Table Of ContentConnecting the generalized robustness and the geometric measure of entanglement
Daniel Cavalcanti
Departamento de F´ısica, Caixa Postal 702, Universidade Federal de Minas Gerais, 30123-970, Belo Horizonte, MG, Brazil∗
The main goal of this paper is to provide a connection between the generalized robustness of
entanglement (Rg) and the geometric measure of entanglement (EGME). First, we show that the
generalized robustness is always higher than or equal to the geometric measure. Then we find a
tighter lower bound to Rg(ρ) based only on the purity of ρ and its maximal overlap to a separable
state. Aswe will see it is also possible to express this lower bound in terms of EGME.
PACSnumbers: 03.67.-a,03.65.Ud,03.67.Mn
Since it was first noted [1, 2] the issue of quantum these quantifiers are related is not obvious a priori, for
8
correlationshas been largely studied and debated. How- these functions rely on distinct geometrical interpreta-
0
0 ever, it was not until entanglement was recognized as tions. While EGME measures the minimum angle be-
2 a physical resource that this theme got a solid status. tween an entangled state and a separable one, Rg can
From this point of view, entanglement was shown to al- be treated as a kind of “distance” between an entangled
n
lowseveraltaskssuchasquantumcryptography[3],tele- state and the set of unentangled states. Furthermore
a
J portation [4], and quantum algorithms [5]. On the other both quantifiers are able to deal with the various types
7 hand, entanglement has also given us new insights for of entanglement that a multipartite system can present.
understanding important physical phenomena including Letusfirstpresentthelanguageweshalladopttotalk
4 superconductivity[6],super-radiance[7],quantumphase about multipartite entanglement. Suppose a state ρ can
v transitions [8, 9, 10, 11, 12], and the appearance of clas- be written as a convex combination of states which are
1
sicality [13]. product of k tensor factors. The state ρ is then said
3
0 One of the greatest challenges concerning entangle- to be a k-separable state. One should note that in a
2 ment is how to properly quantify this resource. Al- systemofnparts,n-separabilityisseparabilityitselfand
0 thoughthisproblemiswellunderstoodforbipartitepure that every state is trivially 1-separable. The set of k-
6 states, in a more complex scenario (multipartite systems separable states will be denoted by Sk. It is clear that
0 or mixed states) a complete theory on the quantification Sn Sn−1 ... S1 = D, where D denotes the set of
h/ of entanglement is still lacking. den⊂sity oper⊂ators⊂.
p Among the difficulties ofdealing with multipartite en- We are now able to understand why Rg and EGME
- can distinguish the types of entanglement a system con-
t tanglementis the factthat systemscomposedby various
n tains. The geometric measure is a pure-state entangle-
parts can exhibit many kinds of entanglement. This is
a ment quantifier given by:
u because one may be interested in the entanglement ac-
q cording to a specific partition of the whole system. So Ek (ψ)=1 Λ2(ψ), (1)
: a state can present some entanglement in relation to a GME − k
v
given partition, while it can be separable according to where
i
X another one.
Λ2 = max φψ 2. (2)
ar In the last years many candidates of entanglement k φ∈Sk|h | i|
quantifiers were proposed. Generically speaking, the
Thus Ek (ψ) measures the sine squared of the mini-
ways of quantifying entanglement can be divided into GME
mumanglebetween ψ andak-separablestate[36]. Itis
twoclasses: quantifierswithageometricalinterpretation, | i
known that this quantity is an entanglement monotone
andthosewithanoperationalmeaning. Inthe firstclass
[28], i.e.: it is a non-increasing function under LOCC.
we cancite the relativeentropyofentanglement[14, 15],
The relation between Ek and the notion of k-
the geometricmeasureofentanglement[16, 17], the neg- GME
entanglement witnesses [29] (observables with positive
ativity [18, 19, 20], and the robustness of entanglement
meanvaluetoallk-separablestates,butnegativetosome
[21, 22, 23]. The entanglement cost [24, 25], the distill-
ρ / S ) (see Ref. [17]) has also been determined. This
able entanglement [24, 26], and the singlet fraction [27] ∈ k
results from the fact that one can always construct a k-
are examples of operational measures.
entanglement witnesses Wk for a pure state ψ of the
The purpose of this letter is to point out a connec- | i
type
tionbetweentwowelldiscussedentanglementquantifiers,
the generalized robustness (R ) [22, 23] and the geo- Wk =λ2 ψ ψ . (3)
g
−| ih |
metric measure of entanglement (E ) [16, 17]. That
GME
As this operator must have a positive mean value for
every k-separable state, the relation
λ2 max φψ 2 =Λ2 (4)
∗Electronicaddress: dcs@fisica.ufmg.br ≥|φi∈Skkh | ik k
2
As the witness (5) obviously satisfies the condition
Wk I we can attest the following:
≤
Rk(ψ) Ek (ψ). (8)
g ≥ GME
Some points concerning the inequality (8) should be
stressed at this stage. First, it is a relation valid to
all kinds of multipartite entanglement. Moreover this
relation will be strict whenever the witness (5) is a
k-entanglement witness which solves the minimization
problem in (7). Finally, one could argue that relation
(8) may be, in fact, a consequence of standard results
frommatrixanalysisrelatingdifferentdistancemeasures
between operators (as commented, both Rk and Ek
g GME
are relatedto such distances). However,it must be clear
that Rk(ψ) is not simply the distance between ψ and its
g
closest state σ S , but one should keep in mind that
FIG. 1: Geometrical interpretation of Rgk. The straight line this distance is∈takken with relation to the state π as a
represents the convex combination ρ+sπ. We see that for a
1+s reference [37] (recall figure 1). This makes the closest
given state π and a value of s this combination becomes k-
separable. Rgk(ρ)isdefinedastheminimums,consideringall k-separable state usually different for Rgk and EGkME.
possible states π. In fact, it is possible to give a tighter relationbetween
Rk and Ek . Recall the lemma 1 shown in ref. [32].
g GME
We now give a clearer proof of it, and interpret it as a
must hold. Thus the optimal entanglement witness of lower bound to Rk.
g
the form (3) is reached when λ=Λ2, and we can write
k Lemma 1. For every state ρ D,
Wk =Λ2 ψ ψ . (5) ∈
opt k−| ih | Tr(ρ2)
Hereoptimalityisdefinedinthesenseofgettingthehigh- Rgk(ρ)≥ max Tr(ρσ) −1. (9)
est value to ψ Wk ψ . σ∈Sk
|h | | i|
In a different fashion, the robustness of entanglement Proof First of all let us show that max Tr(ρσ)
of a state ρ quantifies how robust the entanglement of ρ σ∈Sk
is equal to the minimum value of λ (λ ) such that
min
is under presence of noise. Thus the robustness of ρ in
W =λI ρ is a k-entanglement witness. As Tr(Wσ)
relation to the state π, R(ρ π), is the minimum s such − ≥
k 0 σ Sk,
that the state ∀ ∈
ρ+sπ Tr[(λI ρ)σ]=λ Tr(ρσ) 0. (10)
σ = (6) − − ≥
1+s
It is thus straightforward to see that λ =
min
is k-separable. We will be interested in an extension of
max Tr(ρσ).
the relative robustness, namely the generalized robust- σ∈Sk
Note that
ness. This entanglement quantifier is obtained by the
minimization of the relative robustness over all states π W ρ
W′ = =I <I. (11)
[22]. Recently, an interesting operational interpretation λ − λ
min min
toRkwasgivenintermsofthepercentualincreaseastate
g
can provide to teleportation processes [30]. The general- So it is possible to see that Rk(ρ) Tr(W′ρ), from
ized robustness can also be viewed as a “distance” of ρ which follows the required resulgt. (cid:3) ≥ −
tothesetSk inthespaceofstates(seefigure1)[31],and The lower bound to Rgk expressed by (9) can be easily
thus allow both a geometrical and an operational inter- interpreted: Trρ2 measures the purity of ρ, and Tr(ρσ)
pretation. MoreoverRkwasusedtoinvestigatetheshape
g is the Hilbert-Schmidt scalar product between ρ and σ.
of entangled states sets [31] and was shown to exhibit a
It is expected that the more mixed ρ is, the lower the
kind of polygamy of entanglement [32]. value of Trρ2 gets, and the state becomes less entangled.
As well as the geometric measure, Rk is intimately
g Similarly, the larger maxσ∈SkTr(ρσ) is, closer to the set
connected to the notion of entanglement witnesses. In
S ρ gets, and the system will show less entanglement.
k
fact, Rk can be expressed as
g Butnowwenotethatinthe specialcaseofpurestates
therelationsTr(ρ2)=1andmax Tr(ρσ)=Λ2(ρ)
Rk(ρ)=max 0, min Tr(Wkρ) , (7) σ∈Sk(H) k
g { −Wk∈M } hold and therefore we have the general relation
where is the set of operators M I and Wk is a 1
k-entanMglement witness [33]. ≤ Rgk(ψ)≥ Λ2(ψ) −1. (12)
k
3
and we can see the relation we are looking for: 1
Ek
Rk(ψ) GME . (13)
g ≥ 1 Ek 0.8
− GME
It is interesting that two entanglement monotones
with different geometric interpretation are actually re- 0.6
lated, and furthermore this relation allows an analytic
lower bound to the generalized robustness for all states
whenever Λ2(ρ) can be analytically computed. This is 0.4
k
the case, for example, of completely symmetric states,
Werner states, and the isotropic states [17, 34].
0.2
We can furthermore see from (12) that
log (1+Rk) 2log Λ . (14)
2 g ≥− 2 k 0 0.2 0.4 0.6 0.8 1
The left side of this expression is the logarithmic ro- p
bustness of entanglement (LRk), another entanglement
g FIG. 2: (Color online) Red crosses: logarithmic general-
quantifier with interesting features [33]. Curiously, this
ized robustness of entanglement. Blue diamonds: relative
isexactlythesamelowerboundexpressedtotherelative
entropy of entanglement. Black line: lower bound given in
entropyofentanglement(ERk)in[34]. Numericalandan- Eq. (14).
alytical results (see, for example, Figure 2 and Table I)
suggest that LRk Ek, in general, but at the moment
g ≥ R |S(2,1)i |S(3,2)i |S(4,3)i |S(4,2)i
this is just a conjecture. En 0.5 0.55 0.58 0.625
GME
For bipartite pure states all the quantities considered Rn 1 1.25 1.36 1.65
g
so far can be analytically computed. While the relative
entropy of entanglement equals the entropy of entangle- TABLEI:Acomparisonamongmultipartiteentanglementof
ment(givenby thevonNeumannentropyofthe reduced somestates(17),givenbygeometricmeasureofentanglement
state) [15], the generalized robustness is given by (EGnME) - see Ref. [17] - and therobustness of entanglement
(Rn) - see Ref. [35].
g
Rk(ψ)=( c )2 1, (15)
g i −
Xi
En (S(n,k) ) (i.e.: the geometric measure of
being ci the spectrum of Schmidt of ψ [22]. In this SG(nM,Ek)| withreilationtothecompletelyseparablestates)
contex{t it}can be noted that Λk is given|byi the modulus [|17]. Adiditionally, in this case it was shownthat the rel-
of the highest Schmidt coefficient of ψ [17]. To visual- ative entropy of entanglement is exactly equal the lower
| i
ize and compare these entanglement measures we calcu- bound given in Eq. (14) [34], and moreoverit equals the
latetherelativeentropyofentanglement,thelogarithmic logarithmic robustness of entanglement [35]. So, for the
generalizedrobustness,andthelowerboundexpressedin states (17), it is possible to compare analytically these
(14) for the state entanglement quantifiers. Some examples are shown in
Table I.
ψ(p) =√p00 + 1 p11 . (16)
| i | i − | i In brief, we have shown some relations between the
p
geometric measure of entanglement and the generalized
The plots are available in figure 2.
As the presented relations between Rk and Ek are robustness of entanglement. We reached a lower bound
g GME to Rk with nice interpretations and wrote it in terms of
also valid to multipartite entanglement it would be use- g
ful to illustrate the results in this context as well. We EGkME. These relations also allowed us to compare two
choose to study some completely symmetric states for other entanglement quantifiers, the logarithmic general-
this aim. These states are referred to as Dicke states or, izedrobustnessandtherelativeentropyofentanglement.
sometimes,asgeneralizedW statesandappearnaturally Examples were given to illustrate the results.
as eigenstates of various models such as the η-pairing Because many entanglement quantifiers exist it is im-
model[6]andthe Dicke model[7]. Followingref. [17]we portanttounderstandtheirrelationandthis,webelieve,
will label these states according to the number of 0’s, as should be a major goal in the theory of entanglement.
follows: We hope that this discussion can help in this sense.
k!(n k)!
S(n,k) = − S 000..011..1 , (17)
| i r n! | i Acknowledgments
k n−k
| {z }|{z}
where S is the total symmetrization operator. Wei I would like to thank Dr. Marcelo Terra Cunha and
and Goldbart showed an analytical expression to Flavia Tenuta for useful comments on this paper. I spe-
4
cially thank Dr. Miguel G. Cruz for helpful criticisms, andDamianMarkhamforbringingref. [35]tomyatten-
Fernando Branda˜o for help with some numerical results, tion. FinancialsupportfromCNPqisalsoacknowledged.
[1] E. Schr¨odinger, Die gegenw¨artige Situation in der (2002).
Quantenmechanik, Naturwissenschaften 23 807- [20] K. Audenaert, M. B. Plenio, and J. Eisert, Phys. Rev.
812, 823-828, 844-849 (1935). See a translation at Lett. 90, 027901 (2003).
http://www.tu-harburg.de/rzt/rzt/it/QM/cat.html. [21] G. Vidal and R.Tarrach, Phys.Rev. A 59141(1999).
[2] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev 47, [22] M. Steiner, Phys.Rev.A 67, 054305 (2003).
777 (1935). [23] A. W. Harrow and M. A. Nielsen, Phys. Rev. A 68,
[3] A.K. Ekert,Phys. Rev.Lett. 67, 661 (1991). 012308 (2003).
[4] C. H. Bennett etal., Phys.Rev. Lett. 70, 1895 (1993). [24] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W.
[5] M.A.Nielsen andI.L.Chuang,QuantumComputation K. Wootters, Phys.Rev.A 54, 3824 (1996).
andQuantumInformation(Cambridge: CambridgeUni- [25] P.M.Hayden,M.Horodecki,andB.M.Terhal,J.Phys.
vesity Press, 2000). A: Math. Gen. 34, 6891 (2001).
[6] V.Vedral, New J. Phys. 6, 102 (2004). [26] E. M. Rains, Phys. Rev.A 60, 173 (1999).
[7] N.Lambert,C.Emary,andT.Brandes,Phys.Rev.Lett. [27] M. Horodecki, P. Horodecki, and R. Horodecki, Phys.
92, 073602 (2004). Rev. A 60, 1888 (1999).
[8] T. J. Osborne and M. A. Nielsen, Phys. Rev. A 66, [28] G. Vidal, J. Mod. Opt.47, 355 (2000).
032110 (2002); [29] M. Horodecki, P. Horodecki, and R. Horodecki, Phys.
[9] A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature Lett A 223, 1-8 (1996).
(London) 416608(2002); [30] F. G. S.L. Brand˜ao,Preprint quant-ph/0510078 (2005).
[10] J. I. Latorre, E. Rico, and G. Vidal, Quant. Inf. and [31] D. Cavalcanti , F. G. S. L. Brand˜ao, and M. O. Terra
Comp. 4, 048 (2004); Cunha, Preprint quant-ph/0510068 (2005).
[11] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. [32] D. Cavalcanti , F. G. S. L. Brand˜ao, and M. O. Terra
Rev.Lett. 90, 227902 (2003); Cunha, Phys. Rev.A 72040303(R)(2005).
[12] D. Cavalcanti, F. G. S. L. Brand˜ao, and M. O. Terra [33] F. G. S.L. Brand˜ao, Phys. Rev.A 72, 022310 (2005).
Cunha,Preprint quant-ph/0510132 (2005). [34] T. -C. Wei, M. Ericsson, P. M. Goldbart, and W. J.
[13] W. H.Zurek,Preprint quant-ph/0306072 (2003). Munro, Quant.Inf.Comp. 4, 252 (2004).
[14] V.Vedral,M.B.Plenio,M.A.Rippin,andP.L.Knight, [35] M. Hayashi, D. Markham, M. Murao, M. Owari, and S.
Phys.Rev.Lett. 78, 2275 (1998). Virmani, arXiv:0711.4286 (2007).
[15] V. Vedral and M. B. Plenio, Phys. Rev. A 57, 1619 [36] The extension of Ek (ψ) to mixed states is made
GME
(1998). through the convex-roof construction, which makes this
[16] H. Barnum and N. Linden, J. Phys. A: Math. Gen. 34, quantifier hard to be computed in general. Furthermore
6787 (2001). thisconstructionmakesthegeometricalinterpretationof
[17] T.-C.WeiandP.M.Goldbart,Phys.Rev.A68,042307 Ek not so clear. Thus we will focus on pure states
GME
(2003). unless otherwise specified.
[18] Eisert J 2001 PhD thesis (Universityof Potsdam). [37] Besides that there is a minimization among all possible
[19] G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314 states π.
