The capacity of hybrid quantum memory

10 Pages·0.14 MB·

by  Greg Kuperberg

The capacity ofhybridquantum memory
Greg Kuperberg1,∗
1UC Davis
The general stable quantum memory unit is a hybrid consisting of a classical digit with a quantum digit
(qudit)assignedtoeachclassicalstate. Theshapeofthememoryisthevectorofsizesofthesequdits,which
maydiffer. WedeterminewhenN copies ofaquantum memoryA embedinN(1+o(1)) copiesofanother
quantummemoryB. ThisrelationshipcapturesthenotionthatBisasatleastasusefulasA forallpurposes
inthebulklimit. Weshowthattheembeddings existifandonlyifforall p≥1, the p-normoftheshapeof
A doesnotexceedthe p-normoftheshapeofB. Thelogofthe p-normoftheshapeofA canbeinterpreted
asthemaximumofS(r )+H(r )/p(quantumentropyplusdiscountedclassicalentropy)takenoverallmixed
statesr onA.Wealsoestablishanoiselesscodingtheoremthatjustifiestheseentropies.Thenoiselesscoding
theoremandthebulkembeddingtheoremtogethersaythateitherA blindlybulk-encodesintoBwithperfect
3 fidelity,orA admitsastatethatdoesnotvisiblybulk-encodeintoBwithhighfidelity.
0 Inconclusion,theutilityofahybridquantummemoryisdeterminedbyitssimultaneouscapacityforclassical
0 andquantumentropy,whichisnotafinitelistofnumbers,butratheraconvexregionintheclassical-quantum
2 entropyplane.

n
a
J 1. INTRODUCTION ofthehybridtrittoanyotherquantummemory:Itisbetween

9 a qubit and a qutrit, more than a classical trit, less than any
2 Many questions in quantum information theory involve largermemorythatcontainsaqubit,andneithermorenorless
both quantum and classical information. The usual compu- thanaclassicaldigitwithatleast4states.
3 tationalmodelforsuchdualinformationisindependentquan- Itturnsoutthatthereismorethanonenotionbywhichone
v
tum and classical memory. The measurement algebra of a memoryunithasmorecapacitythananother. (Atypically,all
5
0 combinedmemoryconsistingofana-statequditandab-state such notions are equivalentfor the hybridtrit.) The strictest
1 classicaldigitis relevant relationship between memories is given by algebra
3 embeddings. If A ֒→ B is an algebra embedding (which
0 b need not be unit-preserving, or unital), then the memory B
2 Ma⊗Cb= Ma, can simulate the memory A. In other language, an algebra
0 Mk=1 embeddingisablind,perfect-fidelitydecoding.Section2also
/
h explainsthat althoughother blind, perfect-fidelityencodings
where M is the set of a×a matrices. But this is not the
p a arepossible,anysuchencodingcanbereplacedbyanalgebra
mostgeneralpossiblehybridofclassicalandquantummem-
- embedding. AsSection3.1explains,thequestionofwhether
t ory. Rather the measurement algebra A of a finite memory
n A embeds in B is a computable (but NP-hard) bin-packing
couldbeanydirectsumofmatrixalgebrasofpossiblydiffer-
a problem.
u entdimensions:
Inthisarticlewewillconsideramorerelaxedcomparison,
q
n namely whether many copies of A embed in slightly more
:
v A ∼= Ml . copiesof B. More precisely we say that A bulk-embedsin
Xi Mk=1 k B, orA ֒→b B, ifforeveryrationale >0, thereexistsanN
r The partition (i.e., non-negative integral vector) l = l (A) suchthat
a
is a list of the dimensions of the matrix algebras called the A⊗N ֒→B⊗N(1+e).
shape of the memory A. Section 2 discusses why this is a
reasonablygeneralquantummemorymodel. If A bulk-embedsin B, there is no reason to pay more for
For example, the simplest hybrid memoryis a hybrid trit, A thanBwhenbuyinglargequantitiesofthetwomemories
withshape(2,1). Itconsistsofmatricesoftheform withequalperformance. Ourfirstmainresultisacharacteri-
zationofwhenA bulk-embedsinB:
∗ ∗ 0
Theorem 1.1. If A and B are two hybrid memories, then
 ∗ ∗ 0.
0 0 ∗ A ֒→b Bifandonlyif
 
||l (A)|| ≤||l (B)||
Thismemorymodelsa three-statesystem inwhichonestate p p
isobservedbythe environmentbuttheothertwo remainco- forall p∈[1,¥ ].
herentrelativetoeachother.Itiseasytocomparethecapacity
One direction of Theorem 1.1 is straightforward. The p-
normofapartitionl isdefinedas
1/p
∗Electronicaddress:[email protected];SupportedbyNSFgrantDMS ||l || = (cid:229) l p .
#0072342 p (cid:18) k(cid:19)
k
2
Itiseasytocheckthatthe p-normismultiplicative: ishalfofthedensecodingcapacityofA.
Theorem1.2impliesthatthesetofpossiblepairs
||l (A ⊗B)|| =||l (A)|| ||l (B)||
p p p
for any pair of memories A and B. On the other hand the (HA(r )+t,SA(r )−t),
bin-packingmodelimpliesthatifA embedsinB,then where0≤t≤SA(r ),formsaconvexcapacityregionC(A)
||l (A)|| ≤||l (B)|| . in the first quadrant of the plane. Figure 1 shows an exam-
p p
ple. The constantt expresses the fact that quantum entropy
ItfollowsthatthisinequalityalsoholdswhenA bulk-embeds canbe usedclassically. Since theS-interceptoftheline tan-
inB. TheproofoftheotherdirectionofTheorem1.1isthe genttoC(A)withslope−1 islog ||l (A)|| ,anotherwayto
p p
topicofSection3. stateTheorem1.1isthatmemoryA bulk-embedsinanother
The p-normhasan interestinginformation-theoreticinter- memoryB ifandonlyifC(A)⊆C(B). Inotherwords,A
pretation. In Section 4 we will define the classical entropy bulk-embeds in B if and only if it has no state r with too
H(r ) and the quantum entropy S(r ) of a state r of a quan- muchentropytofitinB.
tummemoryA. Theirdefinitionsarejustified bya capacity
Our second main result is the following noiseless cod-
estimate, Theorem 1.2, and by a noiseless coding theorem,
ing theorem, whichgeneralizesa result of Barnum, Hayden,
Theorem1.3.
Jozsa, and Winter [1]. The terms of the theorem and a self-
Theorem1.2. Everystater ofamemoryA satisfiesinequal- containedproofappearinSection4.2.
ity Theorem1.3. LetA beaquantummemorywithastater and
HA(r )+SA(r )≤log||l (A)||p, lneotisBelebsesacondoitnhgerseqquuaenntucemmemory. Then there is a reliable
p
where r has classical entropy H(r ) and quantum entropy Y X
N N
SA(r ). Foreachp≥1thereexistsar thatachievesequality. A⊗N B⊗N(1+e) A⊗N
Any non-negativepair(H,S)satisfying the inequalityfor all
pcanbeexpressedas for every rational e > 0 if and only if (HA(r ),SA(r )) ∈
C(B). Here “reliable” means that the complete fidelity
(H,S)=(HA(r )+t,SA(r )−t) F(r ⊗N,X ◦Y )→1asN→¥ .
n n
forsomer andsomet∈[0,1].
The “no-go” direction of Theorem 1.3 depends on an in-
teresting Ho¨lder inequality for fidelity of encodings, Theo-
rem4.1. Insimplifiedform,ourinequalitysaysthatif
Y X
y 0.75 A B A
p
o
r
ent aretwoquantumoperationsand 1+1 =1,then
m p q
u 0.50
nt H+3S=log 10 Tr(X ◦Y)≤||l (A)|| ||l (B)|| .
a q p
u
q
S= 0.25 C(A) Thisinequalityisabroadgeneralizationofthefollowingele-
mentarycombinatorialfact: Ifa(uniformly)randomnumber
x from 1 to a is encoded into a random number from 1 to b
withb<aanddecodedbackagain,thentheprobabilitythat
0
xisrecoveredisatmost b.
0 0.50 1.00 1.50 2.00 a
In conclusion, Theorem 1.3 is an important converse to
Theorem 1.1. Together they say that if A and B are two
H=classicalentropy
hybridquantummemories,then,theneitherA blindlybulk-
encodesintoB with perfectfidelity, orA has a state r that
Figure1: ThecapacityregionofamemoryA withshape(2,1,1), doesnotvisiblybulk-encodeintoBwithhighfidelity.
andits3-normboundingline.
Notethatthethreemostcommon p-normsarealso signif- 2. MEMORY
icant for quantum information theory. The logarithm of the
1-norm,log||l (A)|| , is the purelyclassicalcapacityof A.
1 As explained in the introduction, the first question is
The logarithm of the ¥ -norm, log||l (A)||¥ , is the purely whether our model of a hybrid memory is adequately gen-
quantumcapacity.Andthelogarithmofthe2-norm,
eral. Onejustificationcomesfromviewingaquantumsystem
log dimA notasaHilbertspace,butasanabstractoperatoralgebraA.
log||l (A)||2= , If A is infinite-dimensional, it should satisfy some analytic
2
3
axiomsin orderto be usefulforquantumprobabilitytheory; Certainlyanyfinite-dimensional∗-algebraA isthereliable
usuallyitisassumedtobeeitheraC∗-algebraoravonNeu- memoryretainedbysomequantumoperationonamatrixal-
mannalgebra[8,9]. Butifitisfinite-dimensional,itsuffices gebraM . Intheminimalconstruction,letd=||l (A)|| be
d 1
torequirethatA bea(positive-definite)∗-algebra;itisthen thetotalsizeofallblocksofA. WerealizeA ⊆M asma-
d
also a C∗-algebra and a von Neumann algebra. This means trices with a diagonal block of size l (A) for each k. The
k
that in addition to the fact thatA is a complexvectorspace algebraM hasaPOVMwhosekthelementP istheidentity
d k
withassociativemultiplication,ithasanabstract∗-operation of the kth summandA . Thecorrespondingquantumopera-
k
whichisanti-linear,product-reversing,andsuitablypositive- tion
definite:
n
P(A)= (cid:229) PAP
(l AB)∗=l B∗A∗ A∗A=0 =⇒ A=0. k k
k=1
Positivedefinitenessleadstoanimportantpartialorderingon
is a projection, meaning P2 =P, and its image is A. If
A. BydefinitionX ≥Y ifX−Y =A∗AforsomeA.
the thermal evolution of M is given by P, the algebra A
Forexample,thematrixalgebraM isa∗-algebra. d
n measurestheretainedinformation.
Despite theirabstraction, ∗-algebrashaveallof theneces-
Conversely, the following two results show that if E is a
sarystructureforquantuminformationtheory. Theelements
(decay)quantumoperationonafinite-dimensional∗-algebra,
ofa∗-algebraA oftheformA∗Aarecalledpositive. Astate the information retained by En in the limit n →¥ is mea-
r ona∗-algebraA isdefinedasadualvectorr ∈A∗which
sured by a smaller ∗-algebra of effective observables. (See
is positive on positive elements and which is normalized by
alsoZurek[14].)
r (I)=1. Consequently we write r (A) for the expectation
of A rather than Tr(r A). (The latter notation is of course Theorem2.1. LetE :M →M beanSUCPmaponafinite-
equivalent when A is a matrix algebra; it expresses r as a dimensional ∗-algebra M. Then there exists a sequence of
density operator.) A quantumoperationfroma system with integersnk→¥ suchthatEnk convergestoa uniqueprojec-
∗-algebraA toasystemwith∗-algebraBisdefinedasauni- tionP.
tal,completelypositive(UCP)linearmapE :A →B.Here
completely positive means that E sends positive elements to Proof. (Sketch) Choose a basis of M that puts E in Jordan
positive elements after tensoring with the identity on a third canonical form. Since En is SUCP, its matrix entries are
∗-algebra. NotethatthetransposeET :B∗→A∗ isthecor- bounded. Therefore E has no eigenvalues l with |l |>1,
respondingmaponstates. Itiscompletelypositiveandtrace- andif|l |=1,thel -isotypicpartofE isdiagonal. Choosea
preservingifwetaker (I)tobethetraceofr . sequenceof exponentsnk →¥ such thatthe phasesof these
It will be useful to consider a larger class of maps than diagonal entries of Enk are aligned with 1 in the limit. The
traditional quantum operations. A completely positive map restofthematrixofEn decaysto0asn→¥ . ThemapP is
E :A →B is subunital (or SUCP) if E(I)≤I. Whereas uniquebecauseifthephasesdonotalignwith1,thelimiting
a UCP mapconservesprobability,an SUCP mapeither con- mapisnotaprojection.
servesordiminishesit. AnSUCPmapcanbephysicallyre-
FinallyaresultofChoiandEffros[5,pp.166-7]completes
alized in the same way as a UCP map, with the extra inter-
ourjustificationforthe∗-algebramodel.
pretation that missing probability corresponds to ending the
experiment. AnSUCPmapcanalsobecalledadecayquan- Theorem 2.2 (Choi, Effros). If M is a finite-dimensional
tumoperation. ∗-algebraandP isanSUCPprojectiononM,thentheim-
Astandardclassificationtheorem[3]saysthateveryfinite- age ofP is a ∗-algebraA with a modifiedproductA◦B=
dimensional∗-algebraA isadirectsumofmatrixalgebras, P(AB).
n
A ∼= Mlk. hoTldhsefnoornC-t∗r-iavligaelbpraarst)oifsTthheeofraecmt 2th.2at(wthheicmhomdoifireedgepnreordaullcyt
Mk=1
A◦Bisassociative. Themodifiedproductstructureisconsis-
Thusaquantummemoryofshapel isthemostgeneralpos- tentwithapplyingP betweenanytwocomputationalmanip-
siblefinite-dimensionalcomplexalgebraofobservablessatis- ulationsofM. Technicallyspeaking,ChoiandEffrosprove
fyingreasonable algebraicaxioms. (HoweverabandoningC Theorem2.2forUCPmaps,buttheproofforSUCPmapsis
asthefieldofscalarsleadstootherpossibilities[4].) thesame.
Anotherjustificationcomesfromtheinteractionofaphysi- A quantum operation X : B → A is a blind, perfect-
calmemorywithitsenvironment.Consideraphysicaldevice fidelityencodingifithasarightinverseY :A →B,whichis
whosestateisdefinedbya∗-algebraM. RealisticallyM is thencalledthedecoding.Inthiscasethereversecomposition
verylarge,butalmostallofitisthermallycoupledtotheenvi- Y ◦X is a CPU projection P. Moreover, Y identifies A
ronment.Itsdecoherenceonthethermaltimescaleisgivenby withthe Choi-EffrosalgebrastructureonP. Thisconstruc-
somedecayquantumoperationE :M →M. Ifthethermal tion is reversible: Given P, we can define A to be imP
timescaleismuchshorterthanthecomputationaltimescale, withitsChoi-Effrosstructure. CertainlyifY embedsA into
thentheinformationretainedbyEn inthelimitn→¥ isthe B, then a correspondingX exists. (If Y is notunital, then
reliablememoryofM. itis a decayquantumoperation,butX can alwaysbemade
4
non-decay.)Generally,evenwhenA andBareabelian,Y is for all k. (Bratteli diagrams often describe unital homomor-
notanalgebraembedding,butanotherargumentofChoiand phisms,whichrequireequality.) Thehomomorphism f isan
Effros[5,pp.202-3]saysthatitalwaysyieldsone. embeddingifandonlyifeachsummandofA hasatleastone
edge,orequivalentlythat
Theorem2.3(Choi,Effros). IfM isafinite-dimensional∗-
algebraandP isanSUCPprojectiononM,thenimcPalso (cid:229) G ≥1
j,k
embeds(non-unitally)asasubalgebraofM. k
Theorem2.3moregenerallyholdsforvonNeumannalge- forall j.
bras.TheproofadjustsPinacanonicalway.Itisnothardto Thuswe canthinkofA asa setof1-dimensionalblocks,
showthateveryalgebraembeddingisablind,perfect-fidelity Basasetof1-dimensionalbins,andtheembeddingasaway
decodingY;thereexistsanX tomatchit. topacktheblocksofA inthebinsofB. Thepackingmight
repeatsomeofthesummandsofA,butifthereisanyembed-
ding,thereisonewithnorepetition. (Repetitioninthissense
3. EMBEDDINGS hasnothingto dowithcloningasintheno-cloningtheorem.
Inrepresentationtheorythiskindofrepetitionisusuallycalled
multiplicity.)
3.1. Binpacking
Lemma3.1. IfA ֒→B,thenl (A)4 l (B). If2l (A)4
S S
Besides embeddability and bulk embeddability, we will l (B),thenA ֒→B.
alsocomparememoriesusingapartialorderingonpartitions
Proof. Bothstatementsfollowbyinductiononthenumberof
which resembles dominance[13, Ch.7], or majorization, but
isstricter. Thepartitionl supermajorizesthe partition m , or partsofl (A). Theybothholdtriviallywhenl (A)isempty.
m 4 l , if for every n, the sum of all parts of l that are at Toprovethefirstassertion,supposethatinsomeembedding,
S
least n exceeds the same sum for m . Lemma 3.1 below and A1 embedsinBk. LetA beA withA1 removedandletB
Theorem 1.1 imply that supermajorization lies between em- be B with Bk reduced by l (A)1, or removed if l (B)k =
c b
beddabilityandbulkembeddability: l (A) . Byconstruction,A ֒→B. Thusbyinduction,
1
A ֒→B =⇒ l (A)4 l (B) =⇒ A ֒→b B l (A)c≥x≤lb(B)≥x
S
A ֒→b B=6 ⇒l (A)4 l (B)=6 ⇒A ֒→B. forallx≥1. BythedeficnitionofAbandB,
S
Wecanviewthepartsofapartitionl asanunorderedmul- l (A)≥x=l (A)≥cx−l (Ab)1
tiset {l k}. It is sometimes convenient to assume a specific l (Bc)≥x≤l (B)≥x−l (A)1
orderontheparts.Inthiscasewefollowtheusualconvention
thatthepartsofl arenon-increasing: forx≤l (A) ,whiblel (A) vanishesforx>l (A) . Thus
1 ≥x 1
l 1≥l 2≥···≥l n≥1. l (A)≥x≤l (B)≥x,
Givenapartitionl ,letl denotethesumofallpartsofl asdesired.
≥x
thatareatleastx. Thusl 4Sm meansthat To prove the second assertion, suppose that 2l (A) 4S
l (B),orequivalentlythat
l ≤m
≥x ≥x 2l (A) ≤l (B)
≥x ≥x
forallx. Obviouslyintegervaluesofxsuffices,butitwillbe
for all x. We can greedily put A in any B in which it fits
convenientlatertoallownon-integervalues. Alsoℓl denotes 1 k
l witheachpartrepeatedℓtimes. (Thisisnottobeconfused and make A and B as before. (In this greedy algorithm it
is importantto start with the largest summand of A, not an
withmagnifyingeachpartbyafactorofℓ.)
arbitraryonec.)Ifl (bB) ≤2l (A) ,then
In order to analyze bulk embeddings and prove Theo- k 1
rem1.1,wefirstanalyzeordinaryembeddings[3]. IfA and
B arefinite-dimensional∗-algebras,thenanyalgebrahomo- l (A)≥x=l (A)≥x−l (A)1
morphism f :A →B ischaracterizedbyaBrattelidiagram l (Bc)≥x≥l (B)≥x−2l (A)1
G whoseverticesarethesummandsofA andB. LetA be
k
Bthe. kIfthwseudmemnoatnedthoefaAdj,acseontchyatmAatkri∼=xoMfGlkb,yanG dalsikwewelils,ethfeonr fOonr athllexo≤thelr(hAan)d1,biwf lhi(lBe l)(A≥)2≥lx(vAan)is,htehsenfobrixn>k rle(mAai)n1s.
k 1
thediagram’sinterpretationisthat f embedsG j,kcopiesofAj largerthananyblockevenaftecrblock1issubtracted. Inthis
inBk. (ThematrixG istheadjacencymatrixofthediagram case
G .) ThematrixG mustsatisfytheinequality
l (A) =l (A) −l (A)
≥x ≥x 1
(cid:229) G j,kl (A)j≤l (B)k l (Bc)≥x≥l (B)≥x−l (A)1
j
b
5
forallx≤l (A) . Thus Theequalityusestheidentities
1
¥
2l (A)4Sl (B) xeb xdm ∗n(x)= enℓ(b ) ′=nℓ′(b )enℓ(b )
Z0
eitherway,sothebinpackicngexistsbbyinduction. ¥ (cid:0) (cid:1)
x2eb xdm ∗n(x)= enℓ(b ) ′′= nℓ′′(b )+n2ℓ′(b )2 enℓ(b ).
Z0
(cid:0) (cid:1) (cid:0) (cid:1)
3.2. Largedeviations Thisestablishesthelowerbound,Crame´r’stheorem.
ProofofTheorem1.1. Inbrief,withoutlossofgenerality
The proof of Theorem 1.1 combines Lemma 3.1 with the
Chernoff-Crame´r theorem on large deviations [6]. The the- ||l (A)|| <||l (B)||
orem is usually stated in terms of sums of independent ran- p p
dom variables, butit is more convenienthere to formulateit for all p∈[1,¥ ]. In this case we apply Theorem 3.2 to the
intermsofconvolutionsofmeasures.
measures
aTnheinotreermval3[.02,u(C],hleetrnoff, Crame´r). Let m be a measure on m A =(cid:229) l k(A)d loglk(A)
k
ℓ(b )=log ¥ eb xdm (x) m B =(cid:229) l k(B)d loglk(B),
Z0 k
bethelogarithmoftheLaplacetransformofm andlett >0. where d denotes a delta function (or atom) at x. For suf-
x
Thenforalln∈Z+ andallb >0, ficiently large n, Chernoff’sbound for m A and Crame´r’s in-
¥ equalityform B togetherimplythecriterion
dm ∗n≤en(ℓ(b )−b t)
Znt 2l (A⊗n) ≤l (B⊗n)
≥x ≥x
Ifℓ′(0)≤t<uandb minimizes
ofLemma3.1uniformlyforx∈[1,¥ ).
ℓ(b )−b t, Indetail,weassumethat||l (B)||¥ >1;otherwiseA and
BarebothentirelyclassicalandTheorem1.1iseasy.Since
thenforall0<s<t,
||l (A)|| ≤||l (B)||
¥ dm ∗n≥en(ℓ(b )−b t−b s) 1−ℓ′′(b ) . p p
Zn(t−s) (cid:18) ns2 (cid:19) forall p∈[1,¥ ],thenforanyk>1,
||l (A⊗k)|| <||l (B⊗k+1)|| .
Here m ∗n denotes the n-fold convolution of m with itself. p p
Whenℓ′(0)<t<u,theexpression Thee margininTheorem1.1thusallowsustoassumethat
ℓ(t)=minℓ(b )−b t ||l (A)|| <||l (B)||
b p p
b
is the Legendre transform of ℓ(b ). Note that a unique b forall p∈[1,¥ ]byreplacingA byA⊗k andBbyB⊗k+1.
achievesthe minimumbecause the minimandis concaveup, Themeasurem A isdefinedsothat
increasesasb →¥ ,anddoesnotincreaseatb =0.
m A∗n=m A⊗n
Proof. (Sketch)Foranyb ,
and
¥ ¥
dm ∗n≤e−nb t eb xdm ∗n(x) ¥
Znt Z0 l (A)≥ex = dm A(x),
=e−nb tenℓ(b ). Zx
Thisestablishestheupperbound,Chernoff’sinequality. andlikewise for m B. ThereforebyLemma3.1, it sufficesto
Ifb ischosentominimizeℓ(b )−b t,thent=ℓ′(b ). Inthis showthatthereexistsannsuchthatforallt≥0,
case ¥ ¥
2 dm ∗n≤ dm ∗n. (1)
A B
¥ dm ∗n≥e−nb (s+t) n(t+s)eb xdm ∗n(x) Znt Znt
Zn(t−s) Zn(t−s) AsinthestatementofTheorem3.2,let
≥e−nb (s+t)Z0¥ (cid:18)1−(x(−nsn)2t)2(cid:19)eb xdm ∗n(x) ℓA(b )=logZ0¥ eb xdm A(x)=log||l (A)||bb ++11
=e−nb (s+t)(cid:18)1−ℓ′n′(sb2)(cid:19)enℓ(b ). ℓB(b )=logZ0¥ eb xdm B(x)=log||l (B)||bb ++11.
6
ObservethatℓB(b )isasmooth,concavefunction,andthat Letr bea(mixed)stateonA;asexplainedaboveweviewr
asadualvectoronA ratherthanasanelementofA. Let
ℓ′ (b )
bli→m¥ Bb =log||l (B)||¥ <¥ . r k=r |Ak
Itfollowsthatℓ′B′ (b )hasafinitemaximumCforb ∈[0,¥ ). betherestrictionofr toAk. Diagonalizeeachr k andletrk,j
Notealsothat with1≤ j≤l beitsdiagonalentries. (Ingenerala state r
k
ℓB(b )−ℓA(b ) on matrices is diagonalif and only if r (A) dependsonly on
thediagonalentriesofA. Equivalentlyinthepresentcasewe
b
caninterpretr asadensityoperator.)Let
achievesapositiveminimum,since
l
bli→m¥ ℓB(b )−b ℓA(b ) =||l (B)||¥ −||l (A)||¥ rk=r k(I)= j(cid:229)=k1rk,j
lim ℓB(b )−ℓA(b ) =¥ . bethetotaldensityofr inAk;evidently
b →0 b
n
(cid:229)
Temporarilysupposethatt≥ℓ′ (0)andthatb =b (t)min- rk=1.
B
imizesℓB(b )−b t.Let k=1
Wealsodefinethenormalizedstater ′ onA by
2C k k
s= . r
r n r ′ = k,
k r
Then k
¥ withdiagonalentries
dm ∗n≤en(ℓA(b )−b t+b s)
A
Zn(¥t−s) r′ = rk,j.
dm ∗n≥en(ℓB(b )−b t−b s)−log2. k,j rk
B
Zn(t−s)
Theclassicalentropyofthestater onA isdefinedas
Ifnislargeenoughthat
n
2log2 2C 2log2 ℓB(b )−ℓA(b ) HA(r )=−(cid:229) rklog rk.
2s+ n =2r n + n ≤mbin b , k=1
Thequantumentropyofr isdefinedas
then
2Zn(¥t−s)dm A∗n≤Zn(¥t−s)dm B∗n. SA(r )=−(cid:229) n (cid:229)lk rk,jlog rk′,j.
k=1j=1
Thusforsomee >0,inequality(1)holdsforallt>ℓ′ (0)−e .
Ift≤ℓ′ (0)−e ,letu=ℓ′ (0)andletb =0. TheBn (Note that in the literature H is also sometimes used to de-
B B notequantum,orvonNeumann,entropy. Herewefollowthe
¥ ¥ conventionofNielsenandChuang[10].)Thesetwoentropies
dm ∗n≤ dm ∗n=enℓA(0),
Znt A Z0 A are supportedby a numberof elementary justifications: The
classical entropy of r is the Shannon entropy of the restric-
while tionofr tothecenterofA,whichisaclassicalsystem. The
¥ ¥ quantumentropyof r is the expectedvalue of the von Neu-
dm ∗n≥ dm ∗n≥enℓB(0)−log2
Znt B Zn(u−s) B mann entropy of r k, where the index k is chosen randomly
withprobabilityr . Finallythetotalentropy
provided that s ≤ e . Since ℓA(0) < ℓB(0), inequality (1) k
holdswhennislargeenough. HA(r )+SA(r )=−(cid:229) n (cid:229)lk rk,jlog rk,j
k=1j=1
4. ENTROPY
hasthesame formulaas boththeShannonandthe vonNeu-
mannentropy.
4.1. Capacity
TheproofofTheorem1.2isbasedonfindingthermalstates
of A with respect to a certain Hamiltonian. We define the
LetA beafinite-dimensional∗-algebra,whereasbefore energyE ofthe summandA asthenegativeofitscapacity
k k
n n forquantumentropy:
A =Mk=1Ak∼=Mk=1Mlk. Ek=−logl k(A).
7
Weretaintheparameterb fromSection3.2,settingp=b +1, We are interested in reliable noiseless coding, or in other
and we also define the temperature T =1/b . The thermal wordshigh-fidelity,visiblebulk-encoding.Butarigorousdef-
state r attemperatureT hasthe propertythatits restriction initionofreliabilityisnotobvious.SupposethatE isadecay
T
r toeachA isuniform. Ifr isanystatewiththisproperty, quantumoperationfromamemoryA toitself,andthatA has
k k
then its energy EA(r ) is, by definition, the negative of its astater . IfC isanothermemory,wedefinetheC-fidelityof
quantumentropy: A tobe
EA(r )=−SA(r ). FC(r ,E)= min 1−D s ,(id.⊗XT)(s ) , (2)
s ∈(C⊗A)∗
s 7→r (cid:0) (cid:1)
Thefreeenergyofr istherefore
where D is the trace distance on states, and the minimum is
FA(r )=EA(r )−T(HA(r )+SA(r )) taken over states s on C ⊗A that project to the state r on
=−T(HA(r )+pSA(r )). A. In words, the C-fidelity is the complementof the high-
estprobabilitythattheoperationX leavesthelargersystem
Sincethethermalstateminimizesthefreeenergy,wehavede- C ⊗A inanerroneousstate. Wedefinethecompletefidelity
finedenergysothatthethermalstater maximizesquantum F(r ,E) to bethe infimumof C-fidelityoverallC. Itis not
T
entropy plus classical entropy discountedby p. To compute hard to show that complete fidelity agrees with the classical
the maximum, recall that for the thermal state r , the free non-errorratewhenA isclassical,andwithentanglementfi-
T
energyisproportionaltothelogofthepartitionfunction: delitywhenA ispurelyquantum.
Themoredifficulthalf ofTheorem1.3is the no-godirec-
n
FA(r T)=−Tlog ZA(r T)=−Tlog (cid:229) l keb log lk(A) tion. Toreview,theheartoftheno-godirectionoftheclassi-
(cid:18) (cid:19) cal encoding theoremis the following elementaryfact about
k=1
squeezing states: If a state r of a classical memory is en-
n
=−Tlog(cid:229) l b +1=−Tplog ||l (A)|| . codedinto b values, then it cannotbe recoveredwith proba-
k p
k=1 bility greater than b||r ||¥ , where ||r ||¥ is the probability of
the mostlikely valueof r . Or forsimplicity, if r is theuni-
Therefore
formstateonamemorywithavalues,thenthenon-errorrate
HAp(r T)+SA(r T)=log ||l (A)||p, oisfathtimsoinsetqabu.alWitye.wToillstnaeteedita, whyebrreipdlaqcuean||trum||¥ gwenitehraalidziaftfieorn-
entnorm.Ifr isastateonA,definethedense-coding-based
asdesired. supremumofr by
ToprovethefinalclaimofTheorem1.2,observethatevery
pointinC(A)canbewrittenintheform ||r || =maxrk2,j
d
(HA(r T)+t,SA(r T)−s−t) j,k rk
inthenotationofSection4.1.
with 0≤s,t and s+t ≤SA(r T). Starting with thestate r T, Theorem4.1. LetA andB be two hybridquantummemo-
thequantumentropyineachblockcanbedecreasedto0with-
riesandletr beastateonA. If
outchangingthetotalprobabilityofthatblock,hencewithout
changingtheclassicalentropy. Inthiswaywecanabsorbthe
Y X
constants. Theremainingconstantt justmatchesthe onein A B A
theconclusion.
aredecayquantumoperationsand 1+1 =1,then
p q
4.2. Noiselesscoding F(r ,X ◦Y)≤||r || ||l (A)|| ||l (B)|| .
d q p
A final justification for quantumand classical entropiesis Before proving Theorem 4.1, we discuss some special
Theorem1.3,whichweprovehere. Thetheoremisamutual cases. IfA =Ca isclassicalandr istheuniformstate,then
generalizationof, and entirely analogousto, Shannon’sclas- ||r || = 1. Inthiscase,taking p=1,Theorem4.1saysthat
d a
sicalandSchumacher’spurelyquantumcodingtheorems[10,
||l (B)||
Thms. 12.4&12.6][11,12]. F(r ,X ◦Y)≤ 1.
GivenanalgebraA withastater andasecondalgebraB, a
anoiselesscodingisapairofdecayquantumoperations This generalizesthe classical squeezing result, boundingthe
fidelity by the total number of independent states of B
A Y B X A . whetherornotitisclassical. Ontheotherhand,ifA =Ma
ispurelyquantumandr istheuniformstate,then||r || = 1.
d a2
Inthiscase,taking p=¥ ,Theorem4.1saysthat
Sincethesearemapsonalgebrasratherthanstatesspaces,the
second map X is the encoding and the first map Y is the l (B)
decoding. F(r ,X ◦Y)≤ 1 .
a
8
In other words, if A is purely quantum, then the fidelity of Thus
squeezing is bounded by the largest quantum block of B,
regardless of its classical capacity. (But if B =Mb is also ||x||p≤||l (A)||p ||y||q≤||l (B)||q.
purelyquantum,thenitcanbeshownthat
FinallytheHo¨lderinequalityyields
b2
F(r ,X ◦Y)≤ F ≤||r || x·y≤||r || ||x|| ||y||
a2 d d p q
≤||r || ||l (A)|| ||l (B)||
whenr isinform.Inthiscaseifbdividesa,thenmultiplying d p q
Bby ab classicalstatescanboostfidelityto ab.) when 1+1 =1,asdesired.
p q
Proof. TheoperationsX andY admitKrausrepresentations
ProofofTheorem1.3. (Semi-sketch) As in the proofs of
X( B )= (cid:229) X∗ B X Shannon’s and Schumacher’s theorems as presented by
k j,k,ℓ k j,k,ℓ Nielsen and Chuang [10], we first establish the existence of
Mk Mj k,ℓ
ae -typicalsubalgebraA ofA⊗N with respectto thestate
Y( Ak)= (cid:229) Yj∗,k,ℓAkYj,k,ℓ r . (Thee in theproofhetyrpeis notthesame asthe onein the
Mk Mj k,ℓ statementofthetheorem,whichwerenamed .) Wewilltake
e toimplicitlydependonNwithe →0slowlyasN→¥ . We
subjecttothesubunitalconditions
willestablish thatA isapproximatelyrectangularandthat
typ
(cid:229) Xj∗,k,ℓXj,k,ℓ≤I∈Aj therestrictionr typofr ⊗N. Wewillthenconfirmthatif
k,ℓ
(cid:229) Y∗ Y ≤I∈B . (3) (S,H)=(SA(r ),HA(r ))∈C(B),
j,k,ℓ j,k,ℓ j
k,ℓ thenA embedsinB⊗N forsufficientlylargeN; in particu-
N
Recall the definition of r and r ′ in Section 4.1. The mini- lar itreliablyencodes. On theotherhand, if (S,H)6∈C(B),
k k
mum in equation (2) is obtained by lifting the state r to the we will confirm that AN does not reliably encode in B⊗N;
completelycorrelated,completelyentangledstate indeed the fidelity of any encoding-decodingconvergesto 0
exponentially.
s = r y Assumethatthestater onA isdiagonalizedandthatr ,
k k k,j
Mk with 1≤k≤n(A)and1≤ j≤l k(A), areits diagonalen-
tries. Heren(A)denotesthenumberofpartsofl (A). This
on A ⊗A, where y k is a pure state that projects to r k′. By inducesadiagonalizationofthestater ⊗N withadiagonalen-
a computation similar to one in Nielsen and Chuang [10, p.
tryr foreachpairofadmissiblesequences
421],thefidelityisthengivenby K,J
F =F(r ,X ◦Y)= (cid:229) r |r ′(Y X )|2. (4) K=(k1,k2,...,kN) J=(j1,j2,...,JN)
k k j,k,m k,j,ℓ
j,k,ℓ,m
issuchthat
Givenanystates onthematrixalgebraM andanymatri-
cesX∈Mb×aandY ∈Ma×b,theCauchy-Schawarzinequality 1≤ℓ≤N 1≤kℓ≤n(A) 1≤ j≤l kℓ(A).
andpositivitytogethersaythat
Moreover,for eachadmissible K, A⊗N hasan algebrasum-
|s (YX)|2≤s (X∗X)s (YY∗)≤||s ||2¥ Tr(X∗X)Tr(Y∗Y). mand (A⊗N)K. If (K,J) and (K,J′) are two admissible
pairs, the algebrasummand(A⊗N) has an elementaryma-
K
Applyingthistoequation(4),weobtainthebound trixE ;thesematricesthenformabasisofA⊗N. Wewill
K,J,J′
F ≤ (cid:229) ||r || Tr(X∗ X )Tr(Y∗ Y ). (5) consider a set T of admissible pairs (K,J) called the typical
d k,j,ℓ k,j,ℓ j,k,m j,k,m set; momentarilyit canbe anyset. The spanof the matrices
j,k,ℓ,m EK,J,J′ with(K,J),(K,J′)∈T is a subalgebraAtyp. Another
Definethenumbers waytodescribethealgebraAtypistodefinetheprojector
xk,j=(cid:229) Tr(Xk∗,j,ℓXk,j,ℓ) yj,k=(cid:229) Tr(Yj∗,k,mXj,k,m) Ptyp= (cid:229) EK,J,J
ℓ m (K,J)∈T
anddefinethevectorsx=(x )andy=(y ). Thenwecan
k,j j,k andthenlet
restateinequality(5)as
F ≤||r || (cid:229) x y =||r || x·y, Atyp=PtypA⊗NPtyp.
d k,j j,k d
j,k Inthisnotation,themap
whileequation(3)impliesthat
P(X)=P XP
typ typ
(cid:229) x ≤l (A) (cid:229) y ≤l (B).
k,j k j,k j isanSUCPprojectiononA⊗N withimageA .
j k typ
9
Givena >0,saythatanadmissiblepair(K,J)isa -typical Sinced mustbesentto0ande maybesentto0,thefidelity
ifthenumberofoccurrencesN(K,J;k,j)of(k,j)satisfies thereforedecaysexponentiallyifthereexistsa psuchthat
N(K,J;k,j)−ri,j <a . log ||l (B)|| < H +S.
(cid:12) N (cid:12) p p
(cid:12) (cid:12)
(cid:12) (cid:12)
LetT bethesetof(cid:12)alla -typicalpairs.(cid:12)Byrepeatedapplication BythedefinitionofC(B),thisinequalityisequivalenttothe
of Chernoff’s inequality (Theorem 3.2 in a more traditional assumedcondition(H,S)6∈C(B). SinceF(r ,X ◦Y)de-
typ
probabilisticcontext), caysexponentially,itcannotconvergeto1.
r ⊗N(P )= (cid:229) r →1
typ K,J
(K,J)∈T
5. DISCUSSION
foranyfixeda asN→¥ . Moreover
Section2illustratestheprinciplethatclassicalinformation
F(P ,r )≥r ⊗N(P )2,
typ typ theoryistheabelianspecialcaseofquantuminformationthe-
so for anyfixed a , A and A⊗N reliably encodeinto each ory. Many authors maintain a dichotomy between the two
typ
theoriesbyconsideringensemblesofmixedstates. Butsuch
other.Atthesametime,byamessybutstraightforwardcalcu-
lation,ifa issufficientlysmallrelativetoe (anddependingon formalismisultimatelyredundant,becauseanensembleisit-
r butnotonN),A andr havethefollowingproperties: selfaclassicalprobabilisticstate. Moreprecisely,let
typ typ
(log n(Atyp))−HN <Ne r =(cid:229) pkr k∈A
(cid:12)(cid:12)(log l (Atyp)K)−HS(cid:12)(cid:12) <Ne (6) k
(cid:12)(log ||r ||)+H+2S<(cid:12) Ne . be an ensemble of states in a memory A. If the symbol k
(cid:12) d (cid:12) isnotrecorded,thenr encodesallstatisticalinformationthat
Supposethat(H,S)∈C(B). Inthiscase,letC=eN(S+e); canbeextractedfromtheensemble. Butif eachsymbolk is
then recordedasastates k inanothermemoryB,thenwecanlet
l (A ) =0. r ′=(cid:229) p r ⊗s ∈A ⊗B.
typ ≥C k k k
k
Meanwhileequations(6)implythat
If B is abelianand the s ’s are distinctpure states, then the
k
l (AN,e )≥0<eN(H+S+2e). stater ′ denotesanensemblewitharecordofitspreparation.
Theterm “ensemble”also typicallyimpliesthatthememory
ByaderivationusingCra´mer’sboundliketheoneintheproof Bishiddenoruntransmitted. Thistooisonlyaspecialcase,
ofTheorem1.1,
becausememorymaybehiddenwhetherornotitisabelian.
l (B⊗N(1+d )) >2eN(H+S+e) Theorems1.1, 1.2, and1.3 togethersuggestthat all quan-
≥C
tum information can be measured in the bulk limit by two
whenN islargeenough,providedthate issmallcomparedto numbers,classicalentropyHandquantumentropyS.Bycon-
d . ThusbyLemma3.1, AN,e embedsin B⊗N(1+d ) forlarge trast information capacity has more structure than informa-
enoughN,asdesired. tionitself. Thecapacityof a quantummemoryis definedby
Supposethat(H,S)6∈C(B). Inthiscase,supposethat acurvethatrepresentstrade-offsbetweenclassicalandquan-
tumentropy.Thecapacityofageneralquantumchannelcould
Y X beevenmorecomplicated.
A B⊗N(1+d ) A
typ typ There are many interesting partial orderings on quantum
memoriesbesidesembeddability,bulkembeddability,andsu-
aredecayquantumoperationsandthat 1+1 =1. Bythefirst
p q permajorization. One natural example is embeddability in
twoequationsof(6),
the presence of an auxiliary memory, or stable embeddabil-
H ity. Given memories A and B, when is there a memory C
log ||l (Atyp)||q<( +S+2e )N. suchthat
q
CombiningthiswithTheorem4.1andthelastequationof(6), A ⊗C ֒→B⊗C?
weobtainTheorem4.1,
We do not know when A stably embeds in B. Stable em-
log F(r ,X ◦Y)<(e −H−2S)N beddabilityimpliesbulkembeddabilityandisimpliedbyem-
typ
beddability,butwedonotknowhowitcomparestosuperma-
H
+( +S+2e )N+log ||l (B⊗N(1+d ))||p jorizationorder.
q
Theorem1.1isrelatedtoamuchmoregeneralquestionin
=N((1+d )log ||l (B)|| −H −S+3e ). quantuminformationtheory.LetE :A →BandF :C →D
p
p be quantum operations representing two quantum channels
10
between general quantum memories. When are there oper- Finally, it is well-understood that classical and quantum
ationsX andY thatmakethediagram memory are inequivalent resources in quantum complexity
N N
theory. For example there is a quantum algorithm to find a
E⊗N collisionofa2-to-1functionwithwhichusesO(N1/3)classi-
A⊗N B⊗N
cal space (andO(1) quantumspace) [2]. But if the function
e
onlyhasasinglerepeatedvalue,thebestquantumalgorithm
XN YN uses O(N1/4) queantum space [7]. It would be interesting to
F⊗N(1+e) find aen algorithm whose natural space complexity is hybrid
C⊗N(1+e) D⊗N(1+e) quantummemory.
commutewith high fidelity? We can then say thatthe chan-
nelE reliablybulk-encodesinthechannelF. Theorems1.1, Acknowledgments
1.2,and1.3togetheranswerthequestionwhenE andF are
boththeidentitymap,withtherefinementthatperfectfidelity The author would like to thank Daniel Gottesman, Janko
is possible when high fidelity is possible. In light of Theo- Gravner,PatrickHayden,DongseokKim,AlexeiKitaev,and
rem 2.2, the cross-encodingquestion is also settled when E BrunoNachtergaeleforveryhelpfuldiscussions.Thereferees
andF areSUCPprojections. werealsoextremelyhelpful.
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