The Flow of Information in Interactive Quantum Protocols: the Cost of Forgetting

0.53 MB·

by  Mathieu Lauriere

The Flow of Information in Interactive Quantum Protocols :
the Cost of Forgetting ∗
Mathieu LAURIE`RE1 andDave TOUCHETTE2,3
1NYU-ECNUInstituteofMathematicalSciencesatNYUShanghai,
email:[email protected]
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1 2InstituteforQuantumComputing,andDepartmentofCombinatoricsandOptimization,
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UniversityofWaterloo,email: [email protected]
2
3PerimeterInstituteforTheoreticalPhysics
n
a
J

9

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Abstract
-
t
n In the context of two-party interactive quantum communication protocols, we study a recently de-
a fined notion of quantum information cost (QIC), which possesses most of the important properties of
u
itsclassicalanalogue,seeRef.[Tou15]. Notably,itslinkwithamortizedquantumcommunicationcom-
q
[ plexityhasbeenusedinRef.[BGKK+15]toprovean(almost)tightlowerboundontheboundedround
quantumcomplexityofDisjointness. However,theonlyknowncharacterizationofQICwasthrougha
1
notionofpurificationoftheinputstate. Althoughthisdefinitionhastheadvantagetobevalidforfully
v
quantum inputs and tasks, its interpretation for classical tasks remained rather obscure. Also, the link
2
6 betweenthisnewnotionandothernotionsofinformationcostforquantumprotocolsthathadpreviously
0 appearedintheliterature(e.g. inRefs.[JRS03,JN14,KLLGR15])wasnotclear,ifexistentatall.
2 Wesettleboththeseissues: forquantumcommunicationwithclassicalinputs,weprovideanalter-
0
natecharacterizationofQICintermsofinformationabouttheinputregisters,avoidinganyreferenceto
.
1 the notion of a purification of the classical input state. We provide an exact operational interpretation
0
ofthisalternativecharacterizationasthesumofthecostoftransmittinginformationabouttheclassical
7
inputsandthecostofforgettinginformationabouttheseinputs.Toobtainthischaracterization,weprove
1
: agenerallemma,theInformationFlowLemma,assessingexactlythetransferofinformationingeneral
v
interactivequantumprocesses. Specializingthislemmatointeractivequantumprotocolsaccomplishing
i
X classicaltasks,wearealsoabletodemystifythelinkbetweenQICandtheseotherpreviousnotionsof
r informationcostinquantumprotocols.Furthermore,weclarifythelinkbetweenQICandICofclassical
a
protocolsbysimulatingquantumlyclassicalprotocols.
Finally, we apply these concepts to argue that any quantum protocol that does not forget informa-
tion solves Disjointness on n-bits in Ω(n) communication, completely losing the quadratic quantum
speedup. Thisprovidesaspecificsenseinwhichforgettinginformationisanecessaryfeatureofinter-
active quantum protocols in order to obtain any significant improvement over classical protocols. We
alsoapplytheseconceptstoprovethatQICatzero-errorisexactlynfortheInnerProductfunction,and
n(1−o(1))forarandomBooleanfunctiononn+nbits.
∗A one-page abstract of this work will appear in the Proceedings of the 8th Innovations in Theoretical Computer Science
conference(ITCS2017).
1
Contents
1 Introduction 1
2 Preliminaries: QuantumCommunicationandInformation 3
3 InformationFlowLemma 6
4 MakingSafeCopiesoftheInputs 8
5 TheCostofForgetting: aNewCharacterizationofQIC 11
5.1 AlternateDefinitionsofInformationCostsforProtocolswithClassicalInputs . . . . . . 12
5.2 OperationalInterpretationofHICinTermsofCICandCRIC . . . . . . . . . . . . . . . 13
5.3 OperationalInterpretationofQICinTermsofCICandCRIC . . . . . . . . . . . . . . . 14
5.4 QICandCICareAlmostEquivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.5 RunningProtocolsonSuperpositionofInputs . . . . . . . . . . . . . . . . . . . . . . . 16
5.5.1 ProductDistributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.5.2 GeneralDistributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6 ForgettingInformationinClassicalProtocols 19
6.1 ExtendingtheClassicalSetting: aNewCharacterizationofIC . . . . . . . . . . . . . . 19
6.2 ReversibleClassicalProtocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
7 Disjointness: Speed-upforQuantumProtocolsneedsForgettingInformation 22
8 QuantumSimulationofClassicalProtocols 25
9 CleanProtocols,IP,andRandomFunctions 28
9.1 CleanProtocolsandPhaseEncodingoftheOutput . . . . . . . . . . . . . . . . . . . . 28
9.2 RelatingtoQIC(Π,µ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
9.3 InformationLowerBound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
9.4 InnerProductfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
9.5 RandomFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
9.6 Non-ZeroErrorandClassicalProtocols . . . . . . . . . . . . . . . . . . . . . . . . . . 33
References 33
A ProofsforSection7 35
A.1 ProofofLemma38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
A.2 ProofofLemma39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
B TheVariousNotionsofInformationCost 36
1
1 Introduction
Background. Intwo-partycommunicationcomplexity[Yao79],AliceandBobreceiveinputsxand
y andrunaninteractivecommunicationprotocolbyexchangingmessagesinordertocomputef(x,y)
for some function f that depends on both these inputs. Their goal is to minimize the communication
cost(denotedCCandQCCrespectivelyintheclassicalandthequantumsettings),thatis,theamountof
communication(bitsorqubits). Thismodelhasfoundnumerousapplicationsinmanyareasofcomputer
science. Forexcellentintroductionstoclassicalandquantumcommunicationcomplexities,wereferthe
readerto[KN97]and[dW02]respectively.
One question that has received a lot of attention recently is whether it is possible to perform such
protocols without leaking much information. In classical communication protocols, the information
cost (IC) is defined as the information that the transcript reveals to each player about the input of the
otherone. Inquantumcommunicationprotocols[Yao93],theregistersareinaquantumstate,which,in
general,preventstheplayerfromkeepingtrackofthepreviousmessagesduetotheno-cloningtheorem.
Nevertheless, the parties have quantum workspaces, where they may keep information about previous
messages. Thequestionisthentocalculatehowmuchinformationeverynewmessagerevealstothem,
giventhattheyalreadyknowtheirowninputandhavekeptsomeinformationintheirquantumworkspace
accordingtotheprotocol.
Severalnotionsofinformationcostforquantumprotocolshavealreadybeenusedintheliterature,see
e.g.Refs[KNTSZ07,Kla02,JRS03,JRS09,JN14].Eachnotionwassomehowtailor-madeforaspecific
purposeandveryusefulinthatparticularcase. Nevertheless,thesedefinitionsdidnotseemtoprovide
a general understanding of how information behaves in quantum communication. In Ref. [Tou15] has
beenintroducedageneralnotionofQuantumInformationCost(QIC),whichmeasuresthetotalamount
of quantum information about the inputs that is transmitted during the protocol. The corresponding
notion of quantum information complexity of a function (the minimal QIC of a protocol computing
the function) has been shown to exactly characterize the amortized communication complexity of that
function, which is a fundamental property of the information complexity in the classical setting, see
Ref.[BR11]. Moreover,thisnotionofQIChasalreadyfoundmultipleapplications[Tou15,BGKK+15,
NT16].
However,sofartheonlyknowncharacterizationofQICwasthroughanotionofpurificationofthe
input state. Although this definition has the advantage to be valid for fully quantum inputs and tasks,
itsinterpretationforclassicaltasksremainedratherobscure. Also,thelinkbetweenthisnewnotionand
other notions of information cost for quantum protocols that had previously appeared in the literature
wasnotclear,ifexistentatall.
Ourcontributions. InthispaperweshedanewlightontheQuantumInformationCost(QIC),and
settlebothissuesdescribedabovebyrelatingthisquantitytoseveralothernaturalnotionsofinformation
cost,includingtheclassicalIC,andbyproviding,whentheinputsareclassical,anewcharacterizationof
QICwhichhasanoperationalinterpretationanddoesnotrequireanyreferencetoapurificationregister.
The cornerstone of our work is a general lemma, that we call the Information Flow Lemma (see
Lemma3),whichpreciselycharacterizesthetransferofinformationinquantumprocesses,runonarbi-
trary quantum inputs. This result then specializes to the setting we are interested in, namely quantum
communication protocols. We stress that this lemma has already found other applications besides this
work,inparticulartoprovealowerboundonquantuminformationcomplexityoftheAugmentedIndex
functiononauniformdistributionoverthezerosofthefunction[NT16], withcorollariesonthespace
complexity of quantum streaming algorithm for the DYCK(2) problem of well-formed parentheses
overtwopairsofsymbols.
Wethenturnourattentiontoquantumprotocolswithclassicalinputs.Inthisframework,eventhough
some protocols might modify the input register, it is always possible, since the inputs are classical, to
requirethattheplayersstarttheprotocolbymakingacopyoftheirinputsandworkwiththatcopy. We
callprotocolssuchasthese,wheretheinputregistersareleftuntouched,safeprotocols. Thisseemingly
1
insignificantmodificationoftheoriginalprotocolmightdrasticallychangetheinformationcost. How-
ever,weprovethatitcanonlydecreaseit(seeProposition9). Soitisenoughtostudytheinformation
costofsafeprotocolswhenweareinterestedinminimizingtheQICforcomputingataskwithclassical
inputs.
When studying such quantum protocols with classical inputs, a notion of information cost (called
Classical input Information Cost, or CIC) has been introduced in Ref. [KLLGR15], where a first step
was made to understand its relationship with QIC: the former is a lower bound on the latter – that is,
CIC ≤ QIC. Inordertocompletethepicture,weintroducetwonewnotions: theHolevoInformation
Cost(HIC),whichmeasureshowmuchinformationtheplayershaveabouteachother’sinputattheend
oftheprotocol(around-by-roundvariantwasconsideredinRef.[JRS03,JN14]),andtheClassicalinput
ReverseInformationCost(CRIC),whichcountshowmuchinformationabouttheinputsisforgottenat
each round by the player sending the message (this is somehow the dual under time reversal of CIC).
BasedonourInformationFlowLemma,wegivenewoperationalinterpretationstothesequantitiesand,
informallyspeaking,weshowthattheysatisfythetwofollowingverynaturalrelationships: theHolevo
information cost corresponds to the amount of classical information that was learnt and not forgotten
during the protocol, while the quantum information cost captures all of the information transmitted
during the protocol (what was learnt plus what was forgotten). This yields a new characterization of
QICbyCIC,uptoafactorof2. Sothevariousnotionsofinformationcostintroducedinthispaperare
tightlyrelated,namely(seePropositions16,17and19):
MainResult1:Wehave:HIC=CIC−CRIC,QIC=CIC+CRIC.Moreover,CIC≤QIC≤2·CIC.
TheserelationshipsemphasizetheimportanceofCRIC,thecostofforgettinginformation. Thislast
quantitywouldalwaysbezeroinclassicalprotocols: implicitly, classicalinformationisalwaysclone-
able, hence players can memorize the whole history of the protocol and never forget information. To
understand the link with quantum protocols forgetting information, we introduce a model of classical
reversiblecomputing,endowingclassicalprotocolswiththeabilitytoforgetinformation. Weshowthat
this feature can only increase their information cost, and, as such, forgetting information is somehow
a wasteful phenomenon that should be avoided in the context of classical communication (see Theo-
rem 33). However, in quantum protocols, cloning is not possible in general. This raises the question
whetherthepropertyofforgettinginformationisonlycostlyandshouldstillbeavoidedinsomesense.
We answer this in the negative: forgetting information is absolutely necessary to obtain the quantum
communicationimprovementallowedforcomputingcertainfunctions. Indeed,ifnoinformationisfor-
gotteninaquantumprotocol,thenQIC = HICisformallyverysimilartoIC,andthecontinuityinthe
inputdistributionhasnorounddependence,asintheclassicalcase. Thus,therounddependenceinthis
continuity bound for general quantum protocols that do forget information [BGKK+15] can be under-
stoodasbeingduetothefactthatthesameinformationisforgottenandtransmittedmultipletimes.With
thisobservation, weprovethatanyquantumprotocolforDisjointnessthatdoesnotforgetinformation
haslinearquantumcommunicationcomplexity(seeTheorem36). Hence,quantumprotocolsthatdonot
forgetinformationcannotobtainthequadraticquantumspeed-upfortheDisjointnessfunction[AA05],
andthisabilityofquantumprotocoltoforgetinformationisanessentialfeatureofinteractivequantum
communication,notjustsomeodditywecangetaround. Thiscanbesummarizedasfollows:
MainResult2:Forgettinginformationisuselessinaclassicalreversiblesetting, butitisunavoidable
inthequantumsetting: itisanecessaryfeatureofinteractivequantumprotocolstogetsignificantcom-
municationimprovementoverclassicalprotocols.
Thisimportantdistinctionshowsthattheflowofinformationbehavesquitedifferentlyintheclassical
andinthequantumsetting. However,theclassicalcommunicationcomplexityisalwayslowerbounded
by the quantum communication complexity: quantum messages can simulate classical ones. We can
ask the same question in terms of information: is it always possible to quantumly simulate classical
messages while maintaining the information cost? Our next main result provides a positive answer.
We show that to any classical protocol Π corresponds a quantum simulation protocol Π satisfying
C Q
QCC(Π )=CC(Π ),QIC(Π ,µ)=IC(Π ,µ)foranyinputdistributionµ,andimplementingthe
Q C Q C
2
sameinput-outputchannelΠ =Π . Themainissuewedealwithisthepurestatequantumsimulation
Q C
ofprivaterandomnesswithoutalteringtheinformationcost(seeLemma47).
MainResult3: Foranyclassicalprotocol,thereexistsaquantumprotocolwiththesameinput-output
behaviour,andwithcommunicationandinformationcostssmallerthantheclassicalprotocol.
Thisresultletsusconcludethepaperwithonemoreapplication.FortheInnerProductfunction,QIC
atzero-errorovertheuniformdistributionisexactlyn;asimilarlowerboundofn(1−o(1))holdsfor
arandomBooleanfunctiononn+nbits. Furtherusingthequantumsimulationofclassicalprotocols
mentionned above together with the fact that classical IC is continuous at zero-error [BGPW13a], this
shows that, in the limit when the error ε goes to 0, IC of such a random Boolean function is not only
Ω(n)[BW12,KLL+15],butispreciselyn(1−o(1))(suchatightboundfortheICofInnerProductwas
knownfromRef.[BGPW13b]).
Outlineofthepaper. Thispaperisstructuredasfollows. Aftersomepreliminaries(Section2),we
stateandproveourInformationFlowLemma(Section3). InSections4and5,weproveourresultson
safequantumprotocols,andthenintroduceCRIC,HICandmultipleotherquantumnotionsofinforma-
tioncost(atableisprovidedinAppendixBtokeeptrackofdefinitionsandrelationships). Forthesake
ofcomparison,inSection6wedefineICinaclassicalreversiblecomputationparadigmandshowthat
forgettinginformationiswasteful. Incontrast,weproveinSection7thatthereisnoquantumcommu-
nication speed-up for Disjointness when the quantum protocols are not allowed to forget information.
Then,weshowhowtosimulatequantumlyclassicalprotocolsinSection8. Finallyweproveourresults
onInnerProductandrandomBooleanfunctions(Section9).
2 Preliminaries: Quantum Communication and Information
Quantum Communication Model. QuantumcommunicationcomplexitywasintroducedbyYao
in Ref. [Yao93]. The model we use here is closer to the one of Cleve and Buhrman [CB97], with
pre-shared entanglement, but we allow the players to communicate with quantum messages. In this
model, an r-round protocol Π for a given classical task from input registers A = X, B = Y to
in in
output registers A , B is defined by a sequence of isometries U , ···, U along with a pure
out out 1 M+1
state ψ ∈ D(TinTin) shared between Alice and Bob, for arbitrary finite dimensional registers Tin,
A B A
Tin: the pre-shared entanglement. Above, D(A) is the set of all unit trace, positive semi-definite lin-
B
ear operators mapping A into itself. See Refs [Wat15, Wil13]. We need r + 1 isometries in order
to have r messages since a first isometry is applied before the first message is sent and a last one
after the final message is received. In the case of even r, for appropriate finite dimensional quan-
tum memory registers A , A , ···, A , A(cid:48) held by Alice, B , B , ···, B , B(cid:48) held by Bob,
1 3 r−1 2 4 r−2
and quantum communication registers C , C , C , ···, C exchanged by Alice and Bob, we have
1 2 3 r
U ∈ U(A Tin,A C ), U ∈ U(B TinC ,B C ), U ∈ U(A C ,A C ), U ∈ U(B C ,B C ),
1 in A 1 1 2 in B 1 2 2 3 1 2 3 3 4 2 3 4 4
··· , U ∈ U(B C ,B B(cid:48)C ), U ∈ U(A C ,A A(cid:48)), where U(A,B) is the set of uni-
r r−2 r−1 out r r+1 r−1 r out
tary channels from A to B : see Figure 1. We adopt the convention that, at the outset, A = A Tin,
0 in A
B = B Tin, for odd i with 1 ≤ i < r, B = B , for even i with 1 < i ≤ r, A = A and
0 in B i i−1 i i−1
also B = B = B B(cid:48), and A = A A(cid:48). In this way, after application of U , Alice holds
r r+1 out r+1 out i
registerA ,BobholdsregisterB andthecommunicationregisterisC . Inthecaseofanoddnumberof
i i i
messagesr,theregisterscorrespondingtoU ,U arechangedaccordingly.Weslightlyabusenotation
r r+1
andalsowriteΠtodenotethechannelfromregistersA B toA B implementedbytheprotocol,
in in out out
i.e.foranyinputdistributionµonXY andρ encodingµoninputregistersA B ,
µ in in
Π(ρ )=Tr (U U ···U U (ρ ⊗ψ)). (2.1)
µ A(cid:48)B(cid:48) M+1 M 2 1 µ
NotethattheA(cid:48)andB(cid:48)registersarethefinalmemoryregistersthatarebeingdiscardedattheendof
theprotocolbyAliceandBob,respectively.
3
Recall that for a given state, all purifications are related by isometries on the purification regis-
ters. For classical input registers XY distributed according to µ, we consider a canonical purification
|ρµ(cid:105)XRXYRY ofρAµinBin,with
|ρ (cid:105)XRXYRY =(cid:88)(cid:112)µ(x,y)|xxyy(cid:105)XRXYRY . (2.2)
µ
x,y
WethensaythatthepurifyingregistersR R containquantumcopiesofXY. Then,thestateatround
X Y
i,
ρXi RXYRYAiBiCi =Ui···U1(ρXRXYRY ⊗ψTAinTBin) (2.3)
ispure. Also, we requirethatthefinal marginalstate Π(ρAinBinRXRY) on RXRYAoutBout isclassi-
cal. We say that a protocol Π solves a function f with error ε with respect to input distribution µ if
Pr [Π(x,y)(cid:54)=f(x,y)]≤ε,andwesayΠsolvesf witherrorεifmax Pr[Π(x,y)(cid:54)=f(x,y)]≤ε.
µ (x,y)
Wealsomakeuseofthenotionofacontrol-isometry:itisanisometryactingonaclassical-quantum
register by leaving the content of the classical register unchanged. Such a classical register is called a
control-register.
QuantumInformationCost. Themainquantityofinterestinthisworkisthequantuminformation
cost, as introduced in [Tou15]. In quantum communication protocols, there is no clear notion of a
transcript, so this definition counts how much information is exchanged in each round. In the sequel,
we denote the Von Neumann entropy by H, and for a tripartite state ρABC, we denote the conditional
quantummutualinformation(CQMI)betweenAandBconditionedonCbyI(A:B|C)=H(A,C)+
H(B,C)−H(C)−H(A,B,C). We will make use of many properties of CQMI, among which the
following.
Lemma1 Ifρ=ρABC andσ =σDEF aretwostatesondistinctregisters,then
I(AD;BE|CF) =I(A;B|C) +I(D;E|F) .
ρ⊗σ ρ σ
Ifρ=ρABCD =(cid:80) p(c)|c(cid:105)(cid:104)c|⊗ρABD isaclassical-quantumstatewithclassicalregisterC,then
c c
I(A:B|CD) =E [I(A:B|D) ].
ρ c ρc
Ifρ=ρABCD isapurestate,then
I(A;B|C) =I(A;B|D) .
ρ ρ
Letusrecallthedefinitionofquantuminformationcostintroducedin[Tou15].
Definition2 ForaprotocolΠandaninputdistributionµ,wedefinethequantuminformationcostofΠ
oninputµas
(cid:88) (cid:88)
QIC(Π,ρ)= I(C ;R R |B )+ I(C ;R R |A ).
i X Y i i X Y i
i≥1,odd i≥1,even
Foranyfunctionf,anyinputdistributionµ,andanyε>0
QIC(f,µ,ε)=infQIC(Π,µ) (2.4)
Π
wheretheinfimumisovertheprotocolsΠcomputingf witherrorεw.r.tµ.
Thisquantityhasmanyniceproperties(see[Tou15,BGKK+15]);inparticularitcharacterizesthe(quan-
tum)amortizedcommunicationcomplexity. Westressthatthedefinitionisindependentofthechoiceof
purification.
4
R
Ray
A A A A A A A(cid:48)
in 1 2 3 r−1 r
A
U U U out
1 3 f
C C C
1 3 r−1
Alice
Tin
A
··· Π(ρ)
|ρ(cid:105) |φ (cid:105)
1
C
Bob r
Tin
B
C
2 B
U U out
2 r
Bin B2 B3 Br−1
B(cid:48)
Figure1: Depictionofaquantumprotocolintheinteractivemodel,adaptedfromthelongversionof[Tou15,
Figure1].
5
Discussionaboutcompression. Somepreviousnotionsofinformationcostforquantumprotocols
(e.g. inRefs.[JRS03,JN14,KLLGR15])weremoresimilarinspirittoclassicalinputinformationcost
than to quantum information cost. Our results shed new light on why these previous definitions were
restricted to compression results for a single round. In the first round, Alice does not yet possess any
informationonBob’sinput(asidefromwhatshecaninferfromherowninput).Forone-roundprotocols,
itisthenimmaterialwhetheroneusesclassicalinputinformationcostorquantuminformationcost. But
theninsubsequentrounds,generallyAlicehasinherregisterssomeinformationaboutBob’sinput. Itis
thenpossibleforhertoforgetinformationwhilesendingamessage. Wecanevenconstructaprotocol
where, atthethirdround, BobdoesnotlearnanythingwhereasAliceforgetsalotofinformation. For
such a round of communication, the previous definitions of information cost, e.g. CIC introduced in
Ref.[KLLGR15],wouldevaluateto0whereasQICwouldbelarge. Thus,itisimpossibletocompress
suchaquantummessagedowntoitsCIC,thatis,almostatnocost,whilekeeping,inaround-by-round
fashion, the overall state of the protocol almost equivalent to that in the original protocol. Indeed, we
know from our developments that to forget information we must invest communication. As a conse-
quence, we see that for quantum protocols, it is important to take into account the cost of forgetting
information.
The purification register used in the definition of QIC possibly appears artificial when considering
classicalinputs. Inthisdirection, weprovebelow(seeSection5)anarguablymorenaturalcharacteri-
zation(atleastfromaclassicalcorrelationpointofview)ofeachterminthequantuminformationcost
asthesumofhowmuchinformationabouthisowninputapartyissendingplushowmuchinformation
about the other party’s input he is forgetting. However, we argue that there is still virtue in taking the
purificationoftheclassicalinputviewpoint. Firstly,itenablestokeeptrackofaglobalpurestate,which
inmanysituationsisaremarquablypowerfulviewpoint. Secondlyandmorefundamentally,thepurifi-
cation viewpoint has a nice operational interpretation through the task of quantum state redistribution,
whichisusefulwhenaimingatcompressionresults. Indeed,atanypointoftheinteractiveprotocol,the
purequantumstatecanbeseenasa4-partitestateρARASMRconsistingofthereceiver’sandthesender’s
privateregisters(A andA respectively),themessageregisterM andapurificationregisterR. Then,
R S
eachterminQICisoftheformI(R;M|A ),thatis,themutualinformationbetweenthemessageand
R
theinaccessiblepurificationregister,conditionedonthereceiver’ssideinformation. Suchanexpression
is known [DY08, YD09] to quantify the cost of redistributing the message register while maintaining
correlationswiththereceiver’sandthesender’sprivateregistersaswellastheenvironment. TheCIC
terms can also be given such an operational significance for the information about the sender’s input
thatamessagecontains. However,thisviewpointbreaksdownfortheinformationthatisforgotten(see
theoperationalinterpretationgivenatSection5). Indeed, tomeasuretheamountofinformationbeing
forgotten, we condition on the sender’s side information for sending information about the receiver’s
input. Thistermwouldbehardtoaccountforinacompressionviewpoint(unlesswethinkofmessages
goingbackward). Hence,wethinkthatthepurificationviewpointremainsappropriateforcompression
purposes.
3 Information Flow Lemma
In this section, we state and prove the Information Flow Lemma (see Lemma 3 below), which allows
tokeeptrackexactlyoftheflowofquantuminformationinaninteractiveprotocolandiskeytomuch
of our further developments. Moreover, it gives a lower bound on QIC that does not depend on the
numberofround(seeCorollary5),andisused,amongotherthings,togiveanexactmeaningtothecost
of forgetting in interactive quantum protocols. We present here a quite general version of this result.
However, we stress that a more limited version, that is still sufficient to obtain a lower bound on QIC,
hasalreadyfoundsomeapplications;seeRef.[NT16].
Let us consider the more general framework of bipartite interactive quantum processes, of which
the model of quantum communication complexity defined in Section 2 is a special case. This general
frameworkmodelizesadiscretizedquantumprocessinwhichthereisinteractionbetweentwodistinct,
6
E¯F¯
Extension
A¯ A¯ A¯ A¯
0 1 2 r−1
U1 C¯ D¯ U2 C¯ D¯ Ur A¯
1 1 2 r−1 r
Alice
···
ρ ρ
0 r
Bob
V1 D¯1 C¯1 V2 D¯2 C¯r−1 Vr B¯r
B¯ B¯ B¯ B¯
0 1 2 r−1
Figure2: Depictionofaninteractivequantumprocess,adaptedfromthelongversionof[Tou15,Figure1].
localizedparties,andlocalevolutionateachtimestep.
In more details, Alice and Bob start in a joint state ρA¯0B¯0, for which we consider an arbitrary ex-
0
tension ρA0¯0B¯0E¯F¯ (such that TrE¯F¯(ρA¯0B¯0E¯F¯) = ρA¯0B¯0). The process runs for r +1 rounds, with ρi
the state in round i, registers A¯ , B¯ , C¯ and D¯ in each round, with C¯ , D¯ , C¯ and D¯ being
i i i i 0 0 r+1 r+1
trivial registers in the 0-th and r +1-th round, initially and at the end of the process. In round i, for
1≤i≤r,afterbeinggeneratedbyAlice,registerC¯ getscommunicatedfromAlicetoBob,and,after
i
being generated by Bob, register D¯ gets communicated from Bob to Alice. Register A¯ is a quantum
i i
memoryregisterheldbyAlice,andregisterB¯ isaquantummemoryregisterheldbyBob.Theevolution
i
isthroughlocalisometriesU =UA¯i−1D¯i−1→A¯iC¯i onAlice’ssideandV =VB¯i−1C¯i−1→B¯iD¯i onBob’s
i i i i
side: ρA¯iB¯iC¯iD¯iE¯F¯ =(U ⊗V )ρA¯i−1B¯i−1C¯i−1D¯i−1E¯F¯.
i i i i−1
RegistersE¯F¯ areleftuntouchedthroughout, andcanbethoughtofinthefollowingway: wewant
tomeasurehowmuchinformationBobknowsaboutE¯ fromthepointofviewofsomeonewhoknows
F¯. Wegetthefollowingexactcharacterizationoftheflowofinformationfromthispointofview.
Lemma3 (InformationFlowLemma)Givenaninteractivequantumprocessasdefinedabove,thefol-
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lowingholds:
r
I(E¯;B¯ |F¯) −I(E¯;B¯ |F¯) =(cid:88)(I(E¯;C¯ |F¯B¯ ) −I(E¯;D¯ |F¯B¯ ) ).
r+1 ρr+1 0 ρ0 i i ρi i i ρi
i=1
Proof.
WekeeptrackoftheflowofinformationusingthechainruleandlocalisometricinvarianceofCQMI:
I(E¯;B¯ |F¯)=I(E¯;B¯ C¯ |F¯)
r+1 r r
=I(E¯;B¯ |F¯)+I(E¯;C¯ |F¯B¯ )+(cid:0)I(E¯;D¯ |F¯B¯ )−I(E¯;D¯ |F¯B¯ )(cid:1)
r r r r r r r
=(cid:0)I(E¯;B¯ |F¯)+I(E¯;D¯ |F¯B¯ )(cid:1)+I(E¯;C¯ |F¯B¯ )−I(E¯;D¯ |F¯B¯ )
r r r r r r r
=I(E¯;B¯ D¯ |F¯)+I(E¯;C¯ |F¯B¯ )−I(E¯;D¯ |F¯B¯ )
r r r r r r
=I(E¯;B¯ C¯ |F¯)+I(E¯;C¯ |F¯B¯ )−I(E¯;D¯ |F¯B¯ ).
r−1 r−1 r r r r
Applyingrecursivelythesameargumentleadsto
r
I(E¯;B¯ |F¯)=I(E¯;B¯ C¯ |F¯)+(cid:88)(cid:0)I(E¯;C¯ |F¯B¯ )−I(E¯;D¯ |F¯B¯ )(cid:1)
r+1 1 1 i i i i
i=2
=I(E¯;B¯ D¯ |F¯)+I(E¯;C¯ |F¯B¯ )−I(E¯;D¯ |F¯B¯ )
1 1 1 1 1 1
r
+(cid:88)(cid:0)I(E¯;C¯ |F¯B¯ )−I(E¯;D¯ |F¯B¯ )(cid:1)
i i i i
i=2
r
=I(E¯;B¯ |F¯)+(cid:88)(cid:0)I(E¯;C¯ |F¯B¯ )−I(E¯;D¯ |F¯B¯ )(cid:1).
0 i i i i
i=1
Wegetthedesiredresultbyrearrangingterms.
Intheremainderofthiswork,weareconcernedwithquantumcommunicationprotocolsasdefined
inSection2,forwhichaneasycorollaryoftheInformationFlowLemmaisasfollows. Asimilarresult
holdsforAlice.
Corollary4 Given a protocol Π, an input distribution µ and any extension ρAinBinE1E2 satisfying :
0
TrE1E2(ρA0inBinE1E2)=ρAµinBin,
(cid:88) (cid:88)
I(E ;B(cid:48)B |E ) −I(E ;B |E ) = I(E ;C |E B ) − I(E ;C |E B ) .
1 out 2 ρr+1 1 in 2 ρ0 1 i 2 i ρi 1 i 2 i ρi
iodd ieven
CombiningtheaboveresultandasimilaroneholdingforAlice,wegetthefollowinglowerboundon
quantuminformationcost,statedasasumofdifferencesbetweentheamountofcorrelationsofreference
registerswiththeoutputandtheinput.
Corollary5 GivenaprotocolΠ,aninputdistributionµandanytwoextensionsρAinBinE1E2,ρAinBinF1F2
0,B 0,A
satisfying: TrE1E2(ρA0,iAnBinE1E2)=ρµAinBin,TrF1F2(ρA0,iBnBinF1F2)=ρAµinBin,thefollowingholds:
QIC(Π,ρ)≥I(F ;A A(cid:48)|F )−I(F ;A |F )
1 out 2 1 in 2
+I(E ;B B(cid:48)|E )−I(E ;B |E ).
1 out 2 1 in 2
4 Making Safe Copies of the Inputs
Inthissection,weshowthatmakingsafecopiesofclassicalinputsattheoutsetofaquantumprotocol
neverincreasesitsquantuminformationcost. So, whenstudyingthequantuminformationcomplexity
ofafunction,itisalwayspossibletoassumethatprotocolsdonotchangetheinputregisters.
FollowingRef.[JRS03],weintroducethenotionofsafecopiesandsafeprotocols.
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