Family of Concurrence Monotones and its Applications

0.21 MB·English

by  Gilad Gour

A Family of Concurrence Monotones and its Applications
Gilad Gour1,2,∗
1Theoretical Physics Institute, University of Alberta, Edmonton, Alberta T6G 2J1, Canada
2Department of Mathematics, University of California/San Diego, La Jolla, California 92093-0112
(Dated: February 1, 2008)
We extend the definition of concurrence into a family of entanglement monotones, which we call
concurrence monotones. We discuss their properties and advantages as computational manageable
measuresofentanglement,andshowthatforpurebipartitestatesallmeasuresofentanglementcan
bewrittenasfunctionsoftheconcurrencemonotones. Wethenshowthattheconcurrencemonotones
provide bounds on quantum information tasks. As an example, we discuss their applications to
remote entanglement distributions (RED) such as entanglement swapping and remote preparation
of bipartite entangled states (RPBES). We prove a powerful theorem which states what kind of
5 (possibly mixed) bipartite states or distributions of bipartite states can not be remotely prepared.
0 The theorem establishes an upper bound on the amount of G-concurrence (one member in the
0 concurrence family) that can be created between two single-qudit nodes of quantum networks by
2 means of tripartite RED. For pure bipartite states the bound on the G-concurrence can always be

n saturated byRPBES.
a
J PACSnumbers: 03.67.-a,03.67.Hk,03.65.Ud

8
1
I. INTRODUCTION states as

2
v d−1
8 Entanglement is one of the main ingredients of non- Ek(|ψi)= λi , (1)
4 intuitive quantum phenomena. Besides of being of in- Xi=k
1
terest from a fundamental point of view, entanglement
0 where λ λ λ are the Schmidt numbers
0 1 d−1
has been identified as a non-local resource for quantum ≥ ≥ ··· ≥
1 ofthe d d-dimensionalbipartite state ψ , andthenex-
4 information processing [1]. In particular, shared bipar- tendedt×omixedstatesbymeansofthec|onivexroofexten-
0 tite entanglement is a crucial resource for many quan-
sion. Forapurestate ψ thesemeasuresofentanglement
/ tuminformationtaskssuchasteleportation[2],quantum | i
h quantify completely the non-local resource since all the
cryptography[3],entanglementswapping[4],andremote
p Schmidt coefficients of ψ are determined by them. The
- state preparation(RSP) [5, 6, 7, 8] that are employedin entanglement monoton|esidefined in Eq. (1) play a cen-
t
n quantum information protocols. tralroleintransformationsofpurestatesbylocalopera-
a
tions and classical communications (LOCC) [13, 14, 15].
u One of the remarkable discoveries on bipartite entan-
Moreover, each member of the family may quantify the
q glement, is that for pure states, there is a unique and
: single measure of entanglement, called entropy of entan- possibilitytoperformaparticulartaskinquantuminfor-
v
mation processing (for example, E = 1 λ quantifies
i glement[9],thatquantifies,asymptotically, thenon-local 2 − 0
X thepossibilitytoperformfaithfulteleportationwithpar-
resources of a large number of copies of a pure bipartite
tially entangled states [16]).
r state. However,the generalizations of the entropy of en-
a Nevertheless, the family of entanglement monotones
tanglement to mixed states yields, even asymptotically,
E (ρ)isnotenoughtoquantifycompletelytheentangle-
more than one measure of entanglement, such as entan- k
mentofabipartitemixedstateρ. Furthermore,itwillbe
glement of formation and distillation [10]. Despite the
argued here, that if ρ is a d d-dimensional mixed state
enormous efforts that have been made in the last years, ×
with d>4, in general, it is impossible to find analytical
mixedentanglementlacksa completequantification[11].
expression (i.e. an explicit formula like in [17, 18]) for
For a finite number of shared pure states, the en- Ek(ρ) (as well as for the entanglement of formation and
tropy of entanglement is not sufficient, and more mea- other measures of entanglement). Thus, we are moti-
suresofentanglementarerequiredtoquantifycompletely vatedtolookforothersetsofmonotoneswhicharemore
the non-local resources. These are called entanglement computationally manageable.
monotones [12] since they behave monotonically under Suchacomputationallymanageablemeasureofentan-
local transformations of the system. The family of en- glement is the concurrence. The concurrence as a mea-
tanglement monotones E (k = 0,1,2,...,d 1) which sureofentanglementwasfirstintroducedin[17,18]foran
k
−
introduced in [13] were first defined over the set of pure entangledpairofqubitsandlaterongeneralizedtohigher
dimensions [19, 20] (there are other generalizations of
concurrencewhichwewillnotdiscusshere[21]). Already
in [17, 18] the importance of the concurrence monotone
∗Electronicaddress: [email protected] was recognized and the entanglement of formation of a
2
mixed entangled pair of qubits was calculated explicitly In the following definition of the family of concurrence
in terms of the concurrence. In higher dimensions there monotones, the concurrence defined in Eqs. (2,3) is de-
is not yet an explicit formula for the generalized con- noted by C since it is the second member of the family.
2
currence [19], but lower bounds have been found [20].
Recently, it has been shown [22] that the concurrence Definition 1 (a) Consider a d d-dimensional bi-
×
plays also a major role in remote entanglement distri- partite pure state ψ with Schmidt numbers λ
| i ≡
butions(RED)protocolssuchasentanglementswapping (λ0,λ1,...,λd−1). Thedconcurrencemonotones,Ck(ψ )
| i
(ES)andremotepreparationofbipartiteentangledstates (k = 1,2,...,d), of the state ψ are defined as fol-
| i
(RPBES). lows [29]:
In this paper we introduce a family of entanglement
1/k
monotones which we call concurrence monotones. We Sk(λ0,λ1...,λd−1)
C (ψ ) , (4)
k
discuss its properties and show that for pure states all | i ≡ S (1/d,1/d,...,1/d)
(cid:18) k (cid:19)
measures of entanglement can be written as functions of
the concurrence monotones. We show that these con- where Sk(λ) is the kth elementary symmetric function of
currence monotones can serve as a powerful tool to rule λ0,λ1,...,λd−1. That is,
out the possibility of certain tasks in quantum informa-
tion processing. In particular, we find an upper bound S1(λ)= λi , S2(λ)= λiλj ,
on the entanglement that can be produced by tripartite i i<j
X X
RED protocols and show that the protocol given in [22] d−1
for RPBES saturates the bound. The measure of entan- S (λ)= λ λ λ , ...,S (λ)= λ . (5)
3 i j k d i
glement is taken to be one of the members in the con- i<j<k i=0
X Y
currence family, which we give the name G-concurrence,
since for pure states the G-concurrence is the Geometric (b) Consider a d d-dimensional bipartite mixed state
×
mean of the Schmidt numbers. In addition, we provide ρ. The d concurrence monotones, Ck(ρ), of the state ρ
an operational interpretation of the G-concurrence as a are then defined as the average Ck of the pure states of
type of entanglement capacity. the decomposition, minimized over all decompositions of
ρ (the convex roof):
Thispaperisorganizedasfollows. InsectionII wede-
fine the family of concurrence monotones and then dis-
cuss its importance and advantages. In section III we
C (ρ)=min p C (ψ ) ρ= p ψ ψ .
k i k i i i i
discuss its applications to RED protocols and in section | i | ih |!
i i
IV we summarize our results and conclusions. X X (6)
The functions S (λ) and [S (λ)]1/k are Schur-concave
k k
(see p.78,79 in [25]). Moreover,
II. DEFINITION OF CONCURRENCE
MONOTONES
1 d
S (λ) S (1/d,1/d,...,1/d)= , (7)
k ≤ k dk k
In the following, we will use the definition of con- (cid:18) (cid:19)
currence as given in [17, 18] for the 2 2 dimensional sincethevector(1/d,1/d,...,1/d)ismajorizedbyallvec-
×
case,anditsgeneralizationtohigherdimensionsasgiven tors λ = (λ ,...,λ ) with non-negative components
0 d−1
in[19](seealso[20]). Theconcurrenceofapurebipartite that sum to 1. Thus, 0 C (ψ ) 1 and C (ψ ) = 1
k k
normalized state ψ is defined as only when all the Schmi≤dt num| bier≤s of ψ equa|l tio 1/d
| i | i
(i.e. ψ is a maximally entangled state).
| i
d Eq. (4) together with the convex roof extension of C
C(ψ ) (1 Trρˆ2), (2) k
| i ≡ d 1 − r to mixed states (see Eq. (6)) defines an entanglement
r − monotone for each k. To see that, first note that
wherethe reduceddensity matrixρˆ isobtainedbytrac-
r
ingoveronesubsystem. Inthedefinitionaboveweadded Ck(|ψi)=fk(TrB|ψihψ|) , (8)
thefactor d/(d 1)sothat0 C(ψ ) 1. Ford=2
− ≤ | i ≤ where the trace is taken over one subsystem (say Bob’s
Eq.(2)alsocoincideswiththedefinitiongivenin[17,18]
p system)andf (σ) [S (λ(σ))/S (1/d,...,1/d)]1/k(λ(σ)
by means of the “spin flip” transformation. The concur- k k k
≡
is the vector of eigenvalues of the density matrix σ).
rence of a mixed state, ρˆ, is then defined as the average
According to Theorem 2 in [12] C is an entanglement
concurrenceofthepurestatesofthedecomposition,min- k
monotone if f (σ) is a unitarily invariant, concave func-
imized over all decompositions of ρˆ(the convex roof): k
tion of σ. The concavity of f (σ) follows from two facts.
k
First (see p.79 in [25]), for any two vectors x and y with
C(ρˆ)= min piC(|ψii) ρˆ= pi|ψiihψi|! . xi,yi ≥0 (i=0,1,...,d−1)
i i
X X (3) [S (x+y)]1/k [S (x)]1/k+[S (y)]1/k . (9)
k k k
≥
3
Second,fortwoHermitianmatricesAandB,λ(A+B) Such an explicit formula (in terms of a ) is not avail-
ij
≺
λ(A)+λ(B) (see p.245 in [25]). Thus, giventwo density able for most of the measures of entanglement discussed
matrices σ and σ we have (0 t 1) in literature (including the entropy of entanglement, α-
1 2
≤ ≤
entropyorRenyientropy,andthefamilyofentanglement
1/k
monotones given in [13]).
S λ tσ +(1 t)σ
k 1 2
−
f [tσ +(1 t)σ ]= As an example, consider the entropy of entanglement
k 1 − 2  hS(cid:16)(1/d,...,1/d) (cid:17)i
k E(ψ ) = Trρ logρ , where ρ Tr ψ ψ is the re-
| i − r r r ≡ B| ih |
Sk λ tσ1 +λ (1 t)σ2 1/k  dabuocevde,dthenensitiynomradterrixto. cIafl|cψuilaitsegtihveenenitnrotpeyrmofseonftaanijglaes-
−
≥" (cid:2) (cid:0)Sk(1(cid:1)/d,..(cid:0).,1/d) (cid:1)(cid:3)# ment, one must be able to write ρr in its diagonal form.
However, for d > 4, in general, it is impossible to solve
Sk[λ(tσ1)] 1/k+ Sk λ (1−t)σ2 1/k the equation fλ(x) = 0 analytically (fλ(x) is defined in
Eq. (11)).
≥(cid:20)Sk(1/d,...,1/d)(cid:21) " Sk(cid:2)(1(cid:0)/d,...,1/d(cid:1))(cid:3)#
For d 4 the entropy of entanglement can be ex-
=tf (σ )+(1 t)f (σ ). (10) ≤
k 1 − k 2 pressed in terms of the concurrence monotones. For
d = 2, the solution to the quadratic equation f (x) = 0
Thus, Eqs. (4,6) define entanglement monotones. λ
is simple and the entropy of entanglement is given by
Advantages of concurrence monotones 1+ 1 [C (ψ )]2
2
E(ψ )=h − | i , (14)
| i p 2 !
Thereareseveraladvantagesandapplicationsforthese
particularmeasuresofentanglement. First,thefamilyof
where h(x)= xlogx (1 x)log(1 x). This formula
concurrence monotones as defined in Eq. (4,6) is com-
− − − −
holds for mixed states where the concurrence for mixed
plete in the sense that all the Schmidt coefficients of a
states is defined in Eq. (6) and the LHS is replaced by
givenpure statecanbe determinedby thedconcurrence
the entanglement of formation [18].
monotones. To see that, let us define the characteristic
polynomialfλ(x)=(x λ0)(x λ2) (x λd−1)whose For d=3, the solutions to the cubic equationfλ(x)=
− − ··· −
singular values are the Schmidt numbers. It is easy to 0 are more complicated (although possible) and the en-
see that f (x) can be written as tropy of entanglement is given by
λ
d ( 1)k d 1 2
fλ(x)= −dk k xd−k[Ck(λ)]k , (11) E(|ψi)=H 3 + 3 1−[C2(|ψi)]2cos(θ/3),
kX=0 (cid:18) (cid:19) 1(cid:16) 2 p
+ 1 [C (ψ )]2cos (θ+2π)/3 ;
2
where C (λ) 1 and C (λ) λ = 1. Hence, 3 3 − | i
the singukl=a0r valu≡es of f (xk)=(1i.e. t≡he Scihmiidt numbers) 1 3[Cp(ψ )]2+ 1[C (ψ )](cid:0)3 (cid:1)(cid:17)
λ P cosθ − 2 2 | i 2 3 | i , (15)
aredeterminedcompletelybytheconcurrencemonotones ≡ (1 [C (ψ )]2)3/2
Ck. − 2 | i
Furthermore, consider a pure d d-dimensional state
× where H(x,y) = xlogx ylogy (1 x y)log(1
− − − − − −
ψ = a i j , (12) x y). Similarly, for k = 4, it is possible to find the
| i ij| iA| iB so−lutions to the quartic equation f (x) = 0 and express
ij λ
X the entropy of entanglement in terms of the concurrence
where i and j aresomed-dimensionalbasesinAlice monotones.
A B
| i | i
and Bob systems, respectively. The Schmidt numbers The analytical expression for C (ψ ) in terms of the
k
arethenon-zeroeigenvaluesofthematrixA†A(orAA†), reduced density matrix ρ Tr ψ |ψiis given by:
wherethematrixelementsofAarea . Thus,ingeneral, r ≡ B| ih |
ij
ford>4,accordingtoAbel’simpossibilitytheorem(also
C (ψ )=
Galois) there is no analytical expression for the Schmidt k | i
numbers in terms of a . The advantage of our family 1/k
ofconcurrencemonotonijesis thatone canalwaysexpress d 1 ( 1)k− km=1Nm k 1 Trρmr Nm ,
analytically Ck(|ψi) in terms of aij:  kd {XNm} − P mY=1Nm!(cid:18) m (cid:19) 
1/k (cid:0) (cid:1) (16)
TrB(k)
C (ψ )=d , (13)
k | i " d # wherethe sumistakenoverallthe non-negativeintegers
k
N ,N ,...,N thatsatisfytheconstraintN +2N +...+
1 2 k 1 2
where B(k) is the kth compoun(cid:0)d(cid:1)of the matrix A†A (see kN = k. This expression (see also [26, 27]) follows di-
k
p.502 in [25] for the definition of compound matrices). rectly from multinomial formulas given in [28]. As an
4
example, for k =2,3,4 Eq. (16) gives First, for pure bipartite states we have the inequalities
(cf p.224 in [25])
d
C2(|ψi)= d 1(1−Trρ2r) [C2(ψ )]2 [C3(ψ )]3 [Cd(ψ )]d [Gd(ψ )]d .
r − | i ≥ | i ≥···≥ | i ≡ | i(23)
d2 1/3
C (ψ )= 1 3Trρ2+2Trρ3 Second, given a mixed bipartite state ρ we have [33]
3 | i (d 1)(d 2) − r r
(cid:20) − − (cid:21)
d3 (cid:0) (cid:1) Gd(ρ) Ck(ρ) k =1,2,...,d. (24)
C (ψ )= ≤ ∀
4
| i (d 1)(d 2)(d 3)×
h − − − Note that the relations in Eqs. (23,24) may be useful in
1/4
1 6Trρ2+8Trρ3 6Trρ4+3(Trρ2)2 . finding lower bounds on measures of entanglement such
− r r− r r as entanglement of formation. In addition, as we will
(cid:0) (cid:1)i (17) seeinthefollowingsection,theG-concurrencemonotone
plays a central role in tripartite RED protocols.
We can see that for k = 2 Eq. (16) is reduced to the
expression for the concurrence given in [19]. Note also
thatC (ψ )=0ifk isgreaterthentheSchmidtnumber
k
| i III. REMOTE ENTANGLEMENT
of ψ .
| i DISTRIBUTION
The G-concurrence monotone Asmentionedinthe introduction,sharedbipartiteen-
tanglement is a crucial shared resource for many quan-
tuminformationtaskssuchasteleportation[2],entangle-
The last member of the family C is of a particu-
k=d
ment swapping [4], and remote state preparation (RSP)
lar importance and we denote it by G since it is the
d
[5, 6, 7, 8] that are employed in quantum information
geometric mean of the Schmidt numbers
protocols.
G (ψ ) C (ψ )=d(λ λ λ )1/d. (18) Remote preparation of bipartite entangled states [22]
d k=d 0 1 d−1
| i ≡ | i ···
(RPBES) is another important quantum information
Note that for d = 2 the G-concurrence coincides with task in which a quantum network (QNet) have a single
the original definition of concurrence given by Hill and supplier (named “Sapna”) who shares entangled states
Wootters [17]. with nodes via quantum channels, then performs LOCC
The G-concurrence has several interesting features: to produce pairwise entangled states between any two
A computational manageable measure of entanglement: nodes, say, Alice and Bob. A crucial feature of RPBES
•
for the d d bipartite pure state ψ in Eq. (12), the is that Alice and Bob end up sharing a unique bipartite
× | i
G-concurrence is given simply by [30] (cf Eq. (13)) entangled state. A more general scheme, in which Alice
andBobendupsharingadistributionofentangledstates
G (ψ )=d Det A†A 1/d , (19) is called remote entanglement distribution [22](RED).
d
| i
The scheme for tripartite RED, introduced in [22],
where the matrix elements(cid:2)of A(cid:0)are (cid:1)a(cid:3)ij. commences with a four-way shared state, ρˆ1234 = ρˆ12
•Multiplicativity: first,givenad1×d1 (d2×d2)bipartite ρˆ34 withρˆ12 andρˆ34 bipartiteentangledstates,andwit⊗h
entangled state, ψ1 (ψ2 ), we have Sapna (the supplier) holding shares 2 and 3, and Alice
| i | i
andBobholdingshares1and4,respectively. Eachshare
G (ψ ψ )=G (ψ )G (ψ ). (20)
d1d2 | 1i⊗| 2i d1 | 1i d2 | 2i has a corresponding d-dimensional Hilbert space. The
three parties Alice, Bob and Sapna perform LOCC to
Note that although in both sides of the equation above
create a set of outcomes
we take the geometric means of the Schmidt numbers
of the relevant states, G = C is not the same
d1d2 k=d1d2 σˆj =Tr σˆj ,Q ;j =1,...,s (25)
measure of entanglement as Gd1 = Ck=d1 [31]. Second, O ≡{ 14 23 1234 j }
givenabipartitestate ψ ,acomplexnumber
candoperators(compl|exim∈aHtrAic⊗esH)ABˆ∈HA andBˆ ∈HB mwiitxhedQsjtattheeσˆpjrowbahbicihlitiys othbatatinAeldicebyanredduBcoinbgsthhaerefotuhre-
we have [32] 14
way shared state σˆj over Sapna’s shares. In general
1234
G (cψ )= c2G (ψ ) (21) RED,the states σˆj maybe inequivalentunder LOCC
Gd Aˆ dBˆ|ψi =|D|etdAˆ| i2/d Det Bˆ 2/dGd(ψ ), wBohbermeausstinbReBeqEu{SivPa14lte}hnet ustnadteers LσˆO1j4CsCh,arseodAbliyceAalincdeBanodb
⊗ | i | i
(cid:16) (cid:17) (cid:12) (cid:16) (cid:17)(cid:12) (cid:12) (cid:16) (cid:17)(cid:12) (22) can always transform σˆ1j4 into a unique entangled state
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (i.e. independent on j) via LOCC.
where we have used Eq. (19). In this section, we address the issue of which distribu-
A lower bound: the G-concurrence monotone provides tions of states, , can or cannot be created via LOCC
• O
a lower bound for all the other concurrence monotones. by Alice, Bob and Sapna. The G-concurrence monotone
5
plays a major role in the following theorem that estab- with N(j,ll′) r(j,ll′) and
≡ m,m′ mm′
lishes which distributions of states cannot be produced
by RED. r(j,lPl′) λ(l)η(l′) M(j,ll′) 2 . (32)
mm′ ≡ k k′ | kk′,mm′|
Theorem 1 If Alice, Bob and Sapna perform LOCC on Xk,k′
the initial 4-qudit stateρˆ ρˆ with O (in Eq. (25)) the
12 34 ThedensitymatrixsharedbetweenAlice,BobandSapna
⊗
resultant distribution of states shared between Alice and
after outcome j occurs is
Bob, then
G14 ≡ s QjGd(σˆ1j4) ≤ G12G34, (26) σˆ1j234 = Q1j Xl,l′ plql′N(j,ll′)|Φ(j,ll′)i1234hΦ(j,ll′)|, (33)
j=1
X where
with G G (ρˆ ) and G G (ρˆ ).
12 d 12 34 d 34
≡ ≡ Φ(j,ll′) = 1 λ(l)η(l′)
(In the next subsection we will show that the equality in | i1234 √N(j,ll′) k k′
the above equation can alwaysbe achievedby RBESP if Xk,k′mX,m′q
ρˆ12Parnodofρˆ:34Leatreupsuwreri.t)e ρˆ and ρˆ in their optimal de- ×Mk(jk,′l,lm′)m′|k(l)k′(l′)i14|m(l)m′(l′)i23 (34)
12 34
compositions Tracing over Sapna’s subsystems yields
ρˆ =d2−1p ψ(l) ψ(l) , ρˆ =d2−1q χ(l) χ(l) ; σˆ1j4 = Q1 plql′rm(j,mll′′)|φ(mj,mll′′)i14hφm(j,mll′′)|, (35)
12 l| i12h | 34 l| i34h | j l,l′ m,m′
X X
l=0 l=0
X X
(27) where
we can always choose optimal decompositions with no
m|χo(lr)ei34thaerne gdi2veenleimnetnhtesir[S3c4h].midTthdeecsotamtepsos|iψti(ol)ni:12 and |φm(j,mll′′)i14 ≡ rm(1j,mll′′) Xk,k′qλ(kl)ηk(l′′)Mk(jk,′l,lm′)m′|k(l)k′(l′)i14 .
d−1 q (36)
ψ(l) = λ(l) k(l)k(l) FromthedefinitionoftheG-concurrenceformixedstates
| i12 k | i12 (i.e. the convex roof extension), it follows that G (σˆj )
Xk=0q d 14
d−1 cannotexceedthe averageofthe G-concurrenceoverthe
χ(l) = η(l) k(l)k(l) , (28) decomposition in Eq. (35). Thus,
| i34 k | i34
wχi(tlh) λk(,l)raenspdecηtk(il)vetlhye. STXkch=he0mqiinddtecxoelffiincietnhtessotfat|ψes(l)i1k2(la)nd Gd(cid:16)σˆ1j4(cid:17)≤ Q1j Xl,l′ mX,m′plql′rm(j,mll′′)G(cid:16)|φ(mj,mll′′)i14(cid:17). (37)
34 i
| i {| i }
representsd2 differentbasesforeachsystemi=1,2,3,4. Using Eq. (19) we find
Note that in this notation
d2−1 1 (j,ll′) d dk−=10λ(kl)ηk(l′) 1/d Det M(mj,mll′′) 2/d
G12 =d Xl=0 pl(cid:16)λ(0l)λ(1l)···λ(dl−)1(cid:17)d G(cid:16)|φmm′ i14(cid:17)= (cid:16)Q (cid:17)rm(j,mll(cid:12)(cid:12)(cid:12)′′) (cid:16) ((cid:17)3(cid:12)(cid:12)(cid:12)8) ,
G34 =dd2−1ql η0(l)η1(l)···ηd(l−)1 d1 . (29) wMh(ejr,lel′)th.e dT2huesl,emsuebnsttsituotfinegacthhismraetsruixlt Minm(Ej,mlql′′.) (3a7re)
Xl=0 (cid:16) (cid:17) kk′,mm′
yields
SincetheentanglementbetweenAliceandBobremains
zero unless Sapna perform a measurement, we assume s d−1 1/d
tdheasctrtibheedfibrsyttmheeaKsurareumseonpterisatpoerrsfoMˆrm(je)danbdy Sthaepinracoamndpois- G14 ≡Xj=1QjG(cid:16)σˆ1j4(cid:17)≤dXl,l′ plql′ kY=0λ(kl)ηk(l′)!
nents
M(j,ll′) m(l)m′(l′) Mˆ(j) k(l)k′(l′) , (30) × Det M(mj,mll′′) 2/d . (39)
mm′,kk′ ≡23h | | i23 Xj mX,m′(cid:12) (cid:16) (cid:17)(cid:12)
(cid:12) (cid:12)
with k,k′,m,m′ =0,1 and l,l′ =1,2,3,4. Now, from the(cid:12) geometric-ari(cid:12)thmetic inequality we have
The probability to obtain an outcome j is thus
Qj ≡Tr(Mˆ(j)ρˆ12⊗ρˆ23Mˆ(j)†) =d2−1d2−1plql′N(j,ll′), mX,m′(cid:12)(cid:12)Det(cid:16)Mm(j,mll′′)(cid:17)(cid:12)(cid:12)2/d ≤ d1mX,m′Tr(cid:16)Mm(j,mll′′)†Mm(j,mll′′)(cid:17)
Xl=0 lX′=0 (cid:12) (cid:12) = 1Tr(Mˆ(j)†Mˆ(j)). (40)
(31)
d
6
Hence, from Eq. (39) and Eq. (29) we get (k =1,2,...,N) is shared between party k 1 and party
−
k. IftheN+1partiesperformLOCContheinitialstate
1
G14 ≤ d2G12G34Xj Tr(Mˆ(j)†Mˆ(j)). (41) ρof0,1st⊗atρe1s,2b⊗etw··e·e⊗nρpNar−t1y,N0wanitdhNthedreensoutletdanbtyd{isPtrji,buσˆt0jiNon}
(P is the probability to have the state σˆj ), then
Thus, from the completeness relation, Mˆ(j)†Mˆ(j) = j 0N
j
I,CwoenosbidtearinnEowq.t(h2e6)f.ollowing LOCC: aPfter Sapna’s first G0N ≡ PjGd(σˆ0jN) ≤ G01G12···GN−1N , (45)
j
measurement, she sends the result j to Alice and Bob. X
Basedonthis result,Alicethenperformsameasurement with G G (ρ ) (k =1,2,...,N).
k−1k d k−1,k
representedbytheKrausoperatorsAˆ(k)andsendsthere- Theorem1≡andits corollarysuggestanoperationalin-
j
sult k to Bob and Sapna. Based on the results j,k from terpretation of the G-concurrence as a form of entangle-
Sapna and Alice, Bob performs a measurement repre- ment capacity. In the following subsection we show that
sentedbytheKrausoperatorsBˆj(nk) andsendtheresultn if both ρˆ12 and ρˆ23 are d × d-dimensional pure states,
toSapna. Inthelaststepofthisscheme,Sapnaperforms thantheequalityinEqs.(26,45)canalwaysbeachieved.
a second measurement with Kraus operators denoted by
Fˆ(j) and send the result i to Alice and Bob. The fi-
jkn
naldistributionofentangledstatessharedbetweenAlice An optimal protocol for RPBES
andBobisdenotedby N ,σjkni ,whereN isthe
{ jkni 14 } jkni
probability for outcome j,k,n,i and σˆjkni = Tr σˆjkni In this section we show that by LOCC Sapna can pre-
14 23 1234
with pare a bipartite pure state between Alice and Bob with
anyvalueoftheconcurrencemonotoneGwhichislessor
1
σˆ1j2k3n4i = Njkni Aˆ(jk)⊗Fˆj(ki)nMˆ(j)⊗Bˆj(nk) [ρˆ12⊗ρˆ34] etoqcuoallftoorGR1B2GE3S4P. Fthoratthhiasspubrepenosefi,rswteiinnttrroodduucceedthine[p2r2o]-.
(cid:16) (cid:17)
† In this protocol the supplier Sapna shares the initial
Aˆ(jk)⊗Fˆj(ki)nMˆ(j)⊗Bˆj(nk) . (42) (d×d)-dimensionalpure states|ψi12 = dk−=10√λk|kki12
Sincethe G-concur(cid:16)renceofanybipartitesta(cid:17)tesatisfies and |χi34 = dk−=10√ηk|kki34 (which aPre expressed in
Eq. (22), the analog of Eq. (41) for this LOCC protocol the Schmidt decomposition)with Alice and Bob,respec-
P
is therefore, tively.
The steps of the protocol are as follows:
G (i) Sapna performs a projective measurement
14
≡
1 2/d
NjkniG σˆ1j4kni ≤ d2G12G34 Det(Aˆ(jk)) Pˆ(j,j′) =|P(j,j′)i23hP(j,j′)|, j,j′ =0,1,...,d−1, (46)
j,Xk,n,i (cid:16) (cid:17) Xj,k (cid:12) (cid:12)
(cid:12) (cid:12)
Det(Bˆ(n)) 2/d Tr Mˆ(j)†Fˆ(i)†(cid:12)Fˆ(i) Mˆ(j)(cid:12) . with
× jk jkn jkn
Xn (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) Xi (cid:16) (cid:17)(43) |P(j,j′)i23 ≡ d1 d−1 ei[2dπ2(dj+j′)(dm+m′)+θmm′]|mm′i23 ,
m,m′=0
Moreover,fromthegeometric-arithmeticinequalitywe X
(47)
have with θmm′ R chosen freely. Note that the d2 states
Det(Bˆj(nk)) 2/d ≤ d1 TrBˆj(nk)†Bˆj(nk) =1 (44) |θPm(mj,′j.′)i23 a∈re orthonormal, regardless of the choice of
Xn (cid:12) (cid:12) Xn (ii)Aftertheoutcomesj,j′havebeenobtained,thestate
and a si(cid:12)(cid:12)milar relat(cid:12)(cid:12)ion for Aˆ(jk). These results, together ofthesystemcanbewrittenas|P(j,j′)i23|φ(j,j′)i14,where
withthecompletenessrelation Fˆ(i)†Fˆ(i) =1,leadus d−1 d−1
back to Eq. (41). As we can see,ialljkonperjakntions that are φ(j,j′) 14 = λmηm′
P | i
performed by Alice, Bob and Sapna after the first mea- m=0m′=0
X X p
surementby Sapna cannotincreasethe boundonC14 (cid:3). e−i[2dπ2(dj+j′)(dm+m′)+θmm′] mm′ 14 . (48)
Theorem1concernsonesupplierandtwonodes,butin × | i
fact applies to one supplier and any pair of nodes; thus, (iii)Sapnasendstheresultsj andj′ toBob(2log dbits
2
the result of Theorem 1 is applicable to an arbitrarily of information) and the result j′ (log d bits of informa-
2
large QNet with one supplier and many nodes. In fact tion) to Alice. Bob then performs the unitary operation
Theorem 1 can be extended to more than one supplier,
as stated in the following corollary.
Corollary: ConsideranalignchainofN mixedbipar- Uˆ(j,j′) m′ =exp i2π(dj+j′)m′ m′ , (49)
tite states, ρ0,1, ρ1,2,...,ρN−1,N, where the state ρk−1,k b | i4 d2 | i4
(cid:18) (cid:19)
7
and Alice performs the unitary operation rence. Wehaveshownthatforafinitenumberofcopiesof
purestates(i.e. thedeterministiccase)thefamilycharac-
Uˆ(j′) m =exp i2πj′m m . (50) terizes completely the non-local resource. We have also
a | i1 d | i1 discussed the advantage of the concurrence monotones
(cid:18) (cid:19)
overothermeasuresofentanglement(suchastheentropy
(iv) The final state shared between Alice and Bob is of entanglement, the Renyi entropies, etc.) and showed
that for a given bipartite state, ψ = a i j , the
d−1 d−1 | i ij ij| i| i
concurrence monotones can always be expressed analyt-
|Fi14 = exp(−iθmm′) λmηm′|mm′i14, (51) ically in terms of the coefficients aij. PWe also gave an
mX=0mX′=0 p analytical expression of the concurrence monotones (for
pure states) in terms of the reduced density matrix (see
(which is separable for θmm′ =0).
Eq. (16)).
We will show now, that by choosing the phases θmm′
appropriately, Sapna can prepare the state F with
14
| i
any value of G(F ) in the range [0,G G ]. For this
| i14 12 34 We then discussed a particular member of the family
purpose, we define the square (d d) complex matrix A
× which we called the G-concurrence. The G-concurrence
with elements amm′ =√λmηm′exp(−iθmm′). Thus, for pure states is defined as the geometric mean of the
Schmidt numbers. It has several unique properties that
G(F )=d Det A†A 1/d
14 makes it extremely useful. In particular,we have proved
| i
=G(cid:2) G (cid:0) Det(cid:1)(cid:3)V†V 1/d , (52) a powerful theorem that establishes an upper bound on
12 34
the amount of G-concurrence that can be created be-
where G = d(λ λ(cid:2) λ(cid:0) )1(cid:1)/(cid:3)d G = tween two single-qudit nodes of quantum networks by
12 0 1 d−1 34
d(η η η )1/d and the m··a·trix elements of V meansofRED.Thetheoremalsosuggestsanoperational
0 1 d−1
··· interpretationoftheG-concurrenceasatypeofentangle-
aθmremv′m=m2′π=memxp′/(d−tihθemmm′a)t/r√ixdV. Nisoutenitthaartyfaonrdththeecrheofoicree mtoensattcuarpaatecittyh.eWGe-choanvceuprrreonvceedbthoautnditiinsatlhweaythsepoorsesmiblief
G(|Fi14) = G12G34. For other choices of θmm′, Sapna bothoftheentangledstatesarepure,andalsosuggested
can prepare the final state F with any value of the
| i14 an operational interpretation of the G-concurrence as a
G-concurrence monotone in the range [0,G G ].
12 34 type of entanglement capacity. An open question is left
It is important to emphasize here that the choice
ifitispossibletosaturatetheboundwhenthestatesare
θmm′ =2πmm′/dmaximizes only the G-concurrence. In mixed.
fact, for other measures of entanglement the values of
θmm′ that maximize the entanglement depend explicitly
on the Schmidt numbers λ and η . For example, the
m m The concurrence monotones are defined in terms of
concurrence monotone C of the final state F is
k=2 14 the symmetric functions of the Schmidt numbers (see
| i
Eq.(5)). Thesesymmetricfunctionshavemanyinterest-
C2(F 14)=2 λkλk′ηmηm′ ing mathematical properties which were not introduced
| i
nkX>k′mX>m′ here (some of the properties can be found in [25]) and
2 1/2 which are related to the field of majorization. Thus, we
ei(θkm+θk′m′) ei(θkm′+θk′m) . (53)
× − believethatfurtherinvestigationsofthesemonotoneswill
(cid:12) (cid:12) o contribute to our understanding of entanglement.
Thus, in this c(cid:12)ase we see that the values(cid:12)of θ that
(cid:12) (cid:12) km
maximize C (F ) depend explicitly on the Schmidt
2 14
| i
coefficients λk and ηm. Acknowledgments: Iwouldliketoextendmysincere
gratitude to Barry Sanders, for fruitful discussions and
for reviewing this work in its preliminary stages. The
IV. SUMMARY AND CONCLUSIONS author acknowledges support by the Killam Trust, the
DARPA QuIST program under contract F49620-02-C-
In summary, we have introduced a family of entan- 0010,and the NationalScience Foundation (NSF) under
glement monotones that extend the definition of concur- grant ECS-0202087.
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