c ) E(ψ )]. (28) This means that, if the operation Ud acting on the state
| xyiA2B2 − | iA2B2 Ψ generates the maximum increase in entanglement,
Xxy |thein this operation performed on the state U∗ Ψ∗ de-
d| i
Againapplyingthe continuityof the entropyofentan- creases the entanglement by EU. Therefore, the opera-
glement [23], tion may decrease the entanglement at least as much as
itmayincreaseit. Similarlyitis simple toshowthe con-
|E(|ηxyiA12B12)−E(|yiA1|xiB1|cxyiA2B2)| verse, and therefore EU = EU−. Thus we find that, for
T(η , y x c )(n+Kt/2)+Q two-qubit unitary operations, we have the inequality
≤ | xyiA12B12 | iA1| iB1| xyiA2B2
√ǫ(2n+Kt)+Q. (29) 2E CE. (35)
≤ U ≥ +
6
V. COMMUNICATION WITHOUT operationU† is equalto U∗. Ifwe takethe complexcon-
ENTANGLEMENT jugate of the state, then the action of U onthis complex
conjugatestatewillbethesameasthatofU∗ontheorig-
For the case of two-qubit unitaries we may obtain in- inal state. To reverse the communication scheme on the
equalities between the communication that may be ob- complex conjugate state, we simply apply the complex
tainedwithandwithoutentanglement. Toseethis,again conjugate of the sequence of operations in Eq. (38):
consideranypairofrates(R ,R )intheinteriorofSE.
→ ← U
We may select a second pair (R′ ,R′ ) that is closer to (V(0) V(0))TU U(V(τ) V(τ))T. (39)
→ ← A12 ⊗ B12 ··· A12 ⊗ B12
but not on the upper boundary, so that R′ > R and
→ →
R′ >R . Here we can not take the exact complex conjugate of
← ←
Then there exists a communication scheme that com- the state, but we can take the complex conjugate of the
municates n τR′ bits from Alice to Bob and n first term in Eq. (37). To obtain the complex conjugate
τR←′ bits froam≥Bob→to Alice via τ uses of U and wbit≥h of this term, note that we may express |cxyiA2B2, via a
fidelity 1 ǫ. The input state is x y ψ , and Schmidt decomposition, in the form
− | iA1| iB1| iA2B2
the output state η is obtained via a process of the
xy AB
| i
form of Eq. (4). c = λi ϕi χi . (40)
| xyiA2B2 xy| xyiA2| xyiB2
The process described by Eq. (4) consumes entangle-
Xi q
ment resourcesto perform this communication. We con-
sider a strategy for recovering the initial entanglement ThefirstterminEq.(37)isthereforeproportionaltothe
resource state ψ by subsequently performing the state
communication|scihAe2mB2e (4) in reverse. (This approach is
analogous to that applied in Ref. [26].) To achieve this, |yiA1|xiA3|yiA4|xiB1|yiB3|xiB4 λixy|ϕixyiA2|χixyiB2.
Alice and Bob must again retain copies of their inputs Xi q
in auxiliary subsystems A and B . In addition, to en- (41)
3 3
sure that at the end of this process Alice and Bobretain We may take the complex conjugate of this state using
the communicated information, Bob must create a copy the local operations
of the output x and Alice must create a copy of y. We
therefore add auxiliary subsystems A4 and B4 that are |yiA1hy|⊗|xiA3hx|⊗ |ϕix∗yiA2hϕixy|, (42)
initially in the state 0 . After performing the communi- xy i
X X
| i
cation we copy the value y from A1 to A4 and copy the x x y y χi∗ χi . (43)
value x from B to B . | iB1h |⊗| iB3h |⊗ | xyiB2h xy|
1 4 xy i
FromEq.(18),wemayexpressthestateafterthecom- X X
munication process, but before the outputs are copied These are conditional operations with A and A (for
1 3
into A4 and B4, as Alice) and B1 and B3 (for Bob) as the controls. Note
that it is retaining the value of x in A and the value of
3
1−ǫxy|yiA1|xiB1|cxyiA2B2 +√ǫxy|exyiA12B12 y in B3 that makes these operations possible.
x y 0 0 , (36) Applying these operations gives the state
(cid:8)p ⊗| iA3| iB3| iA4| iB4 (cid:9)
where e is some normalized error state. Copy- 1 ǫ y x c∗ y x x y
| xyiA12B12 − xy| iA1| iB1| xyiA2B2| iA4| iB4| iA3| iB3
ingtheresultsofthecommunicationtotheauxiliarysub-
systems A4 and B4 then yields p +√ǫxy|e(x2y)iA124B124|xiA3|yiB3. (44)
Now we have a state that is close to the complex conju-
1 ǫ y x c y x x y
− xy| iA1| iB1| xyiA2B2| iA4| iB4| iA3| iB3 gate of the output state of the communication process.
p +√ǫxy|e(x1y)iA124B124|xiA3|yiB3. (37) Itisthereforepossibletoapproximatelyreversethecom-
munication process by performing the sequence of oper-
Here weareusing superscriptsonthe errorstateto indi- ationsgivenin Eq.(39) to state (44). In this way,via 2τ
cate that it has also been changed under this operation. implementations of U, we obtain the output state
Inordertoreversethecommunicationscheme,wewish
twoasapupselydtthoepienrvfoerrmsetohfetchoemsmequuneinccaetioonf:operations that |xiA1|yiB1|ψ∗iA2B2 −√ǫxy|e(x3y)iA12B12 |yiA4|xiB4
nh i
(V(0) V(0))†U† U†(V(τ) V(τ))†. (38) +√ǫxy|e(x4y)iA124B124 |xiA3|yiB3. (45)
A12 ⊗ B12 ··· A12 ⊗ B12 o
Then we perform local transformations that take the
Itispossibletoapplythe inversesofthelocaloperations
complex conjugate of ψ∗ , giving the final state
V(k) and V(k), because we assume that it is possible to | iA2B2
A12 B12
performarbitrarylocalunitaryoperations. Applyingthe ζ = x x y y ψ y x
inverse of operation U is more difficult, however. When | 1i | iA1| iA3| iB1| iB3| iA2B2| iA4| iB4
the operation U is in the simple form (34), the inverse −√ǫxy|e(x5y)i+√ǫxy|e(x6y)i. (46)
7
Here we have omitted the subscript AB on states in the Given any ǫ > 0, we may perform communication with
0
entire Hilbert space for brevity. fidelity 1 ǫ provided we create the initial state with
A B 0
H ⊗H −
We now consider a sequence of M processes of this fidelity 1 ǫ /2, and 4M√ǫ ǫ /2. The total com-
0 0
type, communicating M messages x to x from Alice municatio−n from Alice to Bob≤is Mn MτR′ , and
1 M a ≥ →
toBobandM messagesy toy fromBobtoAlice. Ifwe the total communication from Bob to Alice is Mn
1 M b
usetheoutputstatefromthefirstcommunicationprocess MτR′ . Thetotalnumberofoperationsdoesnotexcee≥d
←
(46), and perform the same communication process to 2Mτ+Kτ/2E +C(ǫ ,ψ )√Kτ,whichdoesnotexceed
0 ψ 0
communicate the second two messages x2 and y2, then 2Mτ +Kτ/2E0+C(ǫψ,ψ0)√Kτ. The communications
the state we will obtain is per operation from Alice to Bob and Bob to Alice are
therefore at least
ζ = x x x x y y y y ψ y y
| 2i | 1 2iA1| 1 2iA3| 1 2iB1| 1 2iB3| iA2B2| 1 2iA4 MτR′
⊗|x1x2iB4 −√ǫx1y1|e(x71)y1i+√ǫx1y1|e(x81)y1i 2Mτ + Kτ +C(ǫ→/2,ψ )√Kτ,
−√ǫx2y2|e(x52)y2i+√ǫx2y2|e(x62)y2i. (47) 2E0 MτR′0 0
← . (52)
Doing this M times, eachstep generatesanadditional 2Mτ + Kτ +C(ǫ /2,ψ )√Kτ
2E0 0 0
two error terms of the form given in Eq. (46), resulting
These rates will be at least R /2 and R /2 provided
in the final state → ←
R K R √K
|ζMi=|XiA1|XiA3|YiB1|YiB3|ψiA2B2|YiA4|XiB4 R→′ −R→ ≥ 4M→E +C(ǫ0/2,ψ0) →2M ,
0
M
+ √ǫxiyi −|e(x5i,yii)i+|e(x6i,yii)i . (48) R←′ −R← ≥ 4RM←EK +C(ǫ0/2,ψ0)R←2M√K. (53)
Xi=1 (cid:16) (cid:17) 0
Given any ǫ > 0, we may acheive the rate pair
Here we have used X to represent the sequence of mes- 0
(R /2,R /2) with fidelity at least 1 ǫ and without
sages (x ,x ,...,x ) and Y to represent the sequence → ← 0
1 2 M −
initial entanglement by selecting the number of repeti-
of messages (y ,y ,...,y ). In order to perform perfect
1 2 M
tions M to be sufficiently large so that inequalities (53)
communication we would require the state
are satisfied. For this value of M we ensure that the fi-
ζ = X X Y Y ψ Y X . (49) delity 1 ǫ is achieved by selecting the block size τ to
| i | iA1| iA3| iB1| iB3| iA2B2| iA4| iB4 − 0
be sufficiently large so that the inequality 4M√ǫ ǫ /2
0
It is easy to see that the fidelity of the state ζ with ≤
M is satisfied.
| i
respect to the state for perfect communication satisfies
Thus, for two-qubit unitary operations, if the rate
pair (R ,R ) is in the interior of SE, the rate pair
ζ ζ 2 1 4M√ǫ. (50) → ← U
|h | Mi| ≥ − (R→/2,R←/2)is achievablewithout entanglementassis-
Inordertocommunicatealargeamountofinformation tance. AsR→+R← maybe takento be arbitrarilyclose
withoutpriorentanglement,weconstructanapproxima- to C+E, we have the inequality for two-qubit unitaries
tion of the initial resource state ψ using a limited
number of applications of U. We|agiAai2nB2use the fact that C+E ≤2C+. (54)
we are limiting the dimension of the auxiliary state that BHLSderivetheinequalityC+ EU,givingtheordering
≤
may be used. The dimension of ψ may not be of the capacities for two-qubit unitary operations
larger than 2Kτ, which means that|thiAe2eBn2tanglement of CE 2C 2E . (55)
this state cannot be larger than Kτ/2 ebits. It is easily + ≤ + ≤ U
seen that this state may be created with O(Kτ) opera- Similarly, R→ may be taken to be arbitrarily close to
tions. C→E,andR← maybetakentobearbitrarilyclosetoC←E,
In particular, let ψ be a state with entanglement implying the further inequalities for two-qubit unitary
0
E0 >0 that can be c|reaited via a single application of U operations C→E ≤ 2C→ and C←E ≤ 2C←. For two-qubit
from an initially unentangled state. From the theory of unitaries the communication capacities in each direction
entanglement concentration [27], we may create a maxi- are identical [16], so these inequalities may alternatively
mallyentangledstatewithentanglementuptoKτ/2and be expressed as
ficodpeileitsyoaftthleeasstta1te−ǫψψ u.sinHgerKe τC/(2ǫE,0ψ+)Ci(sǫψa,ψco0n)√stKanτt C→E =C←E ≤2C← =2C→. (56)
0 ψ 0
| i Note that these results are crucially dependent on the
thatdependsonthefidelityrequiredandtheinitialstate.
fact that we have limited the dimension of the ancilla
Thus the number of operations requiredis no more than
Kτ/2E +C(ǫ ,ψ )√Kτ, which is of order Kτ. thatmaybe used. The numberofoperationsrequiredto
0 ψ 0
create the initial state scales no faster than O(Kτ), and
If the initial state ψ is created with fidelity 1
| iA2B2 − O(Kτ)/Mτ approacheszero in the limit M . If the
ǫ , then the fidelity of the final state resulting from the
ψ →∞
number of operations required to create the initial state
communication process will satisfy
scaledsuperlinearlywithτ,thenitwouldnotbepossible
ζ ζ 2 1 ǫ 4M√ǫ. (51) to take this limit.
M ψ
|h | i| ≥ − −
8
VI. BIDIRECTIONAL ENSEMBLES j. The information that Alice may communicate to Bob
for each value of j is given by
A powerful means of deriving results for the case of
communication in a single direction is by considering
S p ρB p S(ρB), (62)
the Holevo information of an ensemble of states. When i ij!− i ij
i i
a joint state shared between Alice and Bob, ψ , is X X
i
| i
gbilveen w=ithppir,oψbaibi.litySipmi,ilwarelydewneotme athyisdeansottheetehneseemn-- wavheerraegρeBicjo=mmTruAn|iψciajtii.onAvisereaqguianlgtooverthevaluesofj,the
E { | i}
semble of reduced density matrices possessed by Bob by
E = p ,ρ , where ρ = Tr (ψ ψ ). The index i is
i i i A i i
csihcoasle{innfboyrmA}alticioen,.anIfdamloanygbseequus|eednichteooc|fotmhemsutnaitceaste|ψciliasis- χ→up(E)=Xj qj"S Xi piρBij!−Xi piS(ρBij)#. (63)
givenwithprobabilitiesp ,Alice mayperformcommuni-
i
cation equal to χ(E) [19, 20]. When Alice does not have advance information about
BHLS showthat the communicationcapacityin a sin- the values of j, she will not be able to perform the cod-
gledirectionmaybeevaluatedbyconsideringtheHolevo ing in this way. Nevertheless, it is obvious that it will
information before and after applying the operation U; notbe possible to performmore communicationwithout
that is, we may define priorinformationaboutthevaluesofj. Thereforeχ→up(E)
gives an upper limit to the communication that may be
∆χ→ =sup[χ(Tr U ) χ(Tr )]. (57) performedfrom Alice to Bob using this ensemble. In ex-
U E A E − AE actly the same way, we see that an upper limit to the
communication possible from Bob to Alice is given by
We take the convention that
U p ,U Φ , (58)
Tr E ≡{pi,Tr| i(iΦ} ) , (59) χ←up(E)= piS qjρAij− qjS(ρAij), (64)
XE ≡{ i X | ii } Xi Xj Xj
   
where Tr (ψ ) Tr (ψ ψ ). We use the superscript
“ ” to inXdi|caite≡thatXthi|s iishth|e difference in the Holevo where ρAij =TrB|ψiji.
→ We may obtain a lower bound by assuming that Alice
information for communication from Alice to Bob. We
and Bob code independently of the messages being en-
may similarly define
codedbytheotherparty. IfAlicechoosesmessagei,and
∆χ← =sup[χ(Tr U ) χ(Tr )], (60) has no information about j, she may take Bob’s density
U B E − BE matrix to be ρB = q ρB.
E i j j ij
In order to improve upon this, Bob may apply some
forcommunicationfromBobtoAlice. ItisshowninRef. j-dependent complePtely positive map B to his portion
[16] that C→E =∆χ→U and C←E =∆χ←U. ofthestate. TheideabehindusingthiTsjmapistoreduce
Hereweapplysimilarideastothecaseofbidirectional thej dependenceofBob’sreduceddensitymatrix,sothe
communication. In this case, let i be the message en- averageddensitymatricesρB aremoredistinguishablefor
i
coded by Alice and j be the message encoded by Bob. differentvaluesofi. Withthisoperation,Alice maytake
We may then define a bidirectional ensemble by Bob’s density matrix to be
E ={pi,qj,|ψiji}. (61) ρBi = qjTjB(ρBij). (65)
In general, Bob has a reduced density matrix that de- Xj
pendsoniandj,whichhemayusetoobtaininformation
Via coding over multiple ensembles, Alice may perform
abouti. Similarly,Alicehasareduceddensitymatrixde-
communication to Bob equal to the Holevo information
pendent on i and j, and may perform measurements to of the ensemble p ,ρB . We now take the supremum of
gain information about j. this over the ma{psi iB} , and define
In order to apply the same ideas as for the case of {Tj }
communication in a single direction, it is necessary to
χ→( )= sup χ( p ,ρB ), (66)
consider the bidirectional communication that may be lo E { i i }
{TB}
performed using this ensemble. Here we give upper and j
lower bounds, and we will show in Sec. VIIB that these where ρB is defined as in Eq. (65). This is a lower limit
bounds coincide for a particular type of ensemble. i
tothecommunicationthatmaybeperformedfromAlice
In order to obtain an upper bound, we may assume
to Bob. Similarly a lower limit to the communication
that Alice and Bob use information about the other’s
that may be performed from Bob to Alice is
messageintheir encoding. IfAlice knowsinadvancethe
values ofj that Bobwillbe using, thenshe mayperform χ←( )= sup χ( q ,ρA ), (67)
blockcodingoverthoseensembleswiththesamevalueof lo E {TA} { j j }
i
9
where that E = E− for two-qubit operations.) We then ob-
U U
tainthecodingstatesviaasetofoperations VA VB ,
ρAj = piTiA(ρAij), (68) where ViA and ViB are local operations on{Aliice⊗’s ian}d
Xi Bob’s sides, respectively. We use the notation where (1)
and A is a completely positive map applied by Alice. indicates Alice’s ancilla, (2) indicates Alice’s qubit upon
Ti which U acts, (3) indicates Bob’s qubit upon which U
We will also denote the exact values of the maximum
acts, and (4) indicates Bob’s ancilla. The ancillas may
communication that it is possible to communicate from
AlicetoBobandfromBobtoAlicebyχ→( )andχ←( ), be qubits or arbitrary dimensional qudits. In addition,
E E we use A to indicate Alice’s entire system(1 and 2),and
respectively. From the definitions, it is evident that
we use B to indicate Bob’s entire system (3 and 4). In
χ→( ) χ→( ) χ→( ), χ←( ) χ←( ) χ←( ). this section we indicate the subsystem with superscripts
lo E ≤ E ≤ up E lo E ≤ E ≤ up E
(69) (rather than subscripts) to avoid notational conflicts.
We may also consider the maximum bidirectional com- TheoperationsV VA VB mustsatisfytwocriteria.
i ≡ i ⊗ i
munication that it is possible to perform with the en- (1) V commutes with U. This means that each state
i
semble, χ↔( )=χ→( )+χ←( ). Thisquantity hasthe V Ψ will have its entanglement decreased by E for
i U
E E E | i
upper and lower bounds operation U.
(2) For all states φ ,
χ↔( )=χ→( )+χ←( ), χ↔( )=χ→( )+χ←( ). | i
lo E lo E lo E up E up E up E
Analogous to the case for communication in a si(n7g0le) piTrA(Vi|φi)= 1211⊗ρ(4), (72)
i
direction, we may define the maximum that the classi- X
cal communication capacity of an ensemble may be in- where ρ(4) =TrA3(φ ) is the reduced density matrix for
| i
creased, Bob’s ancilla (4).
As the operation U does not act on the ancilla, ρ(4)
∆χ↔U =sup[χ↔(UE)−χ↔(E)]. (71) will be unchanged under U. That is,
E
This quantityisanalogousto the unidirectionalquantity piTrA(Vi|Ψi)= piTrA(ViU|Ψi)= 2111⊗ρ(4).
∆χ→ introduced by BHLS. However, in contrast to the i i
U X X
unidirectional case, it has not been proven that CE = (73)
+
∆χ↔. This means that the first term on the right side of Eq.
U
(12)willbeunchangedbytheoperationU,andtherefore
that the total one-way communication possible is E .
U
VII. ENSEMBLES FOR For U in the form of Eq. (34), the above condi-
ENTANGLEMENT-ASSISTED tions are satisfied for the operators V = σ(2) σ(3),
COMMUNICATION for i 0,1,2,3 . Here σ are the Paiuli miatri⊗cesifor
i
∈ { }
i 1,2,3 ,σ is the identity, and the superscripts indi-
0
∈{ }
Inthissectionweshowhowtoobtaininitialensembles cate the qubits upon which the operators act. It is easy
for two-qubit operations such that the communication to see that these operators commute with U in the form
possible in one or both directions may be increased by of Eq. (34). To show the second property, let us express
E . Wefirstconsiderthesimplercaseofcommunication thereduceddensitymatrixatBob’sside,ρB =Tr (Ψ ),
U A
| i
inasingledirection,thenextendthistothebidirectional in the form
case.
3
ρB = 1 σ(3) ρ(4). (74)
2 k ⊗ k
A. Communication in a single direction kX=0
It is easily seen that, taking p =1/4, we obtain
i
We consider the case of ∆χ→. As all two-qubitopera-
tionsaresymmetric,theresultUsforthiscasealsoapplyto piTrA(Vi|Ψi)= piσi(3)ρBσi(3) = 21σ0(3)⊗ρ(04).
∆χ←. Note that the second term on the right-hand side i i
U X X (75)
ofEq.(12)istheaverageoftheentanglementofthecod-
Since the trace over qubit (3) gives ρ(4), and the trace
ing states. Therefore, if each of the states of the initial
of each of σ , σ , and σ is equal to zero, we must have
ensemble are chosensuchthat the entanglementofthese 1 2 3
states is decreasedby the maximum EU by operationU, ρ(04) = ρ(4). We therefore obtain Eq. (72), satisfying the
thenthesecondterminEq.(12)willbedecreasedbyEU second condition for the operations Vi. Thus we find
by the operation. If the first term is constant, then the that, by coding with equal probabilities pi = 1/4 each
(2) (3)
total increase in the Holevo information will be E . of the four states σ σ Ψ , it is possible to increase
U i i | i
In order to obtain such an ensemble, let us start with the HolevoinformationbyE . Thuswehaveproventhe
U
an initial state Ψ such that the entanglement is de- inequality ∆χ→ E . Using the equality CE = ∆χ→
creased by the m|axiimum E via operation U. (Recall [16], we have tUhe≥ineqUuality CE E . → U
U → ≥ U
10
B. Bidirectional communication As we have taken the lower limits χ→( ) and χ←( )
lo E lo E
to be the supremums over completely positive maps of
Next we apply a similar coding protocol to the more χ(EB)andχ(EA),andwehaveinequalities(69),wemust
complicated case of bidirectional communication. We have
startwithastate Ψ suchthatU decreasestheentangle- χ→( )=χ→( )=χ→( ), χ←( )=χ←( )=χ←( ).
ment by the maxi|muim E . Alice encodes via the set of lo E up E E lo E up E E
U (84)
four operators σ(2)σ(3) , and Bob encodes via the four This implies that χ↔( )=χ↔( )=χ↔( ).
operators σ(2){σ(3i) .iA}lice’s and Bob’s operators com- Therefore,forthilsospEecifictuyppEeofbidireEctionalensem-
{ j j }
mute, so the order in which these operators are applied ble, the upper andlowerlimits onthe bidirectionalcom-
is irrelevant. We therefore have a total ensemble of 16 munication that it is possible to perform coincide, and
states we have an explicit expression for the bidirectional com-
munication that may be performed. To understand the
E ={pi,qj,σi(2)σi(3)σj(2)σj(3)|Ψi}, (76) reason for this, Bob performs local operations that re-
move the j dependence of his part of the shared entan-
where we take pi = qj = 1/4. Note that each of the gled state. This allows Alice to perform coding without
operators applied by Alice and Bob commutes with U, regard for the value of j. Similarly, Bob may perform
for U in the form of Eq. (34). Therefore, for both the coding without regard for the value of i.
initial and final ensembles, the members of the ensemble Nownotethat,undertheactionoftheoperationU,the
are of the form |ψiji = σi(2)σi(3)σj(2)σj(3)|φi, where |φi = firsttermsinEqs.(80)and(81)areconstant(becausethe
Ψ for the initial ensemble and φ = U Ψ for the final operation does not act upon the ancillas). Also the en-
| i | i | i
ensemble. This simple form allows us to determine the tanglement S[Tr (φ )] = S[Tr (φ )] decreases by E .
A B U
| i | i
exact bidirectional communication that it is possible to Therefore the total increase in the bidirectional commu-
perform. nication possible is equal to 2E .
U
Wenowtakethecompletelypositivemaps A and B Aswehavegivenanexplicitschemethatincreasesthe
Ti Tj
to be the simple unitary operations bidirectional communication that it is possible to per-
formby2E ,wehaveproventheinequalityfortwo-qubit
U
A(ρ)=σ(2)ρσ(2), B(ρ)=σ(3)ρσ(3). (77) operations2E ∆χ↔. Usingtheotherinequalitiesde-
Ti i i Tj j j U ≤ U
rived in the preceding sections, we have the sequence of
This gives inequalities CE 2C 2E ∆χ↔.
For the case+s o≤f the+c≤notUan≤d swaUp operations, these
ρB = q B[σ(3)σ(3)Tr (φ )σ(3)σ(3)]
i jTj i j A | i j i inequalities are equalities. A potential way of proving
Xj equalitiesinthegeneralcaseistoshowthatensemblesof
= q [σ(3)Tr (φ )σ(3)] the form of Eq. (76) can be preparedusing averagebidi-
j i A | i i rectional communication of χ↔( ) per ensemble. Such
Xj preparation is efficient in the seEnse that the communi-
=σ(3)Tr (φ )σ(3), (78)
i A | i i cation required to prepare the ensembles is the same as
what may be performed using the ensembles.
and similarly
In the unidirectional case, it is known that ensembles
ρA = p A[σ(2)σ(2)Tr (φ )σ(2)σ(2)] can be prepared efficiently [28, 29]. BHLS apply this
j iTi i j B | i j i result to show that, given an ensemble such that the
i E
X Holevo information is increased by ∆χ = χ(Tr U )
=σj(2)TrB(|φi)σj(2). (79) χ(TrAE), the entanglement-assisted communicaAtionEca−-
pacity in a single direction is at least ∆χ. This proof is
We then obtain the Holevo informationof the ensembles
based upon a communication scheme where initial com-
EB = p ,ρB and EA = q ,ρA as
{ i i } { j j } munication is obtained by some finite number of opera-
tions, and used to create initial ensembles. The Holevo
χ(EB)=S(1211⊗ρ(4))−S[TrA(|φi)]=χ→up(E), (80) information of these ensembles is increased via applica-
χ(EA)=S(12ρ(1)⊗11)−S[TrB(|φi)]=χ←up(E). (81) tmiounniocfattihoen.opTehriasticoonmUmuannidcautsieodntmoaoybbtaeinusfeudrtthoerobctoamin-
Here we have used further initial ensembles, and the process is repeated.
This process may be repeated a large number of times,
TrA σi(2)σi(3)σj(2)σj(3)|φi ∝11⊗ρ(4), (82) sothattheaveragecommunicationperoperationisequal
i to ∆χ.
X (cid:0) (cid:1)
TrB σi(2)σi(3)σj(2)σj(3)|φi ∝ρ(1)⊗11, (83) It is clear that a similar communication scheme could
be applied in the bidirectional case, if it were possible
j
X (cid:0) (cid:1) to efficiently create initial ensembles. Not all bidirec-
where ρ(1) =Tr (φ ) is the reduced density matrix for tional ensembles can be prepared efficiently. For exam-
2B
| i
Alice’s ancilla. ple, consider the ensemble = 1/2,1/2, ψ where
ij
E { | i}