Quantum information becomes classical when distributed to many users

0.14 MB·English

by  G. Chiribella

Quantum information becomes classical when distributed to many users
Giulio Chiribella∗ and Giacomo Mauro D’Ariano†
QUIT Group, Dipartimento di Fisica “A. Volta” and INFM, via Bassi 6, 27100 Pavia, Italy‡
(Dated: February 1, 2008)
AnyphysicaltransformationthatequallydistributesquantuminformationoveralargenumberM
of users can be approximated by a classical broadcasting of measurement outcomes. The accuracy
of the approximation is at least of the order O(M−1). In particular, quantum cloning of pure and
mixedstatescan beapproximatedviaquantumstateestimation. Asan example,foroptimalqubit
cloning with 10 output copies, a single user has error probability perr > 0.45 in distinguishing
classical from quantum output—avalueclose to theerror probability of the random guess.
PACSnumbers: 03.67.Hk,03.65.Ta
7
0
0 Differently from classical information, which can be the classical channels are almost indistinguishable, with
2 perfectlyreadoutandcopied,quantuminformationcan- probability of error approaching the random guess value
n not, since nonorthogonal quantum states can be neither 1/2 at rate at least α/M, α constant. More generally,
a perfectlydistinguished[1],norperfectlycopied[2]. Since foranygroupofk users,the coherentandtheincoherent
J ideal distribution of quantum information is impossible, schemes produce the same reducedstate within anaccu-

1 one is then interested in the performance limits of op- racykα/M. Thisalsoimpliesthatentanglementbetween
3
timal distribution, and such interest has focused much theoutputcopiesasymptoticallydisappearsatanygiven

attention in the literature to the problem of optimal orderk: forlargeM onlythek-partiteentanglementwith
3
v cloning[3]. Optimalcloningconsistsinfindingthephysi- k =O(M)cansurvive. ThescalingM−1 isageneralup-
7 caltransformationthatconvertsN copiesofapurestate, per bound holding for all physical transformations that
0 randomly drawn from a given set, into the best possible equally distribute quantuminformationamong M users,
0 approximationofM >N copiesofthe samestate. More including pure state cloning and mixed state broadcast-
8
recently, the analogous problem for mixed states (opti- ing. Of course for specific transformations the actual
0
6 malbroadcasting)hasbeen considered[4]. Inboth cases scaling can be even faster.
0 ofpure andmixedstates,the optimaltransformationre- The mathematical description of a quantum channel
/ quires a coherentinteractionofthe input systems with a that transforms states on the Hilbert space H into
h in
p set of ancillae. On the other hand, classical incoherent states on the Hilbert space Hout is provided by a com-
- schemes,suchas the measure-and-prepare—where the N pletely positive trace-preserving map E. Since here we
t
n initialcopiesaremeasuredandM copiesofanestimated focus on channels that distribute quantum information
a state are prepared—are suboptimal for any finite M. to M users, we have H = H⊗M, with H denoting
out
u
When cloning pure states, the measure-and-prepare the single user’s Hilbert space. Moreover, since we re-
q
: schemebecomesoptimalintheasymptoticlimitM →∞ quire the information to be equally distributed among
iv in all known kinds of cloning. This leads to conjecture all users, for any input state ρ on Hin the state E(ρ)
X that pure state cloning is asymptotically equivalent to must be invariant under permutations of the M output
r quantum state estimation [5, 6], a conjecture recently spaces. Invariance under permutations implies that any
a
proved in Ref. [7]. Essentially, the line of proof is that a group of k users will receive the same state
symmetriccloningtransformationwithM =∞,whenre-
strictedtosingleclones,mustbeanentanglementbreak- ρ(k) =Tr [E(ρ)] , (1)
out M−k
ing channel, whence it can be realized by the measure-
and-prepare scheme [8]. Such an argument, however, Tr denoting partialtraceovern outputspaces,nomat-
n
doesnotprovideanyestimateofthegoodnessoftheclas- terwhichones. Inparticular,eachsingleuserreceivesthe
sical scheme for finite number M of output copies, the same state ρ(1) =Tr [E(ρ)]. In the following, we will
out M−1
situation of interest for applications and experiments. name a channel with the above properties a channel for
In this letter we analyze the general class of quan- symmetric distribution of information (SDI-channel, for
tum channels that equally distribute quantum informa- short). OurgoalwillbetoapproximateanySDI-channel
tion to M users, producing output states that are in- E with a classical channel E, corresponding to measure
variant under permutations. This class contains cloning theinputandbroadcastthemeasurementoutcome,with
as a special case. We will show that for M sufficiently each user preparing locallyethe same state accordingly.
large any channel of the class can be efficiently approx- Such channels have the special form
imated by a classical measure-and-prepare channel. In-
deed, we will show that from the point of view of sin- E(ρ)= Tr[P ρ] ρ⊗M , (2)
i i
gle users the states produced by the quantum and by
i
X
e
2
where the operators {P } represent the quantum mea- ρ(k) denoting the reduced state ρ(k) =Tr [ρ].
i M−k
surement performed on the input (P > 0, P =
i i i
11in), and ρi is the state prepared conditionally to the Proof. The identity in the totally symmetric subspace
P
outcome i. The accuracy of the approximation is given H⊗n ⊂H⊗n can be written as
+
by the trace-norm distance ||ρ(1) −ρ˜(1)|| = Tr|ρ(1) −
out out 1 out
ρ˜(1)| between the single user output states. The trace- 11+ =d+ dψ P (ψ) , (7)
out n n n
normdistancegovernsthedistinguishabilityofstates[1], Z
namely the minimum error probability perr in distin- where P (ψ) = |ψihψ|⊗n. Using Eq.(7) with n =
n
guishing between two equally probable states ρ1 and ρ2 M − k, we can write ρ(k) = dM−k dψ ρ (ψ), where
+ k
is given by ρ (ψ) = Tr ρ 11⊗k⊗P (ψ) . On the other
k M−k M−k
R
1 1 hand, the reduced state ρ˜(k) can be written as ρ˜(k) =
perr = 2 − 4||ρ1−ρ2||1 , (3) d+ dψ P (ψ) ρ(cid:2)(ψ) P (ψ). Then,(cid:3)the difference be-
M k k k
and for small distances it approaches the random guess tween ρ(k) and ρ˜(k), denoted by ∆(k), is given by
R
value p = 1/2. In our case, a small distance ||ρ(1) −
gρ˜u(o1ui)st|h|1inmgererbaentswteheanttahseinogultepuustesrohfatshleitttwleochchaannceneolfsdEoisuattinnd- ∆(k) =d+M−kZ dψ"ρk(ψ)− d+Md+M−kPk(ψ)ρk(ψ)Pk(ψ)#.
E by any measurementonhis localstate. In addition, to
discuss the multipartite entanglement in the state ρ(k), Notice that the integrand on the r.h.s. has the
out form A − BAB, with A(ψ) = ρ (ψ) and B(ψ) =
wee will consider the distance ||ρ(k) −ρ˜(k)|| . Since the k
out out 1 d+/d+ P (ψ). Using the relation
state ρ˜(k) coming from E in Eq.(2) is separable, a small M M−k k
out
distance means that any group of k users has a little q
A−BAB =A(11−B)+(11−B)A−(11−B)A(11−B) (8)
chance of detecting entaenglement.
The key idea of this letter is to get the approximation we obtain
of SDI-channels exploiting the invariance of their out-
put states under permutations. In fact, permutationally ∆(k) =d+ C+C†−D , (9)
M−k
invariant states have been thoroughly studied in the re-
search about quantum de Finetti theorem [9], where the where (cid:0) (cid:1)
goal is to approximate any such state ρ on H⊗M with
a mixture of identically prepared states ρ˜= p ρ⊗M. C = dψ A(ψ)[11−B(ψ)] , (10)
i i i
In particular, as we will see in the following, the recent Z
P
techniquesofRef. [10]provideaveryusefultooltoprove D = dψ [11−B(ψ)] A(ψ) [11−B(ψ)] . (11)
our results. For simplicity, we will first start by consid- Z
ering the special case of SDI-channel with output states The operator C is easily calculated using the relation
in the totally symmetric subspace H⊗M ⊂ H⊗M, which
+
is the case, for example, of the optimal cloning of pure dψ ρ (ψ) P (ψ)= dψTr [ρ P (ψ)]
k k M−k M
states. Then, all results will be extended to the general
Z Z
case of arbitrary SDI-channels. Tr [ρ] ρ(k)
M−k
= = ,
In order to approximate channels we use the following d+ d+
M M
finite version of quantum de Finetti theorem, which is
provedwiththesametechniquesofRef.[10],withaslight which follows from Eq. (7) with n=M. In this way we
improvement of the bound given therein [11]: obtain C = sM,k/d+M−k ρ(k) . Since C is nonnegative,
Lemma 1 For any state ρ on H⊗M ⊂ H⊗M, consider we have ||C||1 = Tr[C] = sM,k/d+M−k. Moreover, due
+
to definition (11) also D is nonnegative, then we have
the separable state
||D|| = Tr[D] = Tr[C + C†], as follows by taking the
1
ρ˜= dψ p(ψ) |ψihψ|⊗M , (4) trace on both sides of Eq.(9). Thus, the norm of D is
||D|| =2||C|| . Finally, taking the norm on both sides of
Z 1 1
where the probability distribution p(ψ) is given by Eq. (9), and using triangular inequality we get ||∆(k)||6
4d+ ||C|| =4s , that is bound (6). (cid:4)
p(ψ)=Tr[Π ρ], Π =d+ |ψihψ|⊗M, (5) M−k 1 M,k
ψ ψ M Since the dimensionofthe totallysymmetric subspace
.
where dψ denotes the normalized Haar measure over the H⊗n is given by d+ = d+n−1 , d = dim(H), for M ≫
+ n n
pure states |ψi ∈ H, and d+M = dim(H+⊗M). Then, one kd the ratio d+M−k/d+M(cid:0)tends (cid:1)to 1− k(dM−1). Therefore,
has
Lemma 1 yields
||ρ(k)−ρ˜(k)||1 64sM,k, sM,k =. 1−sd+Md+M−k , (6) ||ρ(k)−ρ˜(k)||1 6 2(dM−1)k, M ≫kd, (12)
3
i.e. the distance between ρ(k) and the separable state taking E as an optimal pure state cloning, the channel E
ρ˜(k) vanishes as k/M. yields an approximation of E based on state estimation
With the above lemma, we are ready to prove the ap- (the measurement outcomes of P are in one to one core-
ψ
proximationtheoremforSDI-channelswithoutputinthe respondence with the pure states on H). On one hand,
totally symmetric subspace: when applied to a pure state |φi, the optimal cloning
Theorem 1 Any SDI-channel E with output states in gives fidelity Fclon = hφ|ρ(o1u)t|φi. On the other hand,
the totally symmetric subspace H⊗M ⊂H⊗M can be ap- since the measurement Pψ gives a possible estimation
+ strategy, the fidelity of the optimal estimation F can-
proximated by a classical channel est
not be smaller than hφ|ρ˜(1)|φi. Therefore, the difference
out
between the two fidelities can be bounded as
E(ρ)= dψ Tr[P ρ] |ψihψ|⊗M , (13)
ψ
Z 2(d−1)
06F −F 6||ρ(1) −ρ˜(1)|| 6 , M ≫d,
where Pψ eis a quantum measurement (Pψ > 0 and clon est out out 1 M
dψ P = 11 ). For large M, the accuracy of the ap- (16)
ψ in
proximation is namely it approaches zero at rate 1/M. A part from a
R constant,thisistheoptimalrateonecanobtaininagen-
||ρ(k) −ρ˜(k)|| 6 2(d−1)k, M ≫kd. (14) eralfashionholdingforanykindofpurestatecloning. In
out out 1 M fact,1/M istheexactrateinthecaseofuniversalcloning,
whereF −F = N(d−1) (see[12]forthesingle-clone
Proof. Consider the channel E∗ in the Heisenberg pic- clon est M(N+d)
fidelity). In addition, from Eq. (16) it immediately fol-
ture, defined by the relation Tr[OE(ρ)] = Tr[E∗(O)ρ]
lows that any quantum cloning map for large numbers
for any state ρ on H and for any operator O on
in N of input copies is approximated by state estimation,
H . Since the channel E is trace-preserving, E∗ is
out since for cloning one has M > N, and M is necessarily
identity-preserving, namely E∗(11 ) = 11 . Apply-
out in large. In this way we proved the asymtotic equivalence
ing Lemma 1 to the output state ρ = E(ρ), we get
out beteween cloning and state estimation for any kind of
ρ˜ = dψ Tr[Π E(ρ)] |ψihψ|⊗M. Since Tr[Π E(ρ)] =
out ψ . ψ cloning (see also the following Theorem 2 for the gen-
Tr[E∗(Π )ρ], by defining P = E∗(Π ), we immediately
obtain Rtψhat ρ˜ =E(ρ), wiψth E as inψEq. (13). The op- eralcase of Hout 6=H+⊗M), for either largeN or large M
out (see open problems in Ref. [6]). We emphasize that the
erators{P } representa quantum measurement onH ,
ψ in M = ∞ result of Ref. [7] cannot be used to prove the
sincetheyareobtaineedbyappleyingacompletelypositive
large N asymptotics.
identity-preserving map to Π , which is a measurement
ψ All results obtained for SDI-channels with output in
on H . Finally, the bound (14) then follows from Eq.
out the totally symmetric subspace can be easily extended
(12). (cid:4)
to arbitrary SDI-channels, exploiting the fact that any
The above theorem proves that for large M the quan-
permutationally invariant state can be purified to a to-
tum information distributed to a single user can be ef-
tally symmetric one [10]:
ficiently replaced by the classical information about the
measurement outcome ψ. In fact, the single user output Lemma 2 Any permutationally invariant state ρ on
statesofthechannelsEandEbecomecloserandcloser— H⊗M can be purified to a state |Φi ∈ K⊗M ⊂ K⊗M,
+
and therefore less distinguishable—as M increases. For where K=H⊗2.
large M, the error probabiliety in distinguishing between
ρ(1) and ρ˜(1) has to satisfy the bound Once the state ρ has been purified, we can apply
out out
Lemma 1 to the state |Φi, thus approximating its re-
1 d−1 duced states. The reduced states of ρ are then obtained
p > − , (15)
err
2 2M by taking the partial trace over the ancillae used in the
purification. This implies the following
namely it approaches1/2atrate M−1. For example, for
qubitsEq. (15)givesalreadywithM =10anerrorprob-
Lemma 3 For any permutationally invariant state ρ on
ability p >0.45,quite close to the errorprobabilityof
err H⊗M, purified to |Φi ∈ K⊗M, K = H⊗2, consider the
a purely random guess. More generally, the bound (14) +
separable state
implies that for any group of k users there is almost no
entanglementin the state ρ(k), since it is close to a com-
out ρ˜= dΨ p(Ψ) ρ(Ψ)⊗M (17)
pletely separable state. As the number of users grows,
multipartite entanglement vanishes at any finite order: Z
onlyk-partiteentanglementwithk =O(M)cansurvive. where dΨ is the normalized Haar measure over the
Applyingourapproximationtheoremtotheparticular pure states |Ψi ∈ K, ρ(Ψ) is the reduced state
case of pure state cloning, we obtain a complete proofof ρ(Ψ) = Tr [|ΨihΨ|], and p(Ψ) is the probability dis-
H
itsasymptoticequivalencewithstateestimation. Infact, tribution given by p(Ψ) = Tr[Π |ΦihΦ|], with Π =
Ψ Ψ
4
D+ |ΨihΨ|⊗M, D+ =dim(K⊗M). Then, one has such channel can be efficiently approximated by a clas-
M M +
sical one, where the input system is measured and the
||ρ −ρ˜ || 64S , S =. 1− DM+−k . (18) measurement outcome is broadcast, and each user pre-
k,A k,A 1 M,k M,k s D+ pares locally the same state accordingly. The approxi-
M
mating channel can be regarded as the concatenation of
a quantum-to-classical channel (the measurement), fol-
Proof. Applying Lemma 1 to τ = |ΦihΦ|, we get the
lowedbyaclassical-to-quantum channel(thelocalprepa-
state τ˜= dΨ p(Ψ) |ΨihΨ|⊗M. The state ρ˜is then ob-
ration). Actually, the latter channel is needed only for
tainedbytracingouttheancillaeusedinthepurification,
R the sake of comparisonwith the originalquantum trans-
namely it is given by Eq.(17). Since partial traces can
formation to be approximated, since, due to the data
only decrease the distance, the bound (18) immediately
processing inequality, this additional stage can only de-
follows from the bound (6). (cid:4)
crease the amount of information contained in the clas-
It is then immediate to obtain the following:
sical probability distribution of measurement outcomes.
Theorem 2 Any SDI-channel E can be approximated by Therefore, asymptotically, there is no broadcasting of
a classical channel quantum information, but just an announcement of the
classical information extracted by a measurement. In
E(ρ)= dΨ Tr[P ρ] ρ(Ψ)⊗M , (19) synthesis, we cannot distribute more information than
Ψ
Z what we are able to read out.
where P isea quantum measurement, namely P > 0
Ψ Ψ
and dΨ P = 11 . For large M, the accuracy of the
Ψ in Acknowledgments.— This work has been founded by
approximation is
R MinisteroItalianodell’Universit`aedellaRicerca(MIUR)
2(d2−1)k through PRIN 2005.
||ρ(k) −ρ˜(k)|| 6 , M ≫kd2. (20)
out out 1 M
ThistheoremextendsTheorem1andallitsconsequences
to the case of arbitrary SDI-channels. In particular, it
proves that asymptotically the optimal cloning of mixed ∗ Electronic address: chiribella@fisicavolta.unipv.it
state can be efficiently simulated via mixed states esti- † Electronic address: [email protected]
mation. The results of the measurement P are indeed ‡ URL:http://www.qubit.it
Ψ
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