Table Of ContentEntanglement measures and approximate quantum error correction
Francesco Buscemi∗
ERATO-SORST Quantum Computation and Information Project, Japan Science and Technology Agency,
Daini Hongo White Bldg. 201, 5-28-3 Hongo, Bunkyo-ku, 113-0033 Tokyo, Japan
(Dated: December 20, 2007)
It is shown that, if the loss of entanglement along a quantum channel is sufficiently small, then
approximate quantum error correction is possible, thereby generalizing what happens for coherent
information. Explicit bounds are obtained for the entanglement of formation and the distillable
entanglement, and their validity naturally extends to other bipartite entanglement measures in
between. Robustness of derived criteria is analyzed and their tightness compared. Finally, as a
byproduct,weproveaboundquantifyinghowlargethegapbetweenentanglementofformationand
8 distillable entanglement can be for any given finite dimensional bipartite system, thus providing a
0 sufficient condition for distillability in terms of entanglement of formation.
0
Keywords: approximate quantum error correction, entanglement of formation, distillableentanglement, in-
2
formationallycompletePOVMs
n
a
J I. INTRODUCTION question whether the loss of entanglement provides not
5 only a condition for exact correction, but also a condi-
1 tion for approximate correction. In this paper we will
The possibility of performing quantum error correc-
show that this is actually the case, extending our anal-
tion obviously lies behind and justifies the vast efforts
]
ysis to different entanglement measures, thereby prov-
h made up to now in order to develop quantum computa-
p tion techniques, since it allows fault-tolerant computa- ing that many inequivalent ways to quantify entangle-
- mentleadinfacttoanalogousconditionsforapproximate
tion [1] even when quantum systems—in fact extremely
t
n sensitive to noise—areconsideredasthe basic carriersof quantum error correction. We will moreover obtain, as
a a byproduct, an inequality directly relating the entan-
information. Besideswell-knownalgebraicconditionsfor
u glement of formation with the distillable entanglement
exact quantum error correction, which directly lead to
q
presentinageneralbipartitemixedquantumstate. Such
[ algebraicquantumerrorcorrectingcodes(forathorough
presentation of quantum error correction theory and a inequality makes rigorousthe intuition, that the gap be-
3 tween entanglement of formation and distillable entan-
detailed accountabout the enormousliterature about it,
v glement, which is known to exist generically large for
see e. g. [3, 4]), an information-theoretical approach to
5
generalmixedquantumstates [10], cannotbe completely
1 quantumerrorcorrection[5,6,7]canshedsomelighton
arbitrary, in the sense that, given a finite dimensional
8 the dynamical processes which underlie quantum noise,
bipartite state, whenever the entanglement of formation
1 offering at the same time the opportunity to better un-
. derstand the conditions under which approximate quan- is “sufficiently close” to its maximum value, then also
6
the distillable entanglement has to be “correspondingly
0 tum errorcorrectionis feasible [8]. In the present paper,
large”. (The concepts of “sufficiently close” and “corre-
7 we will be working within the latter scenario.
0 Approximate quantum error correction is not just a spondingly large”, clearly depending on the dimensions
: of the subsystems, will be quantitatively defined below.)
v theoretical issue: in fact, in all practical implementa-
The paper is organized as follows. In Section II we
i tions the experimenter can only rely upon some con-
X recall some basic notions about quantum channels and
fidence level—exact processes exist as abstract mathe-
r maticalconcepts only. Then,conditions forapproximate their purification into the unitary evolution of a larger
a
closed system. In Section III we present some known
quantumerrorcorrectioncanprovideusefulwaysto test
information-theoreticalconditionsforexactaswellasap-
thereliabilityofarealapparatus. InRef.[8],Schumacher
proximate quantum error correction. In Section IV we
andWestmorelandprovedthatanadequateinformation-
review a useful monogamyrelation satisfied by quantum
theoretical quantity to consider is the coherent informa-
and classical correlations in a tripartite pure quantum
tion: the loss of coherent information along a quantum
state. Such a relation will be exploited in Section V to
noisy channel is small if and only if the quantum noisy
show that to have a small loss of entanglement of for-
channelcanbeapproximatelycorrected. Inasubsequent
mation is equivalent to have small classical correlations
paper [9], the same Authors provided another criterion,
betweenthe referencesystemandtheenvironment. This
this time for exact quantum error correction: the loss
simple observation will lead us to the main result stated
of entanglement (of formation) is null if and only if the
as Theorem 1. Section VI extends the same analysis to
channel can be exactly corrected. They left open the
other entanglement measures. In particular, it is shown
that for certain entanglement measures it is possible to
derive the same result as for the entanglement of for-
∗Electronic address: [email protected]; mation, but in a simpler way, moreover greatly improv-
URL:http://www.qci.jst.go.jp/∼buscemi ing the tightness of the bound. This second result, in-
2
dependent of the previous one, is stated as Theorem 2. Neumann entropy of the state σ. We can alwayschoose,
Section VII stresses two remarks by comparing the two without loss of generality, the reference to be isomor-
theorems obtained so far. The first remark shows that phic to the input, so that dimHR = dimHQ. The
theycanbecombinedtoexplicitlyobtaintheabovemen- reference system R goes untouched through the interac-
tioned inequality, regarding the gap between entangle- tion UQE, in such a way that the global state after the
mentofformationanddistillableentanglementforagen- system-environment interaction is pure and given by
eralbipartitemixedstate. Thesecondremarkproposesa
possible connection between different bipartite entangle- ΨRQ′E′ :=(11R UQE) ΨRQ 0E . (1)
| i ⊗ | i⊗| i
ment measures, used here to derive different criteria for
approximate quantum error correction, and correspond- Sinceweclosedthewholesystem,wewillbeabletoplay
inglyinducedtopologiesonthe setofquantumchannels. with entropic quantities exploiting useful identities like
A brief summary (Section VIII) concludes the paper.
IR:Q′(ρRQ′)+IR:E′(ρRE′)=2S(ρR)=2S(ρQ), (2)
II. TRIPARTITE PURIFICATION OF where IA:B(σAB) := S(σA) + S(σB) S(σAB) is the
CHANNELS quantum mutual information [13, 14] b−etween A and B
when the global state is σAB, and ρRQ′ etc are the re-
Let us consider an input quantum system Q whose duced states calculated from the global tripartite pure
stateisdescribedbythedensitymatrixρQdefinedonthe state ΨRQ′E′ in Eq. (1).
(finite dimensional) input Hilbert space HQ. A chan- | i
nel, mapping states on HQ (that is, the set of non-
negative, trace-one operators on HQ, briefly denoted
III. KNOWN CONDITIONS FOR CHANNEL
as S(HQ)) to states on HQ′, can be represented as CORRECTION
a completely positive trace-preserving (CP-TP) linear
map : S(H Q) S(HQ′). We will use the nota-
How welldoes a channel preservequantuminforma-
tion ρEQ′ := (ρQ).→It is a well-known fact that channels tion? Thatis,howwelldoesEitpreservetheentanglement
E
can be written in their so-called Kraus form [11], that is that an unknown input state shares with other systems?
Awaytogiveaquantitativeanswertothisquestionisto
E(ρQ)= EmρQEm† , ∀ρQ, introducetheentanglementfidelity,thatisanonnegative
m quantity, depending on the channel (we now suppose
X E
that the output space coincides with the input one) and
where the Kraus operatorsE satisfy the normalization
m on the input state ρQ, defined as [15]
condition mEm† Em =11Q.
Besides the above mentioned abstract definition, we F(ρQ, ):= ΨRQ (id )(ΨRQ)ΨRQ ,
P
cangiveadifferentdescriptionofchannels,byexploiting E h | ⊗E | i
apowerfulrepresentationtheorem,directconsequenceof where ΨRQ is a purification of ρQ as before. It can be
Stinespring theorem [12], which states that all channels proved that F(ρQ, ) does not depend on the particular
canberealizedbymeansofasuitableunitaryinteraction purificationΨRQ ofEρQ, andit is an intrinsic property of
UQE oftheinputsystemQwithanancilla E (initialized the channel, given the input state. If F(ρQ, ) is close
inafixedpurestate 0E HE),followedbyatraceover E
to unity, then the channel acts almostlike the identity
| i∈
the ancillary degrees of freedom, in formula channelidonthesupportoEfρQ,thatis,everystateinthe
support of ρQ is faithfully transmitted by , along with
E(ρQ)=TrE′ UQE (ρQ⊗|0ih0|E) (UQE)† . its eventual entanglement with other quantEum systems.
(Weputaprimealso(cid:2)onE,becauseingeneralthe(cid:3)output Anotherquantitywhichtellshowmuchagivenchannel
preservescoherence is givenby the coherent information
ancilla system could be different from the input one.)
I (ρQ, ), defined as [5, 16]
Suchapurificationofthechannelcanalwaysberealized, c E
without loss of generality, with dimHE′ dimHQ
dimHQ′ anditisuniqueuptolocalisome≤triesonHE×′. Ic(ρQ, ):=S(ρQ′) S(ρRQ′) S(ρQ),
E − ≤
Sinceinthefollowingwewillconsiderentropicquantities,
where, consistently with the notation introduced in the
such an isometric freedom is completely innocuous.
previoussection,ρQ′ := (ρQ)andρRQ′ :=(id )ΨRQ.
Itisnowconvenienttointroduceathirdreferencesys-
E ⊗E
tem R, which purifies ρQ as The coherent information can be negative and it plays
a fundamental role in quantifying the rate at which a
ΨRQ := Ψ ΨRQ such that Tr [ΨRQ]=ρQ. channel can reliably transmit quantum information [16,
R
| ih | 17, 18].
As before, also this purification is unique up to lo- Between entanglement fidelity and coherent informa-
cal isometries on H R, so that S(ρQ) = S(ρR), where tion there exists a close relation [8] which states that,
ρR = Tr [ΨRQ] and S(σ) := Tr[σlog σ] is the von given an input state ρQ and a channel : S(HQ)
Q − 2 E →
3
S(HQ′),thereexistsachannel :S(HQ′) S(H Q) andtheaboverelationcanholdstrictly (infact,coherent
R →
such that information can easily be negative). Hence we immedi-
ately obtain the analogous of Eq. (4)
F(ρQ, ) 1 2(S(ρQ) I (ρQ, )). (3)
c
R◦E ≥ − − E
q S(ρQ) E(ρRQ′) g(1 F(ρQ, )), (6)
Inotherwords,ifthecoherentinformationisclosetothe − f ≤ − R◦E
input entropy, then the channel can be approximately
that is, the existence of an approximately correcting
corrected [19]. Most important, also the converse state-
channel implies that the entanglement of formation
ment is true, in the sense that a sort of quantum Fano of ρRQ′ iRs close to S(ρQ), [23].
inequality holds [15, 20]
InRef.[9]itwasleftopenthequestionwhetheralsothe
S(ρQ) I (ρQ, ) g(1 F(ρQ, )), (4) conversestatementistrue,namelyiftheentanglementof
− c E ≤ − R◦E formationof ρRQ′ is a robust measure of the correctabil-
for all channels : S(H Q′) S(HQ), where g(x) ity of a channel. Before answering (affirmatively) this
R →
is an appropriate positive, concave (and hence con- question, we have to go back to the unitary realization
tinuous), monotonically increasing function such that of channels and give an alternative interpretation of the
limx→0g(x)=0. In particular, for x 1/2, we can take entanglement of formation.
g(x) := 4xlog (d/x), where d := dim≤H Q [15, 20]. In
2
other words, if a channel happens to approximately
correct the channel , thRen I (ρQ, ) has to be cor-
E c E IV. CLASSICAL, QUANTUM, AND TOTAL
respondingly close to the input entropy. Notice that
CORRELATIONS
Eqs. (3) and (4) are nothing but entropic formulations
of the fact that approximatecorrectionis possible if and
only if the joint reference-ancilla output state ρRE′ is The entanglement of formation Ef(σAB) is a well-
close to being factorized,that is ρRE′ ρR ρE′ (about behaved measure of the quantum correlations existing
≈ ⊗ betweentwoquantumsystemsAandB describedbythe
this point, see also Ref. [21]). In fact,
jointstateσAB. Ontheotherhand,thequantummutual
S(ρQ) I (ρQ, )=IR:E′(ρRE′) information IA:B(σAB) measures the total correlations,
c
− E quantum as well as classical, that a bipartite quantum
=D(ρRE′ ρR ρE′),
system exhibits [24]. Notice that both entanglement of
k ⊗
formation and quantum mutual information are by con-
where D(ρ σ) := Tr[ρlog ρ ρlog σ] is the quantum
k 2 − 2 struction symmetric under the exchange of A and B.
relative entropy and can be understood as a kind of dis-
Onthecontrary,thequantitymeasuringtheamountof
tance between states.
classical correlations in a bipartite quantum state loses
From Eqs. (3) and (4), it is an immediate corollary
that perfect correction(on the supportof ρQ) is possible such a symmetry, and a logical direction of classical cor-
relations seems to naturally emerge. Such a quantity,
if and only if [5]
proposed in Ref. [25], is defined as
I (ρQ, )=S(ρQ).
c
E
CB→A(σAB):=
However, coherent information is not the only quantity
whichenjoys sucha property. By introducing the entan- max S(σA) p S TrB σAB 11A⊗PiB ,
gσlAemBeanst[o2f2]formation, defined for a bipartite mixed state {PiB}i" −Xi i (cid:2) p(cid:0)i (cid:1)(cid:3)!#
E(σAB):= where the maximum is taken over all possible POVMs
f{pi,|φAiBi}i:mPiinpiφAiB=σABXi piE(cid:0)φAi B(cid:1), o{mPneiaBts}huiere(stiuhsbaastysyisstm,emmPieBtBri>ca,n0sdinfcoperiian:l=lgie,TnerarnaσdlBCPPBiBi→P.AiB(σS=AuBc1h1)B=a)
where the minimum is taken over all possible pure state CA→B(σAB),anditiscloselyrelatedt(cid:2)otheas(cid:3)sistedcla6s-
ensembledecompositionofσAB asσAB = p φAB and sical capacity of quantum channels [26].
i i i
E(φAB) := S Tr φAB is the entanglement of the In Ref. [27] it is provedthat for a tripartite pure state
B
pure bipartite state φAB, in Ref. [9] it iPs proved that φABC the relation CB→A(σAB) + Ef(σAC) = S(σA)
perfect correcti(cid:0)on (o(cid:2)n the(cid:3)(cid:1)support of ρQ) is possible if |holds,iwhere σAB etc are the reduced states of φABC .
and only if In the case of a channel, given the global state Ψ| RQ′E′i
| i
in Eq. (1), we correspondingly have
E(ρRQ′)=S(ρQ).
f
CE′→R(ρRE′)+E(ρRQ′)=S(ρQ). (7)
The “only if” part is not surprising, since it is known f
that (for an elementary proof, see Section IV below)
We are now able to easily prove Eq. (5). In fact, since
I (ρQ, ) E(ρRQ′), (5) I (ρQ, ) = S(ρQ) IR:E′(ρRE′), and from Eq. (7),
c f c
E ≤ E −
4
thanks to the monotonicity of quantum relative en- ( X :=Tr X denotes the trace-norm)
|| ||1 | |
tropy under the action of channels, namely D(ρ σ)
k ≥ 2
D(E(ρ)kE(σ)), ∀(ρ,σ,E), we have that ρRE′ ρR ρE′ 2 = p ρR P˜E′ ρR P˜E′
− ⊗ 1 (cid:12)(cid:12) i i ⊗ i − ⊗ i (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)Xi (cid:16) (cid:17)(cid:12)(cid:12)1
CE′→R(ρRE′)≤IR:E′(ρRE′), (8) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ≤(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) pi ρRi −ρR ⊗P˜iE′ 21 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
KXi (cid:12)(cid:12)(cid:12)p(cid:12)(cid:12)(cid:12)(cid:0) ρR ρR(cid:1) 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
which in turn directly implies ≤ i i − 1
i
X (cid:12)(cid:12) (cid:12)(cid:12)
2K p(cid:12)(cid:12)D(ρR ρR(cid:12)(cid:12))
≤ i i k
I (ρQ, ) E(ρRQ′). Xi
c E ≤ f 2KCE′→R(ρRE′)
≤
=2K S(ρQ) E(ρRQ′) .
f
−
(cid:16) (cid:17) (10)
Let us explain one by one all the passages in the above
V. ENTANGLEMENT OF FORMATION AND equation:
APPROXIMATE CHANNEL CORRECTION
(i) In the first line we applied identity (9) to the sub-
systemE′,where PE′ isaninformationallycom-
In this sectionwe will present the main result, that is, plete POVM and{ Pi˜E}′i its dual frame, and de-
tifhethleoscshoafnennetlacnagnlembeenatpopfrofoxrimmaatteiolyncisorsrmecatleldif.aWndeosnalwy PfinEe′d)]/ppi :.=Tr[ρE′P{iE′i] a}nid ρRi :=TrE′[ρRE′ (11R⊗
i i
beforethatapproximatecorrectionispossibleifandonly
if the joint reference-ancilla output state ρRE′ is almost (ii) Inthesecondlineweusedtheconvexityofthefunc-
factorized [8]. We would then like to say that the loss of tion x x2.
aenlmtaonsgtlefamcteonrtizoefdf.ormation is small if and only if ρRE′ is (iii) In the7→third line we defined K := maxi P˜iE′ 2,
1
which is finite because we are considering fi(cid:12)(cid:12)nite(cid:12)d(cid:12)i-
The “if” part has already been written in the form (cid:12)(cid:12) (cid:12)(cid:12)
of Eq. (8). In fact, if ρRE′ ρR ρE′, then S(ρR mensional Hilbert spaces. (cid:12)(cid:12) (cid:12)(cid:12)
ρE′) S(ρRE′) thanks to Fa≈nnes’⊗continuity propert⊗y, (iv) In the fourth line we used Pinsker inequality [30],
which≈implies that IR:E′(ρRE′) 0, and, in turn, that that is ρ σ 2 2D(ρ σ).
CE′→R(ρRE′) 0, or, equivale≈ntly, that E(ρRQ′) || − ||1 ≤ k
S(ρQ) (see Eq.≈(7)). f ≈ (v) CInE′t→hRe(ρfiRftEh′)liisnedefiwneedsiamspalymauxsiemdumtheovefarcatlltphoast-
To prove the “only if” part is a little trickier. We ex- sible measurements on E′.
ploit the existence, proved in Ref. [28] for every dimen-
(vi) In the last line we used Eq. (7).
sion of the Hilbert space, of (rank-one) informationally
completemeasurements,thatarePOVMswhoseelements Summarizing, we obtained that whenever
formabasisfortheoperatorspace. Inotherwords,there CE′→R(ρRE′) 0, or, equivalently, E(ρRQ′) S(ρQ),
f
alwaysexistsaPOVM{Pi}isuchthatTr[XPi]=0forall then ρRE′ ρ→R ρE′ 2 0 correspondingly,→which in
i if and only if X =0. Notice that this is the generaliza-
− ⊗ 1 →
tion of the usual concept of quantum state tomography. turni(cid:12)m(cid:12) pliestheexisten(cid:12)(cid:12)ceofanapproximatelycorrecting
(cid:12)(cid:12) (cid:12)(cid:12)
Informationally complete POVMs have a (generally non chann(cid:12)(cid:12)el , [8]. Notice(cid:12)t(cid:12)hat, as a trivial corollary,we get
unique)dualset P˜ suchthatthefollowingreconstruc- that CB→RA(σAB)=0 if and only if σAB =σA σB.
tion formula hold{s i}i In the sequence of inequalities in Eq. (10), ⊗the most
unpleasant feature is the size of the constantK. In fact,
it is clearly independent of the channel and the input
Tr[XP ]P˜ =X, X. (9) state, however,we did not investigatehow it depends on
i i
∀ the dimensions of the input and output Hilbert spaces
Xi HQ and HQ′. We can give a rough upper bound on K
byconsideringthe (continuousoutcome)informationally
Notice that the dual operators P˜i are generally non pos- complete POVM {Pg}g∈SU(d) defined as
itive, but can always be chosen hermitian [29]. We are 1
nowinpositiontowritethefollowingchainofinequalities Pg := dUgϕUg†,
5
where U is a unitary representationofthe groupSU(d), Corollary 1 If E E, the following inequality holds
g • f
≤
and ϕ is a pure state. In Ref. [28] the canonical dual set
P˜g g has been explicitly calculated, and it holds that F(ρQ,R◦E)≥1− 2(2dd′−1)2ε•, (13)
{ }
where ε :=S(ρQ) E (ρRQ′)p.
P˜g =2d 1, g, • − •
1 − ∀ Proof. Trivial. (cid:4)
(cid:12)(cid:12) (cid:12)(cid:12)
where d is the d(cid:12)i(cid:12)me(cid:12)n(cid:12)sion of the Hilbert space on which Then, thanks to the above mentioned “extremality
(cid:12)(cid:12) (cid:12)(cid:12)
the POVM P is measured, in our case HE′. Since property” enjoyed by the entanglement of formation
g g
{ }
we saw that its dimension can be upper-bounded as among entanglement measures, Corollary 1 can be ap-
dimHE′ dimHQ dimHQ′,weobtainthefollowing pliedtomanydifferentsituations,makingtheconclusions
≤ ×
we drew form Theorem 1 quite general.
ρRE′ ρR ρE′ 2 2(2dd′ 1)2 S(ρQ) E(ρRQ′) , Ontheotherhand,theso-calledhashinginequality[34]
f
− ⊗ 1 ≤ − −
w(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)here d := dimH(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q and d′ := (cid:16)dimHQ′. Anyw(1a(cid:17)1y), Ic(ρQ,E)≤EdR→Q′(ρRQ′) (≤Ed(ρRQ′)), (14)
the only assumption we need about the ancilla POVM where E (σAB) is the distillable entanglement and
d
PE′ is that it is informationally complete. We EA→B(σAB)istheone-way distillableentanglement(i.e.
{ i }i d
could hence use the one, among informationally com- we restrict the classical communication to go from A
plete POVMs, whose dual set minimizes K. How to to B only), implies the converse direction, namely, if
choosesuchan“optimal”informationallycompletemea- EA→B E , then the analogous of Eq. (4),
d ≤ •
surement is left as a wide open question.
At the end we can state the following: S(ρQ) E•(ρRQ′) g(1 F(ρQ, )), (15)
− ≤ − R◦E
Theorem 1 Given an input state ρQ, defined on the holdstrue. Itisworthstressingherethatwhilethecondi-
Hilbert space HQ, and a channel mapping states on tionEA→B E isverygeneral,theconditionE E is
HQ to states on HQ′, let us deEfine ε := S(ρQ) satisfiedd by≤man•y among known entanglement m•e≤asurfes
f
E(ρRQ′). Then, there exists a channel , from state−s but not by allof them (a notable exception is, for exam-
f
on HQ′ to states on H Q, such that R ple, the logarithmic negativity [35]). Nevertheless, it is
knownthat whatever generic entanglement measure sat-
F(ρQ, ) 1 2(2dd′ 1)2ε, (12) isfyingacertainnumberofconditionscanbeprovedtolie
f
R◦E ≥ − − betweenEA→B andE [36]. Henceinequivalententangle-
d f
where d:=dimHQ and d′ :=pdimHQ′. mentmeasures,providedthey behave“sufficiently well”,
lead to equivalent conditions for approximate quantum
Proof. With Eq. (11) at hand, the proof is straight- error correction, generalizing what was already noted in
forward. Itmakesuseofthewell-knownrelationexisting Ref. [9] in the case of exact correction.
between fidelity and trace-distance, that is By further specializing the entanglement measure, we
can say more. If the entanglement measure is chosen to
ρ σ
F(ρ,σ) 1 || − ||1, be “nottoo large”,itis possibleto refine the bound (13)
≥ − 2
as follows. More explicitly, the following result, that we
and of the main result of Ref. [8], thanks to which the state as a second theorem independent from Theorem 1,
existence of a channel such that can be proved
R
F(ρQ,R◦E)≥F2(ρRE′,ρR⊗ρE′) TinhpuetorHeimlber2tLseptacEe bHe aQ.chLanetneEl a(cσtAinBg)obnesatnateesntoanngtlhee-
•
is guaranteed. (cid:4) ment measure such that
IA:B(σAB)
E (σAB) , σAB (16)
•
≤ 2 ∀
VI. OTHER ENTANGLEMENT MEASURES
holds, and define ε := S(ρQ) E (ρRQ′). Then, there
• •
−
Up to now, we considered the entanglement of forma- exists a channel such that
R
tion E as the entanglement measure quantifying quan-
f
tum correlations. Such a choice is motivated by the fact F(ρQ, ) 1 2√ε•. (17)
R◦E ≥ −
thatitisknown[30]thatE isanupperboundtotheco-
f Proof. The proof goes as follows:
herentinformationitselfaswellastomanyothergenuine
entanglement measures E• (among these, for example, 1 ρRE′ ρR ρE′ 2 D(ρRE′ ρR ρE′)
one finds the distillable entanglement [31], the relative
2 − ⊗ 1 ≤ k ⊗
entropy of entanglement [32], and the squashed entan- (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) =2S(ρQ) IR:Q′(ρRQ′)
glement [33], just to cite three of them). The following (cid:12)(cid:12) (cid:12)(cid:12) −
corollary directly stems from Theorem 1 2S(ρQ) 2E (ρRQ′),
•
≤ −
6
where we used again Pinsker inequality and Eq. (2). At Then, the following inequality holds
this point, by the same passages as in the proof of The-
orem 1, we obtain the statement. (cid:4) S(σA) IA→B(σAB) g 2(2d d 1)2ε(σAB) ,
Relation (17) is clearly much tighter than the analo- − c ≤ A B − f
(cid:18)q (cid:19)
gous relation (13), in that here we succeeded in getting (19)
rid of the dependence on the dimensions of the input where g(x) is a function as in Eq. (4), and dA(B) :=
and output Hilbert space. Notice that condition (16) it dimHA(B).
is provedto holdfor the distillable entanglementandfor
Proof. First of all, let us notice that whatever bipar-
thesquashedentanglement[33]. Ontheotherhand,such titemixedstateσAB canbewrittenas(idA B)(ΨAB),
aderivationcannotbeappliedtotheentanglementoffor- for some channel B and some pure ΨA⊗BEsuch that
mationwhichcanbe smallerorlargerthanthe quantum Tr [ΨAB] = σA. TEhis simple observation is in order to
mutual entropy [10]. B
make sure that all equations, previously obtained for bi-
partite states ρRQ′ =(idR Q)(ΨRQ), canbe in partic-
⊗E
ularinterpretedasequationsvalidforallbipartitemixed
VII. RELATION BETWEEN ENTANGLEMENT
states σAB as well, simply paying attention to the direc-
OF FORMATION AND DISTILLABLE
ENTANGLEMENT tionality intrinsic in the definition of coherent informa-
tion. Then,puttingtogetherEqs.(4)and(12),weobtain
the statement (19). (cid:4)
Itisinterestingto directlycomparethe three relations
The large numerical factor multiplying ε in Eq. (19)
(Eqs.(3),(13),and(17))forapproximatequantumerror f
makesitpossibletheabovementionedgenericgapexhib-
correction that we considered throughout the paper:
ited by high dimensional systems, that is, entanglement
F(ρQ, ) 1 2(S(ρQ) I (ρQ, )), of formation can be close to the maximum value, while
c
R◦E ≥ − − E distillableentanglementisnull(oralmostnull). Eq.(19)
q
F(ρQ, ) 1 2 S(ρQ) E (ρRQ′), thensays“howlarge”,forfixedfinite dimensionsdA and
•
R◦E ≥ − − d , the gapcanactually be: in factwe canaffirmthat if
q B
F(ρQ, ) 1 2(2dd′ 1)2(S(ρQ) E (ρRQ′)), theentanglementofformationis“sufficientlyclose”toits
•
R◦E ≥ − − − maximumvalue,thenalsothecoherentinformationand,
q (18)
thankstothehashinginequality(14),theone-waydistil-
where d := dimHQ and d′ := dimHQ′. The first lable entanglement have to be “correspondingly large”.
is proved in Ref. [8], the second holds if E (σAB) Notice moreover that there may be room for a further
•
IA:B(σAB)/2, while the third holds if E (σAB) ≤ improvementof Eq. (19), since we obtained it as coming
•
E(σAB). The numerical factor in front of the “loss fig≤- from a probably over-simplified estimation. To tighten
f
ure” gets larger as we move from coherent-information– the evaluation of the constant K in Eq. (10) could then
loss toward entanglement-of-formation–loss. This fea- be useful in understanding the relationships between en-
ture is reminiscent of the fact that, in general, the gap tanglement of formation and distillable entanglement as
E > E between entanglement of formation and distill- well, besides being an interesting mathematical problem
f d
able entanglement can be generically large [10]. by itself.
Concerning this point, it is interesting to notice that Beforeconcluding,wewouldliketostressonemorere-
our approach can be somehow useful to understand to mark. It is clearfromEq.(18)how we areactually deal-
which extent such a gap can be authentically arbitrary. ing with three differenttopologiesonthe setofquantum
In fact, entanglement of formation and distillable en- channels induced by different measures of bipartite en-
tanglement coincide on pure states, and both of them tanglement [37]. Also this connection definitely deserves
are known to be asymptotically continuous in the mixed further investigation.
neighborhood of every pure state [36]. It is then reason-
able that, sufficiently close to pure states, entanglement
VIII. CONCLUSIONS
of formation and distillable entanglement become equiv-
alent entanglement measures (in the sense that they can
bereciprocallybounded),andthegapbetweenthemcan- In summary, we generalized the information-
notbecompletelyarbitrary. Infactwecansaysomething theoretical analysis of approximate quantum error
more in the form of the following correction based on coherent information given in
Ref. [8], by showing that approximate quantum error
Corollary 2 Foran arbitrarybipartite mixedstateσAB,
correction is possible if and only if the loss of en-
with S(σA) S(σB), let us define the coherent informa-
≤ tanglement along the quantum channel is small. We
tion
consideredexplicitly differententanglementmeasures,in
IA→B(σAB):=S(σB) S(σAB) particular the entanglement of formation and the distill-
c −
able entanglement, showing how equivalent conclusions
and the entanglement of formation deficit
come from inequivalent entanglement measures. We
ε(σAB):=S(σA) E(σAB). moreover showed that the approach used here can be
f f
−
7
applied also to understand the interconnections existing QuantumComputationandInformationProject. Thank
between entanglement of formation and distillable youtoMasahitoHayashiandLorenzoMacconeforuseful
entanglement, even though they are known to behave comments and suggestions.
quite independently, in particular in high dimensional
quantum systems.
ACKNOWLEDGMENTS
The author acknowledges Japan Science and Technol-
ogy Agency for support through the ERATO-SORST
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