Random multiparty entanglement distillation

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by  Ben Fortescue

Random multiparty entanglement distillation
Ben Fortescue∗ and Hoi-Kwong Lo†
Center for Quantum Information and Quantum Control (CQIQC),
Dept. of Electrical & Computer Engineering and Dept. of Physics,
University of Toronto, Toronto, Ontario, M5S 3G4, Canada
(Dated: January 15, 2008)
Wedescribevariousresultsrelatedtotherandomdistillationofmultipartyentangledstates-that
is,conversion ofsuchstatesintoentangledstatesshared betweenfewer parties,wherethoseparties
arenotpredetermined. Inpreviouswork[1]weshowedthatcertainoutputstates(namelyEinstein-
Podolsky-Rosen(EPR)pairs)couldbereliablyacquiredfromaprescribedinitialmultipartitestate
(namely the W state |Wi = 1 (|100i +|010i +|001i)) via random distillation that could not
√3
8 be reliably created between predetermined parties. Here we provide a more rigorous definition of
0 what constitutes “advantageous” random distillation. We show that random distillation is always
0 advantageous for W-class three-qubit states (but only sometimes for Greenberger-Horne-Zeilinger
2 (GHZ)-classstates). WeshowthatthegeneralclassofmultipartystatesknownassymmetricDicke
statescanbereadilyconvertedtomanyotherstatesintheclassviarandomdistillation. Finallywe
n
a show that random distillation is provably not advantageous in the limit of multiple copies of pure
J states.

5
1
I. INTRODUCTION muchless-wellunderstoodthanthoseoftwo-partystates.

For example, there is no single well-defined maximally
]
h Entanglement in quantum information theory is often entangled state in the multiparty case. It appears that
p consideredasaresource[2,3]whichcanbeusedbyphys- multiparty states can divided into distinct classes [7, 8],
-
t icallyseparatedpartiestoperformtaskssuchasquantum andeveninthe three-partycase itis notknownwhether
n
teleportation [4] or superdense coding [5], under the re- or not all entangled states can be reversibly obtained
a
u striction of the parties to local operations and classical through LOCC from a finite selection of other states
q communications (LOCC). Under LOCC the parties can - the “minimal reversible entanglement generating set”
[ perform local quantum operations on their own portions (MREGS) [9].

of the entangled states and exchange classical informa- A topic of interest in the descriptionof multiparty en-
2
v tion with each other through some classical communica- tangled states is the conversionof these states into, gen-
9 tionchannels,butnotperformjointquantumoperations erally, states shared between fewer parties, and specifi-
5 or (equivalently) exchange quantum information. It is cally two-party states. Since there are many results on
0 known that parties cannot increase their shared entan- the operational properties of two-party states, consider-
4 glement under LOCC, which motivates the view of en- ing such a conversion provides useful information in the
9. tanglement as a resource. multiparty case.
0 Determining what may be accomplished with some Several results e.g. [10, 11, 12, 13, 14, 15] exist re-
7 particularentangledstateunderLOCCprovidesanoper- gardingthe conversionofmultipartytotwo-partyentan-
0 ationaldescriptionofthatstate,whichcaninsomecases gled states shared between predetermined parties. In [1]
v: be used as an entanglement measure - for example the we demonstrated that some two-party entangled states
i well-known result [6] that the maximum ratio at which which could not be reliably obtained (i.e. probability
X
maximally-entangled EPR pairs <1)betweenpredeterminedpartiescouldbereliablyob-
r tained(probability 1inthe limitofmany“rounds”of
a →
1 distillation) betweenparties which wererandomly deter-
Φ = (00 + 11 ) (1)
| i √2 | i | i AB mined in the course of a LOCC-protocol - a process we
refer to as “randomdistillation”. Specifically we showed
can be obtained through LOCC is the entanglement en- that one can reliably distill one EPR pair from a single
tropy ES(ψ )=S(ρ ), where W betweenrandomparties,versusdoingsowithaprob-
AB A
ability at most 2/3 between predetermined parties. The
S(ρ)= trρlog2ρ (2) randomdistillationrateexceedseventheasymptoticrate
−
ρA =−trB(|ψABihψAB|). (3) of H2(1/3) ≈ 0.92 EPRs per W between predetermined
parties in the many-copy limit, where H2 is the binary
In general, the properties of multiparty entangled entropy function
states (those shared between more than two parties) are
H2(x)= xlog2(x) (1 x)log2(1 x). (4)
− − − −
Inthispaper,weaddressanumberofquestionsinran-
[email protected] domdistillation. Firstly. ourcriterionin[1]forwhatcon-
∗
[email protected] stituted “advantageous” random distillation was some-
†
2
what problematic, in particular when considering mul- the reduced state of ψ ψ , traced over all A .
j∈/{I,T}
| ih |
tiple copies of states. Here we provide a new criterion For the single-copy analogues of these quantities, for
for advantageous random distillation applicable to any the distillation
pure-statecase,includingthatofcollectiveoperationson
multiple copies of a state. We also ask whether random ψ Φ ⊗NAiAj, (11)
distillation gives an advantage in the many-copy limit, | iA1...AM −→ | iAiAj
and demonstrate that it does not. LOCCOij
Secondly, in our previous paper we considered only a
|{z}
small number of specific states. Here we consider the we define
randomdistillationproperties ofgeneralclassesofstates
- specifically, distilling the general classes of three-qubit E (ψ) sup N (12)
pure-state entanglement, the GHZ and W classes. We aIJ ≡ h AIAJi
E (ψ) max(sup N ) (13)
show that all W-class states can undergo advantageous s ≡ ij h AIAJi
randomdistillation,buttheGHZclasscontainsexamples
both of states which can and cannot. E (ψ) sup N (14)
Finally,wepreviouslyconsideredprimarilydistillation t ≡ * AiAj+
ij
X
totwo-partyEPRpairs. Hereweconsideraclassoffinal
states sharedbetween largernumbers of parties. For the
We noted in [1] that for the three-party W state
multiparty entangled states known as symmetric Dicke
states, we briefly demonstrate a class of output states
1
which may be reliably obtained through LOCC only by W = (100 + 010 + 001 ) (15)
ABC
random distillation. | i √3 | i | i | i
it is possible to obtain an EPR through LOCC between
II. DEFINITIONS random parties but not specified parties. Hence even
though (from [13]) Es∞(W) = H2(1/3) ≈ 0.92, we find
For conversion of an M-party pure state ψ to EPR E∞(W) 1. However [1] further noted that the condi-
| i t ≥
pairs Φ through LOCC tionEt >Es couldalsobetriviallysatisfied,forexample
| i
by the state Φ Φ , for which E =2>E =1.
AB BC t s
ψ ⊗N Φ ⊗NAiAj. (5) We would|thierefo⊗re|liike to find a condition that more
| iA1...AM −→ | iAiAj
LOCCOij generallycaptureswhentrue“randomdistillation”isad-
vantageous - that is, one obtains a greater entanglement
we define |{z}
yield due to the nondeterministic nature (in terms of
N which parties receive the final state) of the distillation
E∞ (ψ) sup AIAJ (6)
aIJ ≡ N , rather than there simply being somewhat independent
N→∞
entanglements between different pairs of parties. We
N
E∞(ψ) max sup AIAJ (7) would further like to define such a condition in terms
s ≡ ij N→∞ N of general pure-state bipartite entanglement measures,
E∞(ψ) sup ijNAiAj. (8) rather than solely in terms of the distillable EPR pairs.
t ≡N→∞P N WethusconsidertheLOCC-conversion(viaaprotocol
P) of an initial pure state ψ to final pure multipartite
Thatis,E∞ representsthemaximumrateofEPRdistil-
lationbetwaIeJenpartiesI andJ (withthehelpofallother states ψf with probabilities pf.
parties), E∞ represents the highest distillation rate of
s
EPRpairsbetweenanygivenpairofpartiesandE∞ the ψ P ψ ,p (16)
t f f
−→ { }
highest total EPR distillation rate, irrespective of which
LOCC
parties share them.
In this asymptotic case (though not generally) E∞ |{z}
aIJ and the LOCC conversion (via a protocol Q) of multi-
is equal to the entanglement of assistance [10], with [13]
party states ψ to pure two-partystates ψ with prob-
showing that f gIJ
abilities p
g
E∞ =min S(ρ ),S(ρ ) (9)
aIJ T { AIT AJT } Q
ψ ψ ρ ,p (17)
where the minimization is over the division of parties f −→ { gIJ ⊗ g g}
LOCC
otherthanA andA intotwogroupsT andT (i.e. over
I J
bipartite “cuts” separating all parties into two groups, |{z}
(note that in the above, I and J are not necessarily the
one containing A and one containing A ) and
I J same for every g).
ρ =tr (ψ ψ ), (10) We define, for some bipartite pure-state entanglement
AIT Aj∈/{I,T} | ih |
3
measure E and S(ρ ) is a convex function of q(ψ) in the range
A
0 q 1 , corresponding to 0 S 1.
A (ψ ) sup p E(ψ ) (18) ≤ ≤ ≤ ≤
IJ f ≡ g gIJ
P,QXg Proof: Explicit calculation shows
E (ψ ) maxA (ψ ) (19)
sp f IJ f
≡ IJ 1 1 q(ψ)2
Ernd(ψ)≡sPu,Qp pfEsp(ψf) (20) S(ρA)=f(q)=H2 −p 2− ! (27)
f
X
E∞ (ψ) Ernd(ψ⊗N), N (21) and that
rnd ≡ N →∞
d2f
E∞(ψ) Esp(ψ⊗N), N (22) dq2 ≥0, 0≤S ≤1 (cid:3) (28)
sp ≡ N →∞
where the supremums in the above expressions are over We define qsp, qrnd etc. as analogous quantities to
all possible LOCC protocols P and Q. Esp,Ernd etc., withq as the entanglementmeasure. The
Hence E represents the supremum of the expected quantity q is a useful measure in this case since it is
rnd
entanglement(asmeasuredbyE)obtainedbywhichever second-order in the state’s coefficients. Thus, for re-
pair of parties has the highest entanglement once the peated rounds of unitaries and measurements, probabil-
protocol has been performed, while E is the corre- ities and normalisationfactors cancelout when calculat-
sp
sponding quantity for parties chosen before performing ing q , as shown below. Since the Ex (i.e. Ernd, Esp
h i
the protocol. Thus E E in general, and, if etc.) are expectation values for S, it follows from the
rnd sp
E (ψ)>E (ψ), this repre≥sentsgenuine advantageous convexity result that
rnd sp
random distillation as discussed above.
E (ψ) f(q (ψ)). (29)
While as mentioned any bipartite pure-state measure x x
≥
E may in principle be used, for the remainder of this
Note then that by this definition q (ψ) = f−1(E (ψ)),
paper and our results (with the exception of Section IV) x 6 x
in general.
we shalladopt as our measure the entanglement entropy
ES,i.e. theVonNeumannentropyS ofthereducedstate
as noted above. We thus define
A. The W protocol
E(ψ ) ES(ψ ). (23)
AB AB
≡
Wefirstconsidertheprotocolof[1](whichwewillrefer
to as the W protocol) for obtaining an EPR pair from
III. W AND GHZ-CLASS STATES a W state. This consists of all three parties repeatedly
applying the unitary
In [1] we demonstrated advantageous random distilla-
tion for the three-party W and similar states, and that 0 1 ǫ2 0 +ǫ2 , 1 1 (30)
| i−→ − | i | i | i−→| i
random distillation was not advantageous for certain
p
GHZ-likestates. Howevernogeneralresultwasobtained followed by all performing the projection
for the general GHZ and W classes noted in [7], into
F = 0 0 + 1 1, G= 2 2. (31)
one of which any three-qubit pure state with genuine
| ih | | ih | | ih |
tripartite entanglement may be classed. Here we find
If all three parties get outcome F, the protocol is re-
Theorem 1: peated. If exactly one party gets outcome G, the other
two parties have an EPR pair, the expectation value of
For any W-class pure entangled three-qubit state ψ
W theireventualentanglementtending tounityinthelimit
E (ψ )>E (ψ ) (24) of many repetitions and small ǫ. (The probability of the
rnd W s W
protocolaborting due to failure, where two or more par-
Proof:
ties get G, is negligible in this limit).
We make use of the following simple lemma (We alsoshowin[1]thatrandomdistillationis advan-
tageous for a finite number of rounds, with a protocol
Lemma 1: For a general normalised two-qubit pure forwhichtheprobabilityofobtainingarandomly-shared
state EPR pair from a W within R rounds is R . This ex-
R+1
ceeds the single-copy limit (for predetermined parties)
ψ =(c00|00i+c01|01i+c10|10i+c11|11i)AB (25) of 2/3 for R 3 and the asymptotic limit of 0.92 for
≥
R 12.)
theentanglementmeasureS(ρ )increasesmonotonically
A ≥
Note that the W state enjoys a special property that
with the concurrence [16]
makes our previous analysis of random distillation of an
q(ψ)=2c01c10 c00c11 (26) EPR from a W state simple - a failed round (that is,
| − |
4
where all parties obtain outcome F) returns the state to where
a W. Therefore in the limit of many rounds and small
ǫ (where success and failure of this kind are the only k00 = (1−ǫ2)32Rk000 (35)
outcomeswithnon-negligibleprobability)theprotocolis R P ...P
FR F1
“yrseesdeti”nathfteersaemacehwfaaiyl.urIenacnodntervaesrty, trhoiusnids cnaont btheeacnaasle- k01 = (p1−ǫ2)Rk010 (36)
R P ...P
for a general three-qubit pure state. Indeed, whenever a FR F1
rstoautnedboecformanedsoamnedwistsitlalattei.onFofraitlhsias rgeeanseorna,ltthhereaen-aqluybsiist k10R = p(1P−ǫ2.).R.kP100 (37)
of multi-round random distillation for the general three- FR F1
qubitstateisnotentirelytrivial. Inthefollowing,wewill (p1 ǫ2)R2k11
k11 = − 0 (38)
use the properties of the concurrence discussed above to R P ...P
FR F1
perform such an analysis. Before doing so, let us first
demonstrate the evolution of a general three-qubit state andPFN is the probabilitpy of allparties getting F in the
under the W protocol. Nth round of the protocol after having done so in all
previous rounds i.e.
Considerthenapplyingthisprotocoltoageneralthree-
qubit pure state sharedbetween Alice, Bob and Charlie:
PFN =(1−ǫ2) (1−ǫ2)2K00N−1
(cid:16)
+(1 ǫ2)[K01 +K10 ]+K11 (39)
− N−1 N−1 N−1
|ψ1iABC =|0iA k000|00i+k010|01i If the parties performone further roundofunitar(cid:17)ies, the
(cid:16)
+k100|10i+k110|11i BC +|1iA(...) (32) state will be
(cid:17)
|ψR+1iABC =(1−ǫ2)12|0iA (1−ǫ2)k00R|00i
where the (...) represent some additional terms whose +(1 ǫ2)12[k01 01 +k1(cid:16)0 10 ]+k11 11
amplitudeswearenotconcernedwith. WedefineK00 − R| i R| i R| i BC
|k000|2 etc. 0 ≡ +ǫ|2iA (1−ǫ2)k00R|00i (cid:17)
After every party has performed the unitary (30) the +(1(cid:16) ǫ2)12[k01 01 +k10 10 ]+k11 11
state becomes − R| i R| i R| i BC
(cid:17)
+(...) (40)
If the parties then project and Alice alone gets outcome
|ψ1iABC =(1−ǫ2)12|0iA (1−ǫ2)k000|00i G, with probability
+(1−ǫ2)12[k010|01i+(cid:16)k100|10i]+k110|11i BC PGR+1 =ǫ2 (1−ǫ2)2K00R+(1−ǫ2)[K01R+K10R]+K11R
+ǫ|2iA (1−ǫ2)k000|00i+(1−ǫ2)12[k010|01i+(cid:17)k100|10i] the resultan(cid:16)t state will be (41(cid:17))
(cid:16)
+k110|11i(cid:17)BC +(...). (33) P1 ǫ|2iA (1−ǫ2)k00R|00i
GR+1 (cid:16)
If all the parties then perform the projection (31) and p+(1 ǫ2)21[k01 01 +k10 10 ]+k11 11 (42)
all get outcome F the resultant state will differ from the − R| i R| i R| i BC
(cid:17)
initial state. Likewise if these unitaries and projections andBobandCharliewillshareastatewithentanglement
repeat until Alice, say, eventually gets outcome G the (measured by the concurrence q (26))
statethensharedbyBobandCharliewilldependonthe
number of rounds performed up to that point. qRB+C1 =P 1 ǫ2(1−ǫ2)×2|k01Rk10R −k00Rk11R| (43)
In general after R rounds of unitaries and projections GR+1
2
in which all parties get F, the shared state will be = ǫ2(1 ǫ2)2R+1
P P ...P −
GR+1 FR F1
k01 k10 k00 k11 (44)
×| 0 0 − 0 0|
|ψRiABC =|0iA k00R|00i+k01R|01i Thus if we consider applying the W protocol to an
arbitrary three-qubit state we have that for the final ex-
(cid:16)
+k10R|10i+k11R|11i BC +|1iA(...)BC (34) pected concurrence qfBC (26):
(cid:17)
D E
5
B. A random distillation for W-class states
∞ R
qBC lim qBC P P (45) Wewillseethatahigherentanglementthanthe above
f ≥ǫ→0 R+1 GR+1 FN maybeobtainedforaW-classstatebyfirstsymmetrising
R=0 N=1
(cid:10) (cid:11) X Y it and then performing random distillation via the W
∞
=2|k010k100 −k000k110|×ǫl→im0 ǫ2(1−ǫ2)2R+1 purnoittaorcyol. Starting with the state (50) Alice applies the
R=0
X
(46)
2
α α
ǫ2(1 ǫ2) 0 0 + 1 2 , 1 1 (57)
=2|k010k100 −k000k110|×ǫl→im01 (1− ǫ2)2 | i−→ γ| i s −(cid:18)γ(cid:19) | i | i−→| i
− −
(47) producing the state
=|k010k100 −k000k110|. (48) βα δα
α100 + 010 +α001 + 000
| i γ | i | i γ | i
The abovebound concerns only Boband Charlie’s en- (cid:18) (cid:19)ABC
tanglement as a result of Alice eventually getting out- 2
α
come G (andthe others F). Howeverother possible out- +s1− γ |2iA(β|10i+γ|01i+δ|00i)BC (58)
comes are where instead Bob or Charlie gets G resulting (cid:18) (cid:19)
in zero Bob-Charlie entanglement, but some entangle- Alice then projects using (31). If she receives outcome
ment between Alice-Bob or Alice-Charlie. How much G (with probability 1 P ) the protocol terminates,
AF
entanglement depends on the form of the original state, otherwise Bob then app−lies the unitary
but since the W protocol is symmetric (i.e. invariant
with respect to permutation of parties), we see that in β β 2
the special case of a symmetric state ψsymm, the ex- 0 0 + 1 2 , 1 1 (59)
ABC | i−→ γ| i s − γ | i | i−→| i
pected entanglement due to such outcomes must also be (cid:18) (cid:19)
|k010k100−k000k110|=|k0210−k000k110|(sincek010 =k100 producing the state
for symmetric ψ ), for each of Alice-Bob and Alice-
ABC
Charlie. 1 αβ δαβ
(100 + 010 + 001 )+ 000
Thus, consideringonly these outcomes where two par- √PAF"(cid:18) γ | i | i | i γ2 | i(cid:19)ABC
ties share some entanglement and are unentangled with
(60)
the third party, it follows that
2
β δα
+ 1 2 α10 +α01 + 00
B
Ernd(ψAsyBmCm)≥3|k0210 −k000k110| (49) s −(cid:18)γ(cid:19) | i (cid:18) | i | i γ | i(cid:19)AC(6#1)
[7] showed that a general W-class state could be ex-
Bob likewise then projects using (31), the protocol ter-
pressed as
minating if he gets outcome G. If he gets outcome F
(conditional probability P ), the state obtained is
(α100 +β 010 +γ 001 +δ 000 ) (50) BF
ABC
| i | i | i | i
1 αβ δ
where α,β,γ,δ R and α,β,γ > 0, δ 0. We will 100 + 010 + 001 + 000
withou{t loss of g}en∈erality take γ β α.≥ √PAFPBF γ (cid:18)| i | i | i γ| i(cid:19)ABC
(62)
≥ ≥
We find for the state (50) that
which is a symmetric state on which the three parties
perform the W protocol.
S(ρA)=H2(λ), where (51) Thus for the overall protocol
λ2 λ+α2(β2+γ2)=0 (52)
− 2
1 α βγ
Using (27), we find the corresponding concurrences − γ
qrnd(ψW) (1 PAF) 2(cid:18) (cid:16) (cid:17) (cid:19)
≥ − × 1 P
AF
−
q(ρ )=2α β2+γ2 (53) 2
A 1 α α2
q(ρB)=2βpα2+γ2 (54) +PAF(1 PBF) 2(cid:18) −(cid:16)β(cid:17) (cid:19)
− × P (1 P )
q(ρ )=2γpα2+β2 (55) AF − BF
C 2
αβ
p γ
It is straightforward to see that q(ρ ) q(ρ ) q(ρ ) +P P 3
C ≥ B ≥ A AF BF × P(cid:16) P(cid:17)
and thus AF BF
α2 α2β2
=2 1 βγ+2α2+ (63)
Es∞p(ψW)=S(ρB). (56) (cid:18) − γ2(cid:19) γ2
6
We use the Lemma superpositionoftheM-qubitstateswithasingleexcited
qubit)
Lemma 2:
1
W = (0...01 + 0...010 +(permutations))
α2 α2β2 | Mi √M | i | i
q (ψ )=2 1 βγ+2α2+
rnd W − γ2 γ2 (69)
(cid:18) (cid:19) to which applying the W protocolproduces a randomly-
>q(ρB)=2β α2+γ2 (64) shared WM−1 state. Considering a bipartite split of the
initial and final states between one of the parties P who
Proof: See Appendix A. p
shares the final state and all other parties, we see that
Hence from (29) 1 1
S(σPf)=H2 >S(σPi)=H2 (70)
M 1 M
E (ψ ) f(q (ψ ))>f(q(ρ ))=E (ψ ). (cid:3) (cid:18) − (cid:19) (cid:18) (cid:19)
rnd W rnd W B sp W
≥ where i and f denote initial and final states. Thus such
(65)
a distillation cannot be reliably performed for predeter-
mined final parties.
C. GHZ-class states We can also consider a more general class of states
whose entanglement properties are of some interest [18,
19, 20] - the M-party symmetric Dicke states [21, 22].
As noted in [1], the above inequality (E (ψ) >
rnd These are of the form
E (ψ)) is not generally true for GHZ class states, with
sp
1
the GHZ state itself , and more generally states of the
ψ(M,N) = N 1s, (M N) 0s (71)
ifnorgmaαco|0u0n0tier+exβam|11p1lei.(OfonrewmhiigchhtEwsopnd=erEwrnhde)thperrorvaidn-- | i MCN X| − i
where MC are tphe binomial coefficients
dom distillation gives no advantage for any state in the N
GHZ class. Here, we answer this question in the nega- M!
MC (72)
tive. More specifically, we find an explicit example of a N ≡ N!(M N)!
GHZ class state for which random distillation gives an −
and the sum is over all permutations of the individual
advantage over distillation to predetermined parties.
qubits. E.g.
Our example state is
1
ψ(4,2) = (0011 + 0110 + 1100
ψG =α(100 + 010 + 001 )+ǫ111 , ǫ= 1 3α2. | i √6 | i | i | i
| i | i | i | i | i −
(66)
p + 1001 + 0101 + 1010 ). (73)
for0< α,β,γ,δ,ǫ R. Thethree-tangleτ [17]for | i | i | i
ABC
this stat{e is equalt}o∈16ǫα3,andbeing non-zerothe state Considering the Von Neumann entropy of a party P we
is thus [7] GHZ-class. have
By symmetry of ψG, we have Esp(ψG)=H2(α2+ǫ2), 1
and S(σPM,N)=H2 MC (74)
(cid:18) N(cid:19)
f−1(Esp(ψG))= 8α2(1 2α2) (67) and hence any LOCC distillation ψ(M,N)
− −→
ψ(M′,N′) cannot be reliably performed for predeter-
p
From its symmetry and the analysis of section IIIA, ψG mined final parties if M′C <MC .
N′ N
can be randomly distilled to obtain
However, we see that if we apply the W protocol to a
q =3α2. (68) state ψ(M,N) we canreliably obtain either a randomly-
rnd shared ψ(M 1,N) (applying the usual protocol) or
−
It follows that q (ψ ) > f−1(E (ψ )) and hence ψ(M 1,N 1) (applying the W protocol but with 0
rnd G sp G − − | i
E (ψ ) > E (ψ ) for α2 > 8/25. I.e. there exist and 1 states reversed). Essentially the parties can reli-
rnd G sp G | i
ably”drop”eithera 0 or 1 fromthetermsofthestate
GHZ class states for which random distillation is advan-
| i | i
to produce a state randomly shared between one fewer
tageous and (as shown in [1]) others for which it is not.
party.
Given that the parties can also (by all applying a
IV. SYMMETRIC DICKE STATES bit-flip operation) always reliably convert ψ(M,N)
−→
ψ(M,M N), we find that the parties can reliably per-
−
form
While we do not have a general treatment of random
distillation applied to pure states shared between > 3 ψ(M,N) ψ(M′,N′) , or ψ(M′,M′ N′) if
| i−→| i | − i
parties, it is clear that there are such states from which (75)
final states shared between fewer parties can be reliably
M′ M
obtainediffthosepartiesarenotpredetermined. In[1]we ≤
N′ (M′ M)+N
gave the example of the M-party W state, (a symmetric
≥ −
7
many of which output states could not be achieved for √1 ǫ2 0 ⊗N + ǫ2 etc.) will not sustain the advan-
− | i | i
predetermined final parties. tage, since for N copies the probability of success will
fallroughlyasO( 1 ),while notpredeterminingthe par-
3N
ties will at most triple the expected entanglement.
V. RANDOM DISTILLATION IN THE Confirming this idea more generally, we find in the
MANY-PARTY LIMIT limit of large N:
Theorem 2:
In our previous paper [1], we show that random dis-
tillation is useful for the case of a single copy of the W
state. One might wonder whether random distillation
remains advantageous in the limit of many copies of a Ernd(ψ⊗N) Esp(ψ⊗N), N . (80)
−→ →∞
general pure state (including W states). Here we show
In other words, as defined in (21) and (22),
that (according to our current definition) the answer is
no.
E∞ (ψ)=E∞(ψ). (81)
In [1] we showed that one could randomly distill one rnd sp
EPR pair from a single W state compared to 0.92 EPRs Proof:
per W between predetermined parties in the many-copy
This is shown by the result of [23], that for a LOCC
limit. Trivially, it follows that for multiple copies of the
protocol distilling EPR pairs from N copies of a two-
W state we canobtainadvantageousrandomdistillation
party pure state σ ,
AB
inthecontextofE >E -thatis,manycopiesoftheW
t sp
state can produce more EPR pairs in total (summing up σ ⊗N Φ N′ (82)
those between all pairs of parties) than can be obtained | iAB −→ | iAB
between predetermined parties. LOCC
However, this does not tell us whether random distil- the probability of getting|{zN}′ > NS(ρ ) tends to 0
A
lation remains useful for many copies of a pure state in
as N . Specifically the probability shrinks as
our redefined sense of E > E - obtaining more en- → ∞
rnd sp exp(O( N)). Note that this is stronger than the well-
tanglement between only two parties when the two are known−result that optimally N′ = NS(ρ ), since it
A
not predetermined. h i
disallowsimprovingonthe optimumexpected yieldeven
Inwhatfollows,wewilldiscussthecaseoftwocopiesof
some of the time.
W statesandnotethatwefindanadvantageforrandom Consider a process (16), where ψ =φ⊗N , for some
distillation for this case. More concretely, we can easily A1...Am
pure state φ. The optimum distillation to specified par-
devise a simple two-copy analogue to the W protocol,in
ties will be to some pair of parties A ,A . where (from
I J
whichthreepartiessharingtwoW stateseachrepeatedly (9)) the distillation rate is S (A Tφ ) where S denotes
perform the two-qubit unitary φ I IJ φ
Von Neumann entropy of the bracketed parties’ reduced
state of φ, T in general represents some group of par-
|00i−→ 1−ǫ2|00i+ǫ|2i (76) ties not contIaJining AI or AJ and Tiφj is the group that
p minimisesS (A T ),i.e. foranyfixedbutarbitrarypair
(with all other states (01 , 10 , 11 ) mapping to them- φ i ij
| i | i | i of parties A , A .
selves)combinedwith a projectioninto either a 2 state i j
| i
or the SU(2) SU(2) subspace. As with the general
three-qubit sta⊗te, in this case repeated rounds change Sφ(AiTiφj)≤Sφ(AiTij) ∀ Tij. (83)
the overall state. We find, by considering a four-qubit
Thus, as N
measure analogous to q, that →∞
Ernd(W⊗2) 2[ζlog2ζ (0.5 ζ)log2(0.5 ζ)] Esp(ψ)−→NSφ(AITIφJ) (84)
≥− − − −
1.843, where (77)
and
≈
ζ = 1− 1− 89 2 (78) Sφ(AITIφJ)≥Sφ(AiTiφj) ∀ ij (85)
q4
(cid:0) (cid:1)
For E (ψ) > E (ψ), by the definition in (20) we
rnd sp
Hence
require at least one possible output state ψ to have
f
Ernd(W⊗2)>Es∞p(W⊗2)=2H2 31 ≈1.837. (79) Ebyspψ(ψf′f,)a>ndEsopc(cψu)r.riLnegtwusitchonssoimdeerfionxeedsupcrhoψbafb,idlietnyotpe′fd.
(cid:18) (cid:19) Suppose optimal distillation of ψ′ (to specified parties)
f
Hence there is an advantage to random distillation of istopartiesAX andAY withthecorrespondingbipartite
W⊗2, but the proven advantage is very marginal. We cut being between A Tf on one side (using, here and
X XY
see that this extending this protocol in a na¨ıve man- below,f todenotequantitiesdefinedforreducedstatesof
ner to more copies (i.e performing a unitary 0 ⊗N ψ′,analogouslytoφabove)anditscomplementaryseton
| i → f
8
theotherside. SimilartoEqs. (83)and(85),wehavefor advantageous random distillation does not occur in the
each fixed but arbitrary pair i,j, S (A Tf) S (A T ) many-copy limit, and hence is a property specific to in-
forallT andS (A Tf ) S (AfTf)ifoijra≤lli,jf. Tiheinj, dividual quantum states that cannot be considered in a
ij f X XY ≥ f i ij regularisedform,incontrasttomanyotherentanglement
in the many-copy limit
properties.
E (ψ′)=S (A Tf )>E (ψ)=NS (A Tφ ) (86) Clearly we have still only dealt with a limited class of
sp f f X XY sp φ I IJ
statesandtheextremalconditionsofasinglecopyorthe
However, from (83), (85) and (86) we have that many-copy limit. Our quantitative approach does not
readily generalise to all states - e.g. for random distilla-
S (A Tφ ) S (A Tf ) tiontofinalstatessharedbetweenmorethantwoparties,
f X XY ≥ f X XY the lack of a standard entanglement measure makes the
>NS (A Tφ ) NS (A Tφ ) (87)
φ I IJ ≥ φ X XY choice of target state more arbitrary, and an “advanta-
geous” random distillation is less defined by a measure
Consider now a bipartite division of ψ between the
thanby the probabilityofachievinga giventargetstate.
group A Tφ acting as a single party (i.e. we allow
X XY However,as demonstratedwith Dicke states above,two-
joint quantum operations within this group) denoted by
party entanglement measures can be used to determine
A and the group of all other parties acting as a single
whetherornotsuchstatesareachievablebetweenprede-
party B. A and B perform the above LOCC protocol
termined parties.
independently on M copies of ψ. Then with probability
(p )M,they obtainM copiesofψ′. Inthe limit oflarge For distillationto two-partyentanglementfrommulti-
f′ f ple copiesofa state,anopenquestionishow anyadvan-
M,thepartiesAandB can,throughLOCC,distillthese
tage due to random distillation scales with the number
copies to MS (A)>MNS (A) EPR pairs.
f φ
of copies, since we now know such advantage vanishes in
ThusAandBwouldbedistillingmorethanMNS (A)
φ
the many-copy limit.
EPR pairs from MN copies of φ, and from [23] their
success probability must be exp(O( MN)), hence p′ As noted in [1], even when the target states are two-
− f ∼
exp(O( N)). But for E (ψ) > E (ψ) under these partyandthusthefinalentanglementisreasonablywell-
rnd sp
−
circumstances would require S (A Tf ) exp(O(N)), defined,thefull“structure”oftheoutputofrandomdis-
f X XY ∼ tillation would be defined by a probability distribution
which would require a forbidden increase in Schmidt
number across a bipartite split between group A Tf overfinalentanglementsforgivenpairsofparties,rather
X XY than a single number. For example, the W protocol for
and all other parties.
a W state shared between parties A,B,C reliably pro-
Hence in the limit of large N, we cannot have advan-
ducesanEPRpairbetweenoneofthethreepairsofpar-
tageous random distillation of N copies of a pure state.
(cid:3). tiesAB,BC,AC, witheachpairshavingaprobabilityof
1/3 of receiving the EPR. As shown in [1], EPRs can be
reliablyproducedfromsomeW-likestateswhicharenot
symmetric, but in this case the probability of getting an
VI. CONCLUSION
EPR is not the same for each pair. An interesting open
question is what the optimum such probability distribu-
Wehavegeneralisedseveraloftheresultsnotedforspe-
tion (in terms of E ) is for a given state, and how this
cific cases in [1]. We have more carefully defined what rnd
can be determined from the form of the state.
constitutes “random” distillation so that any apparent
advantage in terms of entanglement yield is specifically The authors acknowledge financial support from
due to the final parties not being predetermined. The NSERC, CIFAR, the CRC program, CFI, OIT, PREA,
advantageous random distillation we previously noted MITACS, CIPI and QuantumWorks.
for the W and similar states has been shown to apply
to the general W-class of three-qubit states (and the
GHZ class not to have a consistent property in this re-
spect). We have shown that for the important class of
VII. APPENDIX A
symmetric Dicke states our W protocolcan achieve con-
versionsbetweenstates whicharenotachievable forpre-
determined final parties. Finally we have shown that ProofofLemma 2 canbe done algebraicallyasfollows
9
α2 α4 α4β4 α2 α4β2
q2 q(ρ )2 =4 1 2 + β2γ2+4α4+ +8α2βγ 1 +4
rnd− B − γ2 γ4 γ4 − γ2 γ2
(cid:18) (cid:19) (cid:18) (cid:19)
α2β3 α2
+4 1 4β2(α2+γ2) (88)
γ − γ2 −
(cid:18) (cid:19)
α2β2 α2β4 α2 β3 α2
=α2 12β2+8 +4α2+ +8βγ 1 +4 1 (89)
− γ2 γ4 − γ2 γ − γ2
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
γ β β2 β4 β β3
=α2 β2 8 +4 12 +α2 8 +4+ 8 4 (90)
β γ − γ2 γ4 − γ − γ3
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
γ β β2 2β 2 β 2
=α2 4β2 1 2 +α2 +4 1 (91)
" (cid:18)β − (cid:19)(cid:18) − γ(cid:19) (cid:18)γ2 − γ (cid:19) (cid:18) − γ(cid:19) !#
There are thus 3 terms in the above. We recall that β =γ, but in that case the second term is >0. Thus
0<α β γ. Thefirsttermisclearly 0sinceγ β,
and th≤e ot≤her two terms are clearly 0≥since they≥are qrnd >q(ρB) (cid:3). (92)
≥
squared. Thefirstandthirdtermsarebothequalto0iff
[1] B. Fortescue, H.-K. Lo, Phys. Rev. Lett., 98, 260501 [11] T.Laustsen,F.VerstraeteandS.J.VanEnk,Quant.Inf.
(2007) Comp., 3 No.1, 64 (2003)
[2] I. Devetak, A.W. Harrow, A. Winter, [12] J. A. Smolin, F. Verstraete and A. Winter, Phys. Rev.
arXiv:quant-ph/0512015v1 A, 72, 052317 (2005)
[3] A. Abeyesinghe, I. Devetak, P. Hayden, A. Winter [13] M.Horodecki,J.OppenheimandA.Winter,Nature436,
arXiv:quant-ph/0606225v1 673 (2005)
[4] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, A. [14] G. Gour, Phys.Rev. A, 72, 042318 (2005)
Peres, and W. K. Wootters, Phys. Rev. Lett., 70 1895 [15] G. Gour, R. W. Spekkens, Phys. Rev. A, 73, 062331
(1993) (2006)
[5] C. H. Bennett and S.J. Wiesner, Phys. Rev. Lett., 69 [16] W. K. Wooters, Quant.Inf. Comp. 1, 27 (2001)
2881, 1992 [17] V. Coffman, J. Kundu and W. K. Wootters, Phys. Rev.
[6] C.H. Bennett, H.J. Bernstein, S. Popescu, and B. Schu- A 61 052306 (2000)
macher, Phys.Rev.A,53, 2046 (1996) [18] N. Kiesel, C. Schmid, G. T´oth, E. Solano, and H. Wein-
[7] W. Du¨r, G. Vidal, J. I. Cirac, Phys. Rev. A 62, 062314 furter, Phys.Rev.Lett. 98, 063604 (2007)
(2000) [19] G. T´oth, J. Opt. Soc. Am.B 24, 275 (2007)
[8] F.Verstraete,J.Dehaene,B.DeMoor,andH.Verschelde [20] J. K. Stockton, J. M. Geremia, A. C. Doherty, H.
Phys.Rev.A 65, 052112 (2002) Mabuchi, Phys. Rev.A 67, 022112 (2003)
[9] C. H. Bennett, S. Popescu, D. Rohrlich, J. A. Smolin, [21] R. H.Dicke, Phys.Rev. 03, 99 (1954)
and A. V.Thapliyal, Phys.Rev. A 63, 012307 (2000) [22] L.MandelandE.Wolf,OpticalCoherence andQuantum
[10] D.P.DiVincenzo,C.A.Fuchs,H.Mabuchi,J.A.Smolin, Optics, CUP, Cambridge, UK (1997)
A. Thapliyal, A. Uhlmann, Proc. Quantum Computing [23] H.-K. Lo and S. Popescu, Phys. Rev. A, 63, 022301
and Quantum Communications: 1st NASA Intl. Conf., (2001)
Palm Springs (1998), Springer LNCS 1509, pp. 247-257,
Heidelberg, (1999)