Measurable lower bounds on concurrence

0.18 MB·English

by  Iman Sargolzahi

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1 MEASURABLE LOWER BOUNDS ON CONCURRENCE
0
2

n
a IMANSARGOLZAHI∗,SAYYEDYAHYAMIRAFZALIandMOHSENSARBISHAEI
J
Department of Physics, Ferdowsi Universityof Mashhad
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Mashhad, Iran
2

] Received(receiveddate)
h
Revised(reviseddate)
p
-
t
n We derive measurable lower bounds on concurrence of arbitrarymixed states, for both
a bipartite and multipartite cases. First, we construct measurable lower bonds on the
u purely algebraic bounds of concurrence [F. Mintert et al. (2004), Phys. Rev. lett., 92,
q 167902]. Then, using the fact that the sum of the square of the algebraic bounds is a
[ lowerboundofthesquaredconcurrence,wesumoverourmeasurableboundstoachieve

ameasurablelowerboundonconcurrence. Withtwotypicalexamples,weshowthatour
2
method can detect more entangled states and also can give sharper lower bonds than
v
thesimilarones.
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2 Keywords: Measuringentanglement, Concurrence
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Communicated by: tobefilledbytheEditorial
1
.
0 1. Introduction
1
9 Recently, many studies have been focused on the experimental quantification of entangle-
0
ment [1]. Bell inequalities and entanglement witnesses [1, 2] can be used to detect entangled
:
v states experimentally, but they don’t give any information about the amount of entangle-
i
X ment. In addition, quantum state tomography [3], determination of the full density operator
r ρ by measuring a complete set of observables, is only practical for low dimensional systems
a
since the number of measurements needed for it grows rapidly as the dimension of the sys-
tem increases. Fortunately, several methods have been introduced which let one to estimate
experimentally the amount of the entanglement of an unknown ρ with no need to the full
tomography[1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 29, 19, 30, 20, 26, 27, 28, 21,
22, 23, 24, 25]. A simple and straightforward method is the one introduced in [8, 14, 18] for
finding measurable lower bounds on an entanglement measure, namely the concurrence [31].
These lower bounds are in terms of the expectation values of some Hermitian operators with
respect to two-fold or one-fold copy of ρ. It is worth noting that these bounds work well for
weakly mixed states [32, 8, 14, 18, 5].
∗[email protected],[email protected]
1
2 Title ...
In this paper we will use a similar procedure as [8, 14] to construct measurable lower
bounds on the purely algebraic bounds of concurrence [33, 31]. In addition, using a theorem
inSec. II,weshowthatthesumofourmeasurableboundsleadstoameasurablelowerbound
on the concurrence itself. Then, we show that this method gives better results than those
introduced in [8, 14] for two typical examples.
Thepaperisorganizedasfollows. InSec. II,theconcurrenceanditsMKB (Mintert-Kus-
Buchleitner) lower bounds [33] are introduced. In Secs. III and IV, we propose measurable
lower bounds on the purely algebraic bounds of concurrence [33], which are a special class of
MKB bounds. Thegeneralizationtothe multipartitecaseis giveninSec. Vandweendthis
paper in Sec. VI with a summary and discussion.
2. Concurrence and its MKB Lower Bounds
For a pure bipartite state Ψ , Ψ , concurrence is defined as [31]:
A B
| i | i∈H ⊗H
C(Ψ)= 2[ ΨΨ 2 trρ2], (1)
h | i − r
p
whereρ isthereduceddensityoperatorobtainedbytracingovereithersubsystemsAorB.It
r
isobviousthatiff Ψ isaproductstate,i.e. Ψ = Ψ Ψ ,thenC(Ψ)=0. Interestingly,
A B
| i | i | i⊗| i
C(Ψ) can be written in terms of the expectation value of an observable with respect to two
identical copies of Ψ [31, 11, 12]:
| i
C(Ψ)= Ψ Ψ Ψ Ψ ,
AB AB AB AB
h | h |A| i | i
=4PA PB, (2)
p A − ⊗ −
where PA (PB) is the projector onto the antisymmetric subspace of ( ).
− − HA⊗HA HB⊗HB
A possible decomposition of is
A
= χ χ ,
α α
A | ih |
α
X
χ =(xy yx ) (pq qp ) , (3)
α A B
| i | i−| i | i−| i
where x and y (p and q ) are two different members of an orthonormal basis of the A
| i | i | i | i
(B) subsystem. For mixed states the concurrence is defined as follows [31]:
C(ρ)=min p C(Ψ ),
i i
i
X
ρ= p Ψ Ψ , p 0, p =1, (4)
i i i i i
| ih | ≥
i i
X X
wheretheminimumistakenoveralldecompositionsofρintopurestates Ψ . Itisappropriate
i
| i
to write C(ρ) interms ofthe subnormalizedstates ψ ratherthanthe normalizedones Ψ :
i i
| i | i
C(ρ)=min ψ ψ ψ ψ ,
i i i i
h |h |A| i| i
i
Xp
ψ =√p Ψ , ρ= ψ ψ ; (5)
i i i i i
| i | i | ih |
i
X
since all decompositions of ρ into subnormalized states are related to each other by unitary
matrices [3]: consider an arbitrary decomposition of ρ = ϕ ϕ (As a special case,
j| jih j|
P
Author(s) ... 3
one can choose ϕ = λ Φ , where Φ and λ are eigenvectors and eigenvalues of ρ
j j j j j
| i | i | i
respectively: ρ= λ Φ Φ .),foranyotherdecompositionofρ= ψ ψ wehave[3]:
j j|pjih j| i| iih i|
P P
ψ = U ϕ , U†U =δ . (6)
| ii ij| ji ki ij jk
j i
X X
So Eq. (5) can be written as:
C(ρ)=min U U lmU†U† ,
U ij ikAjk li mi
i sjklm
X X
lm = ϕ ϕ ϕ ϕ . (7)
Ajk h l|h m|A| ji| ki
Fromthe definitionofC(ρ) inEq. (4)itis obviousthatC(ρ)=0iffρcanbedecomposed
into product states. In other words, C(ρ) = 0 iff ρ is separable. In addition, it can be
shownthattheconcurrenceisanentanglementmonotone[34](Anentanglementmonotoneis
a function of ρ which does not increase, on average,under LOCC and vanishes for separable
states [35].). But, except for the two-qubit case [36], C(ρ) can not be computed in general;
i.e., in general, one can not find the U which minimizes Eq. (7). Any numerical method for
finding the U which minimizes Eq. (7) leads to an upper bound for C(ρ). So, finding lower
bounds on C(ρ) is desirable. So far, severallower bounds for C(ρ) have been introduced [33,
31,37,38,39,40,41,42,43,5,8,13,14,18,19,21,22,23,24]. Oneofthemisthatintroduced
byF.Mintertet al. in[33,31]. Now,weredrivetheirlowerboundsinaslightlydifferentform
to make them more suitable for finding measurable lower bounds in the following sections.
AssumethatthedecompositionofρwhichminimizesEq. (5)isρ= ξ ξ ,thenfrom
j| jih j|
Eqs. (3) and (5), we have:
P
C(ρ)= χ ξ ξ 2 χ ξ ξ min χ ψ ψ , (8)
α j j β j j β i i
j s α |h | i| i| ≥ j |h | i| i|≥{|ψii} i |h | i| i|
X X X X
where χ χ ,andtheminimumistakenoveralldecompositionsofρasρ= ψ ψ .
| βi∈{| αi} i| iih i|
Now, using Eq. (6), we have:
P
min χ ψ ψ =min U TβU⊤ =min UTβU⊤ ,
{|ψii} i |h β| ii| ii| U i | jk ij jk ki| U i | ii|
X X X X (cid:2) (cid:3)
Tβ = χ ϕ ϕ . (9)
jk h β| ji| ki
Since Tβ is a symmetric matrix, the minimum in Eq. (9) can be computed and we have [31]:
min UTβU⊤ =max 0,Sβ Sβ , (10)
U | ii| { 1 − l }
i l>1
X (cid:2) (cid:3) X
where Sβ are the singular values of Tβ, in decreasing order. The above expression is what
l
was named purely algebraic lower bound of concurrence in [31, 33] and we will refer to it as
ALB(ρ).
Let us define
τ = z∗ χ , z 2 =1. (11)
| i α| αi | α|
α α
X X
4 Title ...
Obviously, τ is an element of another (normalized to 2) basis of PA PB, χ′ . Then:
| i − ⊗ − {| αi}
τ χ′ ,
| i≡| 1i
= χ χ = τ τ + χ′ χ′ . (12)
A | αih α| | ih | | αih α|
α α>1
X X
Again, as the inequality (8), we have:
C(ρ)= χ′ ξ ξ 2 τ ξ ξ
|h α| ji| ji| ≥ |h | ji| ji|
j s α j
X X X
min τ ψ ψ
i i
≥{|ψii} i |h | i| i|
X
=min U U⊤ =max 0,Sτ Sτ ,
U | T ii| { 1 − l }
i l>1
X (cid:2) (cid:3) X
= τ ϕ ϕ = z Tα , (13)
Tjk h | ji| ki α jk
α
X
whereSτ arethesingularvaluesof ,indecreasingorder. Theaboveexpressionisthegeneral
l T
form of the lower bounds introduced in [33, 31] and we call it LB(ρ).
We end this section by proving a useful theorem: if χ′ be an orthogonal (normalized
{| αi}
to 2) basis of PA PB, i.e. = χ′ χ′ , then:
− ⊗ − A α| αih α|
P
C2(ρ)= χ′ ξ ξ 2 χ′ ξ ξ 2
|h α| ii| ii| |h α| ji| ji|
ij s α s α
X X X
χ′ ξ ξ χ′ ξ ξ
≥ |h α| ii| ii||h α| ji| ji|
ij α
XX
2
= χ′ ξ ξ [LB (ρ)]2 ,
|h α| ii| ii|! ≥ α
α i α
X X X
LB (ρ)= min χ′ ψ ψ . (14)
α {|ψii} i |h α| ii| ii|
X
In proving the above theorem we have used the Cauchy-Schwarz inequality. Obviously, any
entangledρ whichcan notbe detected by LB , can not be detected by [LB (ρ)]2 either;
α α α
i.e., [LB (ρ)]2 is not a more powerful criteria than LB , but, quantitatively, it may lead
α α α P
to a better lower bound for C(ρ).
P
It should be mentioned that the above theorem is, in fact, the generalizationof what has
been proved in [42]. There, it was shown that:
τ(ρ)= C2 (ρ) C2(ρ),
mn ≤
C (ρ)= mXin ψ L L ψ∗ , (15)
mn {|ψii} i |h i| mA ⊗ nB| ii|
X
whereL andL aregeneratorsofSO(d )andSO(d )respectively(d =dim( )),
mA nB A B A/B HA/B
and ψ∗ is the complex conjugate of ψ in the computational basis. In this basis L and
| ii | ii mA
L are [44]:
nB
L = x y y x , L = p q q p .
mA | iAh |−| iAh | mB | iBh |−| iBh |
Author(s) ... 5
For an arbitrary ψ , according to the definition of χ in Eq. (3), it can be seen that:
α
| i | i
ψ L L ψ∗ = χ ψ ψ . (16)
|h | mA ⊗ nB| i| |h α| i| i|
So:
C (ρ)=ALB (ρ),
mn α
ALB (ρ)= min χ ψ ψ . (17)
α α i i
{|ψii} i |h | i| i|
X
So what was proved in [42] is, in fact, the special case of χ′ = χ in expression (14). In
| αi | αi
addition, since ALB can detect bound entangled states [33, 31], this claim of [42] that any
α
state for which τ(ρ)>0 is distillable, is not correct.
3. Measurable Lower Bounds in terms of Two Identical Copies of ρ
As we have seen in Eq. (2) the concurrence of a pure state Ψ can be written in terms of
| i
the expectation value of the observable with respect to two identical copies of Ψ . For an
A | i
arbitrary mixed state ρ , it has been shown that [8]:
AB
C2(ρ ) tr ρ ρ V , i=1,2;
AB AB AB (i)
≥ ⊗
V =4 PA PA PB, V =4PA PB PB , (18)
(1) − − + ⊗ − (cid:0) (2) (cid:1) − ⊗ − − +
where PA (PB) is th(cid:0)e projector(cid:1)onto the symmetric subspaceo(cid:0)f (cid:1) ( ). The
+ + HA⊗HA HB⊗HB
above expression means that measuring V on two identical copies of ρ, i.e. ρ ρ, gives us
(i)
⊗
a measurable lower bound on C2(ρ). It is worth noting that if the entanglement of ρ can be
detected by V , then ρ is distillable [24].
(i)
As one can see from expression (13), the LB of a pure state Ψ can also be written in
| i
terms of the expectation value of the observable τ τ with respect to two identical copies of
| ih |
Ψ . Now,foranarbitrarymixedstateρ,canwefindanobservableV suchthatthefollowing
| i
inequality holds?
LB2(ρ) tr(ρ ρV) , (19)
≥ ⊗
Fortunately for the special case of τ = χ , where χ are defined in Eq. (3), we can do
α α
| i | i | i
so.
AssumethatthedecompositionofρwhichgivestheminimuminEq. (9)isρ= θα θα ,
i| i ih i |
i.e.:
P
ALB (ρ)= χ θα θα . (20)
α |h α| i i| i i|
i
X
In addition, assume that for a Hermitian operator V , which acts on ,
α A B A B
H ⊗H ⊗H ⊗H
and arbitrary ψ and ϕ , ψ and ϕ , we have:
A B A B
| i | i | i∈H ⊗H | i∈H ⊗H
χ ψ ψ χ ϕ ϕ ψ ϕV ψ ϕ . (21)
α α α
|h | i| i||h | i| i|≥h |h | | i| i
Now, from the expressions (20) and (21), we have:
ALB2(ρ)= χ θα θα χ θα θα θα θα V θα θα =tr(ρ ρV ) . (22)
α |h α| i i| i i||h α| ji| ji|≥ h i |h j| α| i i| ji ⊗ α
ij ij
X X
6 Title ...
So, for any V satisfying inequality (21), measuring V on two identical copies of ρ gives a
α α
lowerboundonALB2(ρ). Wecanprovethattheinequality(21)holdsfor(seetheAppendix):
α
V =V = V , V =V = V ,
α (1)α (1) α (2)α (2)
M M M M
= ,
A A B B
M M ⊗M ⊗M ⊗M
= x x + y y , = p p + q q , (23)
A B
M | ih | | ih | M | ih | | ih |
where x , y , p , q are introduced in Eq. (3) (note that χ χ = ). In addition,
α α
| i | i | i | i | ih | MAM
for any V such as
α
V =c V +c V , c 0, c 0, c +c =1, (24)
α 1 (1)α 2 (2)α 1 2 1 2
≥ ≥
inequalities (21) and, consequently, (22) also hold.
According to the definition of V in Eqs. (23) and (24), we have:
α
tr(ρ ρV )=tr(̺ ̺V ) ,
α α
⊗ ⊗
̺= ρ , (25)
A B A B
M ⊗M M ⊗M
which means that if V detects the entanglement of ρ, it has, in fact, detected the entangle-
α
ment of a two-qubit submatrix of ρ. Any ρ which has an entangled two-qubit submatrix is
distillable [45]. So any ρ which is detected by V is distillable.
α
The right hand side of the inequality (18) is invariant under local unitary transforma-
tions [8]:
tr ρ ρV =tr ρ′ ρ′V ,
(i) (i)
⊗ ⊗
ρ′ =U U ρU† U† , (26)
(cid:0) A⊗(cid:1) B (cid:0)A⊗ B (cid:1)
where U andU arearbitraryunitary operators. This is so because U† U†PAU U =
A B A⊗ A ± A⊗ A
PA and U† U†PBU U = PB. So, the choices of local bases in the definition of V
± B ⊗ B ± B ⊗ B ± (i)
in (18) are not important since all the choices lead to the same result. But, according to
the definition of V in Eqs. (23) and (24), the right hand side of the inequality (22) is not
α
invariant under local unitary transformations. It is however expected since the ALB (ρ) is
α
not invariant under such transformations either.
UsingEqs. (23)and(24),itcanbeshownsimplythattherighthandsideoftheinequality
(22) is invariant under the following transformations:
tr(ρ ρV )=tr(ρ′ ρ′V ) ,
α α
⊗ ⊗
ρ′ =u u ρu† u† ,
A⊗ B A⊗ B
u =u , u u† =u†u = ,
MA AMA A A A A A MA
u =u , u u† =u† u = ,
MB BMB B B B B B MB
tr(ρ′) 1. (27)
⇒ ≤
χ is also invariant, up to a phase, under the above transformations, i.e. u u
α A A
| i ⊗ ⊗
u u χ = eiβ χ and 0 β 2π, but it is not so for the ALB (ρ). Consider the
B B α α α
⊗ | i | i ≤ ≤
decompositionof ρ into pure states as ρ= θα θα . From Eq. (27) we know that there is
a decomposition of ρ′ into pure states as ρ′ =i| i ihθ′iα| θ′α , where θ′α = u u θα . So,
P i| i ih i | | i i A⊗ B| i i
using Eq. (20):
χ θ′α θ′α = Pχ θα θα =ALB (ρ). (28)
|h α| i i| i i| |h α| i i| i i| α
i i
X X
Author(s) ... 7
But
χ θ′α θ′α min χ ψ′ ψ′ =ALB (ρ′), (29)
|h α| i i| i i|≥ |ψ′i |h α| ji| ji| α
Xi { j }Xj
where the minimum is taken overall decompositions of ρ′ into pure states: ρ′ = ψ′ ψ′ .
j| jih j|
So:
P
ALB (ρ′) ALB (ρ). (30)
α α
≤
Note that expressions (22), (27) and (30) show that tr(ρ ρV ) bounds the amount of
α
⊗
ALB2(ρ′), for all possible ρ′ in Eq. (27), from below.
α
Now, using inequalities (14) and (22):
C2(ρ) ALB2(ρ) tr(ρ ρV ) , (31)
≥ α ≥ ⊗ α
α α
X X
where the summation is only over those α for which tr(ρ ρV ) 0.
α
⊗ ≥
Example 1. In a d d dimensional Hilbert space, isotropic states are defined as [2]:
×
1 F
ρ = − I φ+ φ+ +F φ+ φ+ ,
F d2 1 −| ih | | ih |
− (cid:0) d 1 (cid:1)
φ+ = i i ,
A B
| i √d| i
i=1
0 F 1, X F = φ+ ρ φ+ . (32)
≤ ≤ h | F| i
The concurrence of ρ is known and we have [34]:
F
2d 1
C(ρ )=max 0, F . (33)
F ( rd−1(cid:18) − d(cid:19))
If we rewrite ρ as
F
1 F Fd2 1
ρ = − I+ − φ+ φ+ gI+hφ+ φ+ ,
F d2 1 d2 1 | ih |≡ | ih |
− −
then:
h2 2
tr ρ ρ V =2d(d 1) dg2 gh . (34)
F ⊗ F (i) − d2 − − d
(cid:20) (cid:21)
(cid:0) (cid:1)
In Eq. (23), if we choose x=p,y =q , then:
{ }
h2 2
tr(ρ ρ V )=4 2g2 gh ,
F ⊗ F α d2 − − d
(cid:20) (cid:21)
and the expectation values of other V are not positive. Since the case x=p,y =q occurs
α
{ }
n= d(d−1) times in a d d dimensional system, we have:
2 ×
h2 2
tr ρ ρ V =2d(d 1) 2g2 gh , (35)
F ⊗ F α α! − (cid:20)d2 − − d (cid:21)
X
8 Title ...
1.2
1
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1
F
Fig. 1. Comparison of Eqs. (34), dotted line, and (35), dashed line, for d=4. The solid line is
the exact valueof concurrence, Eq. (33). The lowerbounds given by V(i) and αVα areset to
zerowhentherighthandsidesofEqs. (34)and(35)arelessthanzero.
P
where the summation is only over those V for which x=p,y =q . For d > 2, Eq. (35)
α
{ }
givesabetterresultthanEq. (34)(Fig. 1). Ford=2bothgivethesameresult,asweexpect
from Eq. (23).
4. Measurable Lower Bounds in terms of One Copy of ρ
From the experimental point of view, any lower bound which is defined in terms of the
expectation value of an observable with respect to two identical copies of ρ, encounters, at
least,twoproblems. First, formeasuringV orV we needto measureinanentangledbasis
(i) α
in both parts A and B.Measuring in an entangledbasis is more difficult than measuring in a
separable one [12]. Second, it is not clear that the state which enters the measuring devices
is really as ρ ρ even if we can produce such state at the source place [46, 10]. So, having
⊗
lower bounds in terms of the expectation value of an observable with respect to one copy of
ρ is more desirable.
Using:
C(ρ)C(σ) tr ρ σV , i=1,2;
(i)
≥ ⊗
1
⇒C(ρ)≥(cid:0)C(σ)tr ρ(cid:1)⊗σV(i) , (36)
(cid:0) (cid:1)
for arbitrary ρ and σ, F. Mintert has introduced the following measurable lower bound on
C(ρ) [14]:
1
C(ρ) tr(ρW ) , W = − tr I σV , (37)
≥− σ σ C(σ) 2 ⊗ (i)
(cid:0) (cid:1)
where σ is a pre-determined entangled state and the partial trace is taken over the second
copy of . If C(σ) is not computable, which is the case for almost all mixed σ, an
A B
H ⊗H
upper bound of C(σ) can be used in the definition of W . From inequality (37), it is obvious
σ
Author(s) ... 9
thatforanyseparablestate: tr(ρ W ) 0. If,atleast,foroneentangledstatetr(ρ W )<0,
s σ e σ
≥
then W is an entanglement witness [2].
σ
We can, also, construct measurable lower bounds in terms of one copy of ρ by using
inequality (21). Suppose that the decomposition of σ which gives the minimum in Eq. (9) is
σ = γα γα , i.e.:
j| jih j |
ALB (σ)= χ γα γα . (38)
P α |h α| ji| ji|
j
X
Using expressions (20), (21) and (38):
[ALB (ρ)][ALB (σ)]= χ θα θα χ γα γα
α α |h α| i i| i i||h α| ji| ji|
ij
X
θα γα V θα γα = tr(ρW′ ) ,
≥ h i |h j | α| i i| ji − σα
ij
X
W′ = tr (I σV ) .
σα − 2 ⊗ α
So:
1
ALB (ρ) tr(ρW ) , W = W′ , (39)
α ≥− σα σα ALB (σ) σα
α
where σ is a pre-determined entangled state for which ALB (σ) >0. Note that, in contrast
α
to C(σ), ALB (σ) is alwayscomputable, soweneverneedto use anupper boundofitinthe
α
definition of W . In addition, it can be shown simply that
σα
tr(ρW )=tr(̺W ) , (40)
σα σα
where ̺ is defined in Eq. (25). So any ρ which is detected by W is distillable. Also, using
σα
inequalities (14) and (39):
C2(ρ) [ALB (ρ)]2 [tr(ρW )]2 , (41)
α σα
≥ ≥
α α
X X
where the summation is over those α for which tr(ρW ) 0.
σα
≤
For isotropic states,using expressions (37) or (41) (by choosing σ = φ+ φ+ ) gives the
| ih |
exact value of C(ρ ) for arbitrary d. In the following, we give an example for which the
F
expression (41) gives better results than the expression (37).
Example 2. Consider a two-qutrit system which is initially in the pure state
Φ = λ 01 + λ 12 + λ 20 , (42)
0 1 2
| i | i | i | i
p p p
and its time evolution is given by the following Master equation [14]:
ρ˙ = ρ,
L
= 1 +1 , (43)
A B A B
L L ⊗ ⊗L
where , for a one-qutrit ρ , is
A/B A/B
L
Γ
= 2γρ γ† ρ γ†γ γ†γρ ,
LA/B 2 A/B − A/B − A/B
(cid:0) (cid:1)
10 Title ...
and γ is the coupling matrix for the spontaneous decay:
0 0 0
γ = √2 0 0 .
 
0 1 0
 
To construct W in expression (39) and W in expression (37), we choose
σα σ
σ = Φ Φ ,
ME ME
|1 ih |
Φ = (01 + 12 + 20 ) . (44)
ME
| i √3 | i | i | i
It can be shown simply that for three χ , for which p=x 1,q =y 1 ( is the sum
α
| i { ⊕ ⊕ } ⊕
modulo 3), ALB (σ) =2/3, and ALB (σ)=0 for other χ . So, using expression (39), we
α α α
| i
can construct three W as (x=0,1,2 and y =x 1):
σα
⊕
W = x,y 1 x,y 1 + y,x 1 y,x 1 x,x 1 y,y 1 y,y 1 x,x 1
σα
| ⊕ ih ⊕ | | ⊕ ih ⊕ |−1| ⊕ ih ⊕ |−| ⊕ ih ⊕ |
= x,y 1 x,y 1 + y,x 1 y,x 1 σxy σx⊕1,y⊕1 σxy σx⊕1,y⊕1 ,
| ⊕ ih ⊕ | | ⊕ ih ⊕ |− 2 1 ⊗ 1 − 2 ⊗ 2
σab = a b + b a , (cid:16) σab = i(a b b a) . (cid:17) (45)
1 | ih | | ih | 2 − | ih |−| ih |
Also, using expression (37), we can show that:
3
1
W = W . (46)
σ σα
√3
α=1
X
As wecansee fromEqs. (45)and(46),the number oflocalobservablesneededformeasuring
W or three W is the same and is equal to 12, which is less than what is needed for a full
σ σα
tomography. Also, note that l,m 1 is an orthonormal basis of . So, at least
A B
{| ⊕ i} H ⊗H
from the theoretical point of view, all the observables l,m 1 l,m 1 can be measured
| ⊕ ih ⊕ |
usingonlyonesetup. Insuchcases,formeasuringW orthreeW ,weonlyneed7different
σ σα
setup of localmeasurements. The comparisonof the results ofinequalities (37)and (41),for
two typical λ , is given in Fig. 2.
i
{ }
5. Extending to Multipartite Systems
In a bipartite system, any Hermitian operator which, for arbitrary ψ and ϕ , satisfies the
| i | i
inequality
C(ψ)C(ϕ) ψ ϕV ψ ϕ , (47)
≥h |h | | i| i
gives a measurable lower bound on C2(ρ), i.e. C2(ρ) tr(ρ ρV) [10]. This can be proved
≥ ⊗
simply by writing ρ in terms of its extremal decomposition ρ = ξ ξ . In [18] it was
j| jih j|
shownhowtouse suchV toconstructmeasurablelowerbounds formultipartiteconcurrence.
P
Following a similar procedure, we construct measurable lower bounds on multipartite con-
currence using V . As the previous sections, we will use the inequality (21) instead of the
α
inequality (47). In other words, we will work with the algebraic lower bounds of C(ρ) rather
than the concurrence itself.
The concurrenceofanN-partitepure state Ψ , Ψ ,isdefined as[31]:
| i | i∈HA1⊗···⊗HAN
C(Ψ)=21−N2 C2(Ψ), (48)
l
s l
X