Coherent Communication of Classical Messages

0.13 MB·English

by  Aram W. Harrow

Coherent Communication of Classical Messages
Aram Harrow
MIT Physics, 77 Massachusetts Ave, Cambridge, MA 02139∗
(Dated: February 1, 2008)
We define coherent communication in terms of a simple primitive, show it is equivalent to the
ability to send a classical message with a unitary or isometric operation, and use it to relate other
resourcesin quantuminformation theory. Usingcoherent communication, weareabletogeneralize
super-densecoding topreparearbitraryquantumstates insteadof onlyclassical messages. Wealso
derive single-letter formulae for the classical and quantum capacities of a bipartite unitary gate
assisted byan arbitrary fixedamount of entanglement peruse.
Beyond qubits and cbits us to exactly describe the power of cobits in terms of
4
The basic units of communication in quantum infor- conventionalresources. Thepurposeofthispaperisthus
0
0 mation theory are usually taken to be qubits and cbits, not to define a new incomparable quantum resource,
2 meaningoneuseofanoiselessquantumorclassicalchan- but rather to introduce a technique for relating and
nel respectively. Qubits and cbits can be converted to composingcommunicationprotocols. Thenwewillapply
n
a each other and to and from other quantum communica- coherent communication to remote state preparation[2].
J tion resources, but often only irreversibly. For example, This will allow us to generalize super-dense coding

9 one qubit can yield at most one cbit, but to obtain a and determine the classical and quantum capacities of
qubit from cbits requires the additional resource of en- unitary gates with an arbitrary amount of entanglement
3
tanglement. Byintroducinganewresource,intermediate assistance.
v
in power between a qubit and a cbit, we will show that
1
9 manyresourcetransformationscanbe modifiedto be re- Sources of coherent communication
0 versible and thus more efficient. Qubitsandcbitsarisenaturallyfromnoiselessandde-
7 If x is a basis for C2, then a qubit channel phasing channels respectively, and can be obtained from
x=0,1
0 {| i}
canbe described as the isometry x x and a cbit anynoisychannelbyappropriatecoding[3,4,5,6]. Sim-
3 | iA →| iB
canbe writtenas x x x . HereAis the sender ilarly, we will show both a natural primitive yielding co-
0 | iA →| iB| iE
/ Alice, B is the receiver Bob and E denotes an inacces- herent bits and a coding theorem that can generate co-
h
sible environment, sometimes personified as a malicious herent bits from a broad class of unitary operations.
p
Eve. Tracing out Eve yields the traditional definition of The simplest way to send a coherent message is by
-
t a cbit channel with the basis x considered the com- modifying super-dense coding[7]. In [7], Alice and Bob
n
a putational basis. {| i} begin with the state Φ = 1 1 x x , which
u Define a coherent bit (or “cobit”) of communication we call an ebit. Alice e|nc2oides a√t2wPo-xb=it0m| eisAs|agieBa a by
q 1 2
as the ability to perform the map x x x . applying Za1Xa2 to her half of Φ and then sending it
v: Since Alice is free to copy or destroy|heirAch→ann|eilAin|piuBt, to Bob, who decodes by applyi|ng2i(H I)cnot to the
Xi 1qubit > 1cobit > 1cbit, where X > Y means that state, obtaining ⊗
the resource X can be used to simulate the resource Y.
r
a We will also write X > Y (c) when X +Z > Y +Z (H I)cnot(Za1Xa2 I)Φ2 = a1 a2
⊗ ⊗ | i | i| i
for some resource Z used as a catalyst and X > Y (a)
when the conversion is asymptotic, by which we mean Now modify this protocol so that Alice starts with a
that X⊗n &Y⊗≈n for n sufficiently large. quantum state |a1a2i and applies Za1Xa2 to her half of
In Prop. 1, we will see that coherent bits come from Φ2 conditioned on her quantum input. After she sends
| i
any method for sending bits using a coherent procedure her qubit and Bob decodes, they will be left with the
(a unitary or isometry on the joint Alice-Bob Hilbert state |a1a2iA|a1a2iB. Thus,
space); hence their name. From their definitions, we see
1qubit+1ebit>2cobits (1)
that cobits can be thought of as cbits where Alice con-
trols the environment, making them like classical chan-
Infact,anyunitaryoperationcapableofclassicalcom-
nels with quantum feedback. Further connections be-
municationisalsocapableofanequalamountofcoherent
tween cobits and cbits will be seen later in this Letter,
communication,thoughingeneralthisonlyholdsasymp-
whereweshowthatirreversibleresourcetransformations
totically and for one-way communication.
areoftenequivalenttoperforming“1cobit>1cbit,”and
in [1] where several communication protocols are “made Proposition 1 If U is a bipartite unitary or isometry
coherent” with the effect of replacing cbits with cobits. such that
In the first half of this paper we will describe how to
obtain coherentbits and then how to use them, allowing U +e ebits>Ccbits (a)
2
then Here( )denotes communicationfromAlice toBoband
→
( ) means from Bob to Alice.
U +e ebits>Ccobits (a) . ←
Uses of coherent communication
HereweconsiderU aresourceinthesenseof[8],which
Bydiscardingherstateaftersendingit,Alicecancon-
considered the asymptotic capacity of nonlocal unitary
vert coherent communication into classical communica-
operations to generate entanglement and to send cbits.
tion, so 1cobit > 1cbit. Alice can also generate en-
By appropriate coding (as in [9]), we can reduce Propo-
tanglement by inputting a superposition of messages (as
sition 1 to the following coherent analogue of Holevo-
in [8]), so 1cobit > 1ebit. The true power of coher-
Schumacher-Westmoreland[3] (HSW) coding.
ent communication comes from performing both tasks—
classical communication and entanglement generation—
Lemma 2 (coherent HSW) Given a set of bipartite
pure states = ψ l , an isometry U such that simultaneously. This is possible whenever the classical
S {| xiAB}x=1 ψ message sent is random and nearly independent of the
U x = x ψ , and an arbitrary probability dis-
ψ| iA | iA| xiAB other states at the end of the protocol (as in the “obliv-
tribution p then
x
{ } ious” condition of [12]).
U χcobits+E ebits (a) Teleportation [13] satisfies these conditions, and in-
ψ
≥
deed a coherent version has already been proposed in
whereE =PxpxS(TrAψx)and χ=S(PxpxTrAψx)− [14]. Given an unknown quantum state |ψiA and an
E. EPR pair Φ , Alice begins coherent teleportation
| 2iAB
not by a Bell measurement on her two qubits but
Proof One can show[3, 5] that for any δ > 0,ǫ > 0
by unitarily rotating the Bell basis into the computa-
and every n sufficiently large there exists a code n tional basis via a CNOT and Hadamard gate. This
C ⊂S
with = exp(n(χ δ)), a decoding POVM Dc c yields the state 1 ij XiZj ψ . Using two co-
with e|Crr|or < ǫ and a−type q with p q 6 l{/n s}u∈chC 2Pij| iA | iB
1 herent bits, Alice can send Bob a copy of her regis-
k − k
that every codeword c := c1...cn has type q (i.e. ter to obtain 1 ij ij XiZj ψ . Bob’s decod-
x, c = x = nq ). By error <∈ǫ,Cwe mean that for 2Pij| iA| iB | iB
j x ing step can now be made unitary, leaving the state
∀ |{ }|
anUyscin∈gCN,ehuψmc|a(Irk⊗’sDthc)e|oψrceim>[110]−, Bǫ.ob can make his de- |Φ2i⊗AB2|ψiB. In terms of resources, this can be summa-
rized as: 2cobits+1ebit 1qubit+2ebits. Since the
coding POVM into a unitary operation U defined by ≥
D ebit that we start with is returned at the end of the
U 0 φ = c √D φ . Applying this to his half of a
coDd|ewi|orid ψPc:|=iψ c| i ψ will yield a state within protocol, we need only use it catalytically: 2cobits ≥
| ci | c1i···| cni 1qubit+1ebit (c). Combining this relationwith Eq.(1)
ǫ of c ψ , since measurements with nearly certain out-
c yields the equality
| i| i
comes cause almost no disturbance.
The communication strategy begins by applying Uψ 2cobits=1qubit+1ebit (c). (4)
to c to obtain c ψ . Bob then decodes unitar-
| iA | iA| ciAB
ily with U to yield a state within ǫ of c c ψ . Thusteleportationandsuper-densecodingarereversible
D | iA| iB| ciAB
Since c is of type q, Alice and Bob can coherently per- solongasalloftheclassicalcommunicationisleftcoher-
mute the states of ψ to obtain a state within ǫ of ent.
c
|ciA|ciB|ψ1iA⊗Bnq1···||ψliiA⊗Bnql. Then they can apply en- AnotherprotocolthatcanbemadecoherentisGottes-
tanglement concentration[11] to |ψ1iA⊗Bnq1···|ψliA⊗Bnql to m(i.aen.’sxmetyhod[15] xfor xsimuylati)ngusaingdiostnreibeubtietdanCdNOonTe
obtain nE ebits without disturbing the coherent mes- | iA| iB → | iA| ⊕ iB
≈ cbit in either direction; i.e. 1cbit( ) + 1cbit( ) +
sage c c .
| iA| iB 1ebit>1cnot. At first glance, this a→ppears comp←letely
There are many cases in which no ancillas are pro- irreversible, since a CNOT can be used to send one cbit
duced, so we do not need the assumptions of Lemma 2 forwardorbackwards,orto createone ebit,but nomore
that communication is one-way and occurs in large than one of these at a time.
blocks. For example, a cnot can transmit one coher- Using coherent bits as inputs, though, allows the re-
ent bit from Alice to Bob or one coherent bit from Bob coveryof2 ebits at the end of the protocol,so 1cobit(
to Alice. Given one ebit, a cnot can send one coherent ) + 1cobit( ) + 1ebit > 1cnot + 2ebits, or usin→g
←
bit in both directions at once using the encoding entanglement catalytically, 1cobit( ) + 1cobit( )
1cnot+1ebit (c). Combined with→Eq. (3), this←yield≥s
(H I)cnot(Ya Zb)Φ = b a (2) another equality:
⊗ ⊗ | 2iAB | iA| iB
This canbe made coherentbyconditioning the encoding cnot+ebit=cobit( )+cobit( ) (c).
← →
on a quantum register a b , so that
| iA| iB Another useful bipartite unitary gate is swap, which is
cnot+ebit cobit( )+cobit( ) (3) equivalent to 1qubit( )+1qubit( ), up to catalysis.
≥ → ← ← →
3
Applying Eq. (4) then yields such that
2cnot=1swap (c) UA|ϕiA|0iA′ =XpAk|ϕiA|kiA′ +pAfail|ϕiA|failiA′
k
which explains the similar communication and entangle-
After applying UA, Alice will perform the two-outcome
ment capacities for these gates found in [8]. Previously,
measurement k k , fail fail on system A. The
the most efficient methods known to transform between {Pk| ih | | ih |} ′
these gates gave 3cnot 1swap 1cnot. probability offailure is less than ǫ andupon success, the
≥ ≥ resulting state is
Remote state preparation n
1
In remote state preparation (RSP), Alice uses entan- √nXRk∗(cid:12)ψ¯(cid:11)A|kiA′Rk|ψiB
glement and classical communication to prepare a state k=1 (cid:12)
of her choice in Bob’s lab [2]. Asymptotically, this rela-
This can be simplified if Alice applies the unitary oper-
tion is
ation k k (RT) (i.e. RT to system A condi-
Pk| ih |A′ ⊗ k A k
tioned on the value of system A). Since RTR =I, this
1cbit+1ebit 1remote qubit (a) ′ k k∗
≥ leaves A in the state ψ¯ , disentangled from the rest of
(cid:12) (cid:11)
where“nremotequbits”meantheabilityofAlicetopre- the system so she can(cid:12)safely discard it. After this, Alice
pareanarbitrary2n-dimensionalpurestateinBob’slab. and Bob share the state √1nPnk=1|kiA′Rk|ψiB. Alice
While the entanglement cost of RSP is shown to be op- now uses logn cobits to transmit k to Bob coherently,
timal in [2], this lower bound does not necessarily apply obtaining √1nPnk=1|kiA′|kiB′Rk|ψiB. Bobperformsthe
tsouberxotuetnidneed. Icnompamrtuincuiclaatri,ounsipnrgotcoochoelrsetnhtactoumsmeuRnSiPcaatisona unAitlaircyePankd|kBiohkb|Bh′a⊗v(eRgk†o)nBeafnrdomenΦdswtiothΦ|Φ,nwiAh′Bic′h|ψiisBa.
d n
allowsallofthe entanglementto be recoveredatthe end slight increase in entanglement, though asymptotically
of the protocol, so that asymptotically insignificant. Thus logncobits logd remote qubits+
≥
log(n/d)ebits (c), which implies 1coherent bit
1cobit 1remote qubit (a) (5) ≥
1 remote qubit (a). However, for the cost of preparing
≥
the inital catalyst to vanish asymptocally, we need to
However, here “n remote qubits” means the slightly
perform RSP in many separate blocks, say √n blocks of
weaker ability to prepare √n states, each with 2√n di-
√n qubits each.
mensions, so we cannot achieve the full range of remote
states possible in [2] without n ebits used as catalysts.
Corollary 3 (RSP capacity of unitary gates) If
To show Eq. (5), we will need to examine the pro-
U Ccbits (a) then U C remote qubits (a).
tocol in [2] more closely. Alice wishes to transmit a
≥ ≥
d-dimensional pure state ψ = ψ ψ to Bob using the
| ih | Corollary 4 (super-dense coding of quantum
shared state Φ = 1 d ii . For any ǫ > 0,
| di √dPi=1| iAB states) 1qubit+1ebit 2 remote qubits (a)
choose n = (dlogd/ǫ2). Ref. [2] proved the existence ≥
O
of a set of d d unitary gates R ,...,R such that for Corollary 4 was first proven directly in [16] and in fact,
1 n
×
any ψ, findinganalternateproofwastheoriginalmotivationfor
this work.
n
ǫ d ǫ Entangled RSP: The states prepared in RSP need not
(cid:16)1− 2(cid:17)I ≤ nXRkψRk† ≤(cid:16)1+ 2(cid:17)I (6) always be completely remote: [2] also showed how Al-
k=1
ice can prepare the ensemble = p ,ψ of bipar-
i i
E { }
For k = 1,...,n define Ak = n(1+dǫ/2)(RkψRk†)T, where tSite=stSa(tes upsiTngr χψ=) aSnd−EE=cbits panSd(TSr ebψit)s., wUhsienrge
the transpose is taken in the Schmidt basis of Φd. Due coherentPcoimimuAnicaition allows χPeibitis to bAe riecovered,
to Eq. (6), we have the operator inequalities (1 ǫ)I
− ≤ so that
A I. Thus, if we define A = I A , then
Pk k ≤ fail −Pk k
A 0, TrA < dǫ and A A ,...,A ,A is a
fail fail 1 n fail χcobits+Eebits (a). (7)
≥ ≡ { }
validPOVM.WhenA I isappliedtoΦ ,theprobability ≥E
d
⊗
Oof,o(uOtcomI)eΦAdfa=il i(sIlesOsTth)aΦndǫ,.mSeianscuerefomrenatnyouotpcoermaetokr Hwhereer,ebAyli≥cen’sEi,nwpeutmiesarnesntruicsteesdoftoUtψh:e|siipAac→e s|piaiAn|nψeidiAbBy
lψeTav=es⊗Aψ¯lice|ψ¯a.nidBob⊗witht|heistateRk∗(cid:12)(cid:12)ψ¯(cid:11)⊗Rk|ψiwhere |i1T··h·einpirofoorf po-ftyEpqi.ca(l7)seiqsujeunsctelsikie1·t·h·einp.roof of Eq. (5),
(cid:12) (cid:11)(cid:10) (cid:12)
We(cid:12)now s(cid:12)how how to apply coherent communication butevensimplersincetheoriginalprotocolin[2]already
to the above procedure. First we apply Neumark’s left Alice’s state independent of k upon success. [The
theorem[10] to convert A into a unitary operation UA same benefits of coherent communication do not apply
4
to the “low-entanglement” version of RSP in [2]; here To proveconcavity,considera probabilitydistribution
entanglementcan only be recoveredfrom the part of the p , a set of ensembles and the new ensemble =
j j
{ } E E
message corresponding to the measurement.] p , jj . Takingthe simultaneoussupremumof
{ j | iAB⊗Ej}
Using super-dense coding with Eq. (7) allows super- χ(U ) χ( )overeach subjecttoE( ) E(U )
j j j j j
E − E E E − E ≤
dense coding of entangled quantum states according to e completes the proof.
j
χ/2qubits+(E +χ/2)ebits , a claim for which no
≥ E Quantum capacities of unitary gates We can also con-
direct proof is known.
sider the ability of unitary gates to send quantum infor-
Coherent RSP of entangled states can also help de-
mation. Define Q to be the largest number such that
e
termine the communication capacity of bipartite uni-
U +e ebits Q qubits. Then from Eq. (4), one can
tary gates as introduced in [8]. For any e R, define ≥ e
∈ obtain
C (U) to be the largest number such that U +e ebits
e ≥ 1
C (U)cbits (a). Fornegativevaluesofe,C corresponds Q = C (10)
e e e 2 e+Qe
to creating e ebits per use of U in addition to com-
municating C−e cbits; we arbitrarily set Ce = when where Ce can be determined from Proposition 5.
U e ebits. −∞ Conclusions
6≥− Coherent communication offers a new way of looking
Proposition 5 at quantum information resources in which irreversible
transformations occur only when coherence is discarded
Ce(U) = ∆χe(U) and not just because we are transforming incomparable
:= sup χ(U ) χ( ):E( ) E(U ) e resources. Whenever classical communication is used in
{ E − E E − E ≤ }
a quantum protocol, either as input or output, and the
E
classicalmessageisnearlyindependentoftheotherquan-
where = p ,ψ is an ensemble of bipartite states,
i i
E { } tumstates,theremaybe gainsfrommakingthe commu-
U = p ,Uψ U and χ and E are defined as above.
i i †
E { } nication coherent.
Thus the asymptotic capacityequals the largestincrease The case of noisy coherent communication remains to
in mutual information possible with one use of U if the be fully solved and it would be interesting to find the
averageentanglementdecreasesby nomorethane. This coherent capacity of a noisy channel, or more generally
was proven for e= by [8]. tofindtheirCeandQetradeoffcurves. Somepreliminary
∞
Achieving C (U) ∆χ (U): results in these directions have been obtained[1].
e e
≥
We base ourprotocolonthe one usedin Section4.3of My thanks go to C.H. Bennett, P. Hayden, D. Leung,
[8] to show C =∆χ . For any ensemble , J.Smolin,A.WinterandespeciallyI.Devetakforhelping
∞ ∞ E mecomeupwithandrefinetheideasinthispaper. Iam
U + U also grateful for the hospitality of the Caltech IQI, IBM
E ≥ E
χ(U )cobits+E(U ) ebits (a) (8) T.J.WatsonresearchcenterandtheERATOcenter. My
≥ E E fundingisfromtheNSAandARDAunderAROcontract
(χ(U ) χ( ))cobits
≥ E − E DAAD19-01-1-06.
+(E(U ) E( )) ebits+ (a) (9)
E − E E
Here Eq. (8) used Lemma 2 and Eq. (9) used Eq. (7).
Nowwemovetheebitstothelefthandsideandneglect
the catalytic use of to get ∗ Electronic address: [email protected]
E
[1] I. Devetak, A.W. Harrow, A. Winter, eprint quant-
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Taking the supremum of the right-hand side over all A. Winter, quant-ph/0307100.
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[4] P.W.Shor,“Thequantumchannelcapacityandcoherent
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col, so at all points we work with pure state ensembles. computation, 2002.
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[8] C.H. Bennett, A.W. Harrow, D.W. Leung and
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5
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