Relations between bipartite entanglement measures

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by  Katharina Schwaiger

Relations between bipartite entanglement measures
K. Schwaiger and B. Kraus
Institute for Theoretical Physics, University of Innsbruck, Innsbruck, Austria
Weinvestigatethepropertiesandrelationsoftwoclassesofoperationalbipartiteandmultipartite
entanglementmeasures,theso-called sourceandtheaccessible entanglement. Theformermeasures
how easy it is to generate a given state via local operations and classical communication (LOCC)
from some other state, whereas the latter measures the potentiality of a state to be convertible to
other states via LOCC. Main emphasis is put on the bipartite pure states, single copy regime. We
investigate which parameter regime is physically available, i.e for which values of these measures
doesthereexist abipartitepurestate. Moreover,wedetermine,givensomestate, whichparameter
regime can be accessed by it and from which parameter regime it can be accessed. We show that
7
this regime can be determined analytically using the Postitivstellensatz. Moreover, we compute
1
the boundaries of these sets and the boundaries of the corresponding source and accessible sets.
0
2 Furthermore, we relate these results to other entanglement measures and compare their behaviors.
Apart from that, an operational characterization of bipartite purestate entanglement is presented.
n
a
J
I. INTRODUCTION ment from an operational point of view might also be
0
relevantis these fields of research. It is preciselythe aim
3
ofthis papertoshinenewlightontothe physicalcharac-
Biparite entanglement, in particular pure state entan-
terization of bipartite entanglement.
] glement, is considered to be very well understood. This
h Here, we first show that there is a very simple way to
p assessment steams mainly from the facts that (i) in the
operationallycharacterizetheentanglementcontainedin
- asymptotic regime the entanglement can be completely
t a bipartite pure 2k 2k system. It consists of consid-
n characterized via the entanglement of formation [1] and ×
ering such a state as a 2k qubit state and determining
a (ii)inthesinglecopycaseacompletesetofentanglement
the entanglement (measured by only one parameter, e.g.
u measuresforpurestatesisknown[2]. Tobemoreprecise,
q theratewithwhichncopiesofapurestate, Ψ Cd Cd thegeometricmeasureofentanglement[5])inallpossible
[ | i∈ ⊗ biparite splittings. We show that these measures deter-
can be transformed asymptotically and reversibly into
mine the Schmidt coefficients of the state uniquely. As
1 the maximally entangled state, Φ+ = ii is given
| i i| i anybipartitestatecanbe embeddedina2k 2k system,
v by the entanglement of formation [1]. In the single copy ×
9 regimeitisknownthattheentanglementPmonotonespre- this interpretation can be applied to any bipartite pure
7 system. Moreover, such a characterizaton is also best
sented in [2] completely characterize the entanglement
6 suited for the case, where, as explained above, a bipar-
contained in the state. That is given the d 1 entangle-
8 − tite splitting of a multiqubit state is considered. After
ment monotones a unique state (up to local unitary op-
0
that we pursue a different approach, which is based on
. erations(LUs)) is characterized. However,in contrastto
1 entanglement measures which have a very clear opera-
the entanglement of formation, the entanglement mono-
0 tional meaning. We recently introduced two classes of
tones do not have a clear physical meaning. Hence, de-
7 entanglementmeasures,which areapplicable to any sys-
1 spite the fact, that the characterizationof bipartite pure
tem size as well as to pure and mixed states, the source
: stateentanglementismathematicallywellunderstood,it
v entanglement, E and the accessible entanglement, E .
is lacking a clear operational meaning. s a
i Wedeterminedthesemeasuresincaseofmultiparitesys-
X
However,suchaclearphysicalpictureasintheasymp- temsin[6]andincaseofbipartitesystemin[7]. Inorder
ar toticcaseisalsohighlydesirableforthesinglecopycase. toexplainthephysicalmeaningletusdenotebyMs(Ψ )
This is not only for the sake of understanding bipartite the set of states which can be transformedinto the s|tatie
entanglement better from an operational point of view, Ψ andbyM (Ψ )thesetofstatesintowhich Ψ canbe
a
but, probably more importantly, also to understand cer- |trainsformedto|viai LOCC.Using then an arbit|rairymea-
tain aspects of correlations in multipartite systems. A surethesourceentanglement,E (Ψ ),measuresthevol-
s
prominent example, where the relevance of bipartite en- ume of the set M (Ψ ). That is,|itimeasures how many
s
tanglement within multipartite systems is shown, is the states can be transf|orimed into Ψ . The more states are
fundamental area law proven within condensed matter inM (Ψ )the less entangledth|e sitate is,asallstatesin
s
physics [3]. It gives a bound on the bipartite entangle- M (Ψ|) aire at least as entangled as Ψ . The accessible
s
ment for ground states of local Hamiltonians (in 1D). enta|ngilementisthengivenbythevolu|mieofM (Ψ ). In
a
This result has been used to prove an efficient descrip- [7] we derivedclosedexpressionsforthe sourceen|taingle-
tion of the ground state of the system in terms of tensor ment and showed how the accessible entanglement can
networkstates,leadingto efficientnumericalsimulations be evaluated.
of such systems, which disclosed new insights about the
It is the aim of this paper to better understand which
underlying physics [4].
values these measures can take and how they are re-
Hence, a better understanding of bipartite entangle- lated to eachother andto other entanglementmeasures,
2
such as the entanglement of formation. We will present II. NOTATIONS AND PRELIMINARIES
several numerical results concerning the allowed region
these measures can take and will then show how these Inthissectionwefirstintroduceournotationsandba-
regions can be determined analytically. To this end we sic properties of bipartite states and restate the source
use the Positivstellensatz, which characterizes those sets and accessible entanglement of bipartite states intro-
of polynomial equations and inequalities which have a ducedin[7]. Thenwereviewatheoremfromrealalgebra,
real solution. Moreover, we will explain how the map- namely the Positivstellensatz, that gives a necessaryand
ping of this problem to a SDP presented in [8] can be sufficientconditionfortheexistenceofasolutionofaset
achievedin our case. The analysisperformedhere might of polynomial equations and inequalities and discuss its
be alsousedinorderto obtainaboundone.g. the value relevance in the context of this work.
of the source entanglement given the geometric measure Every pure state of a bipartite quantum system with
of entanglement. The results presented here show that Hilbert space = Cd1 Cd2 can be (up to LUs) writ-
whenever the considered functions are complete, in the ten as ψ = Hd √λ ⊗ii , i.e. ψ d √λ ii ,
sense that they uniquely characterize a state of interest, | i i=1 i| i | i ≃LU i=1 i| i
where d = min d ,d and λ 0 denote the Schmidt
1 2 i
then the boundaries of these regions can be easily deter- P{ } ≥ P
coefficients with λ = 1. We denote by λ(ψ) =
mined. However, in contrast to previous investigations, i i
(λ ,...,λ ) RId the Schmidtvectorof ψ andwillcon-
where different entanglement measures have been con- 1 d ∈ P | i
sider in the following w.l.o.g. two d-level systems, which
sidered [9], the boundaries are more involved otherwise.
we call from now on d d states. Note that we will of-
Moreover, we investigate the entanglement contained in ×
ten refer to a state via its Schmidt vectors. As can be
the states which belong to the source and the accessible
easilyseenfromthe Schmidtdecompositiongivenabove,
set of some given state. There, again, the boundaries
two d d states, ψ , φ , are LU equivalent if and only
of these sets are easily characterized in case the mea- if (iff)×λ↓(ψ) = λ|↓(iφ)|, wihere here and in the following
sures are unique (as we will see in the case of 3 3 and
× λ↓(ψ) RId, with λ↓(ψ) λ↓(ψ) 0 denotes the
4 4systems). Finallywewillalsoconsiderprobabilistic i i+1
∈ ≥ ≥
× sorted Schmidt vector of ψ . Note that when consider-
transformations in this context.
| i
ingLOCCtransformationsofbipartitestates,weexclude
The outline of the remainder paper is the following. LU-transformations, as they do not alter the entangle-
We first recall the definition of the classes of entangle- ment of the states. That is we pick one representativeof
ment measures we consider. Moreover, we recall an im- eachLU-equivalenceclass,i.e. λ↓(ψ),andconsidertrans-
portant theorem in real analysis, the Positivstellensatz, formations among these representatives. In the context
whichgivesnecessaryandsufficientconditionsfortheex- of LOCC transformations of pure bipartite states, the
istenceofarealsolutiontoasetofpolynomialequations following functions of x=(x ,...,x ) RId,
1 d
∈
and inequalities in n variables. We also explain how a
relaxation of the problem can be solved efficiently for a d
fixed degree using semidefinite programms (SDP). Fur- Ek(x):= xi, k 1,...,d , (1)
∈{ }
thermore, we present an operational characterization of i=k
X
bipartite pure state entanglement. Next, we analyze the
play animportant role. It was shownin [13] that a state
mathematical properties of the entanglement measures
ψ can be transformed into φ deterministi-
and show e.g. that despite the fact that these are entan- |calily∈vHia LOCC iff λ↓(ψ) is majori|zeid∈byHλ↓(φ), written
glementmeasures,theyarenotentanglementmonotones.
λ↓(ψ) λ↓(φ), i. e.
The only other example of such measures are, up to the ≺
knowledge of the authors, the Renyi entropies for α>1, E (λ↓(ψ)) E (λ↓(φ)) k 1,...,d , (2)
k k
α= . After that, we focus in Sec. IVA and Sec. IVC ≥ ∀ ∈{ }
6 ∞
onthepossiblevaluesofEs andEa whichcanbereached with equality for k = 1. A direct consequence of this
from a physical state. We compare these results to pre- criterionis thatthe sourceandaccessibleset,i.e. the set
vious investigations [9], where mainly entropic functions of states that can either reach or be reached by a state
are considered as measures and show the differences to ψ deterministically via LOCC, of ψ are given by
| i | i∈H
the ones investigated here. The values of the entangle-
mentmeasureswhichcanbetakenbystatesinthesource Ms(ψ)= φ s.t. λ(φ) λ(ψ) ,
{| i∈H ≺ }
and accessible set of a given state show a very interest- M (ψ)= φ s.t. λ(ψ) λ(φ) . (3)
a
ing behavior. We then study how the entanglement can {| i∈H ≺ }
be transformedonthe courseofa LOCCprotocol. After Let us now review the idea of measuring the volume of
thepresentationofthesenumericalinvestigationweshow thesourceandaccessibleset,i.e. M (ψ)andM (ψ),and
s a
howthesesetscanbeobtainedanalyticallyusingthePos- thus, obtaining a valid operational entanglement mea-
itivstellensatz in Sec. IV. Moreover, we show how this sure, as mentioned in the introduction. Let µ denote an
theoremcan be usedto provethat a setof entanglement arbitrary measure in the set of LU-equivalence classes.
measures is complete. Finally, we also consider proba- As mentioned above we consider one representative of
bilistic transformations and study the entanglement in each LU-equivalence class in M and M , as we do not
a s
this setting. considerLU-transformations. Then,thesourcevolumeis
3
defined by V (ψ) = µ[M (ψ)] and the accessible volume ofoperationalentanglementmeasures. Forthesegeneral-
s s
by V (ψ) = µ[M (ψ)]. Hence, the accessible and source izationswemeasurethesetofstates, Ψ Ck Ck,that
a a
| i∈ ⊗
entanglement are given by can be converted via LOCC to a d d state ψ , with
× | i
smaller or equal dimensions, i.e. d k. Then we can
Va(ψ) Vs(ψ) identify the state ψ with a state Ψ≤k(ψ) Ck Ck,
E (ψ)= , E (ψ)=1 , (4)
a Vasup s − Vssup whose Schmidt vec|toir is simply given by ad∈ding ⊗k d
(cid:12) (cid:11) −
zeros to the initial, d-dimensional Sc(cid:12)hmidt vector of ψ .
where Vsup (Vsup) denote the supremum of the accessi- | i
a s Thus, we get a whole class of operational entanglement
ble (source) volume according to the measure µ. Note
measures given by the generalizations of the source en-
that for any valid measure µ, E and E are valid en-
s a tanglement,thatareforeachk givenby(see[7]formore
tanglement measures, i.e. they are not increasing under
details)
LOCC. This can be easily verified considering the oper-
ational meaning of Es, Ea. Whereas Es measures how 1
easyitistoobtainacertainstateviaLOCC,Eameasures Esk→d(ψ)= sup E (Ψk(φ))Es(Ψk(ψ)) ,k ≥d.
s
how useful a state is, as this state is at least as powerful |φi∈Cd⊗Cd
asanystateinitsaccessibleset. Forbipartitepurestates (6)
wepresentedin[7]thefollowingclosedexpressionforthe
source entanglement [27] Notethatthedimensionofthevolumescorrespondingto
the generalizations of the source entanglement is higher
E (ψ)=1 ( dk=1σkλk− d+21)d−1, (5) than the dimension of the source volume corresponding
s − d−1(σ σ ) tothesourceentanglementinEq.(5). Thatisthedimen-
σX∈Σd P k=1 k− k+1 sion of the source sets corresponding to the generalized
where Σd denotes the symQmetric group and the sum source entanglement Esk→d is equal to k−1.
runs over all elements of this group. Note that for LetusnowreviewthePositivstellensatz[10],whichisa
the accessible entanglement we have closed expressions fundamentaltheoreminrealalgebraicgeometry. Itstates
for low-dimensional bipartite systems, e.g. for 3 3 that there either exists a polynomial identity, which cer-
12λ λ if λ > 1 × tifiesthatasystemofpolynomialequationsandinequali-
states Ea(ψ) = (12[λ22λ33−11/4(12−2λ1)2] if λ1 ≤ 12 ttioesthhiasssnyostseomlu.tAionfeiwnRdenfionrittihoenrseoefxaislgtsebinrdaiecedobajescotlus,tiio.en.
(see Appendix A). Furthermore, we provided algorithms the ideal and the cone, are in order before we can recall
to compute the accessible entanglement numerically in the theorem.
[7] for higher dimensional states. For 3 3 states the
×
source and the accessible set in terms of the Schmidt Definition 1. The subset I R[x ,...,x ] is an ideal if
1 n
coefficients is shownin Fig.1. We reconsiderthe bound- it satisfies ⊆
ariesofthesesetsinFig.1inthefollowingsections. Note
thatsimilarfigureswerealreadyintroducedin[14],where (a) 0 I,
∈
theauthorsinvestigatedgeometricpropertiesofbipartite
entanglement in terms of the Schmidt coefficients. Note, (b) If a, b I, then a+b I,
∈ ∈
λ3 (c) If a I and b R[x1,...,xn] , then a b I.
1 ∈ ∈ · ∈
A simple example would be the ideal corresponding
to the set of multivariate polynomials h ,...,h , h
1 m i
R[x ,...,x ] i. It is given by { } ∈
1 n
∀
m
1 1 I(h1,...,hm)= h h= tihi, ti R[x1,...,xn] . (7)
λ1 λ2 { | i=1 ∈ }
X
FIG. 1: [7] The source and theaccessible set of a 3×3 state Another example of an ideal is the set of polynomials
with Schmidt vector λ = (0.6,0.37,0.13) in terms of the with a common set of roots.
φ
Schmidt coefficients. The thick, dashed line encloses the set
ofsortedSchmidtvectorsthatisinone-to-onecorrespondence Definition 2. The subset P R[x1,...,xn] is a cone if
⊆
to the LU-equivalence classes. The red (blue) regions depict it satisfies
the source (accessible) set of the quantum state respectively.
Note that the number of vertices of the accessible set can (a) If a,b P, then a + b P,
∈ ∈
change and is in this case equal to four (see [7] for details).
(b) If a, b P, then a b P,
∈ · ∈
further, that in [7] we introduced also the generalization
of the source entanglement, which leads to a whole class (c) If a R[x1,...,xn] , then a2 P.
∈ ∈
4
Hence, the cone of a set of multivariate polynomials wediscussgeneralpropertiesofthesourceandaccessible
f ,...,f , f R[x ,...,x ] i, reads entanglement.
1 s i 1 n
{ } ∈ ∀
m
P(f ,...,f )= f f =s + s f + s f f +
1 s 0 i i ij i j
{ | A. Operational characterization of bipartite
i=1 i<j
X X entanglement via the geometric measure of
s f f f +...s f f ...f , entanglement
ijk i j k 12...s 1 2 s
}
i<j<k
X
(8) In this section we show that the bipartite entangle-
with s{i1,...,is} ∈ R[x1,...,xn] a sum of squares polyno- mLeetntno=f alodg×(dd) saynstdemconcasindebrethcheabraipcatertriitzeedstaastefolψlows.
mial(seealsoSec.IVB).Furthermore,the multiplicative ⌈ ⌉ | iAB
as a 2n-qubit state, i.e. ψ . As we will
monoid M of a set of polynomials g1,...,gk is defined | iA1...AnB1...Bn
{ } showbelow,thedSchmidtcoefficientsof ψ aregiven
by | iAB
by the bipartite entanglement (measured with one func-
l tion) of all possible bipartite splittings of the qubits in
M(g1,...,gl)= giai, ai ∈N. (9) B versusthe restincluding the splitting A versusB.The
i=1 entanglement is measured here with the geometric mea-
Y
WiththisdefinitionswecannowstatethePositivstellen- sure of entanglement, i.e. Eg(ψ) = 1 λ1 with λ1 the
−
satz [10]. largest Schmidt coefficient. The geometric measure of
entanglement[5]operationallyquantifiesentanglementby
Theorem 1. Let f , h , g
befinitefamilies of{poil}yin=o1m,...isals{injR}j[=x11,,.....,.m,xn{]. kT}hke=n1,t..h.,el IttheisdidsetafinnceedoafsaEstat=e |1ψi tomtahxe neaφreψstse2p,awraitbhleφstataen.
g φ
f (x) 0, i=1,...,s − kh | ik | i
i ≥ arbitrary product state. For bipartite pure states it is
set x Rn gk(x)=0, k=1,...,l  is empty iff equivalent to the measure given above. Note that in the
∈ | 6
 hj(x)=0, j =1,...,m multipartite case this measure has also been used in the
context of quantum computing [11].
f P(f1,...,fs),g M(g1,...,gl),hI(h1,...,hm)
∃ ∈ ∈ ∈ Lemma 1. Let the bipartite state ψ =
s.t. f +g2+h=0. (10) √λ i i C2n C2n be considered as a|2in-qubit
i i| iA| iB ∈ A ⊗ B
Note that it is easy to see that if Eq. (10) is fulfilled, state ψ = λ i i ...i i i ...i .
there cannot exist a solution to the set of polynomial PThen,| ψi is ui1n,ii2q,u..e.,liyn (upi1ti2o...LinU|s1)2detenrimAi|n1ed2 bynitBhe
equations and inequalities, as for any solution x0 Rn geometr|icimePasure of enptanglement of all possible bipar-
we have that f(x0) 0, g2(x0) > 0 and h(x0) ∈= 0. tite splittings with k qubits in B versus the rest for all
Thus, f(x0)+g2(x0)≥+h(x0)>0, which is in contradic- k 1,...n .
tion with Eq. (10). For a proof of Theorem 1 see [10]. ∈{ }
The Positivstellensatz thus results in a single equation The proof of the above lemma is given in Appendix
that corresponds to a necessary and sufficient condition B. Note that in order to give this operational character-
fortheexistenceofarealsolutiontoapolynomialsystem ization, the basis,in which the 2n-qubit state is written,
of equations and inequalities. It is a very powerful theo- has to be fixed, e.g. as in Lemma 1. This operational
remthatleadsinmanycasestoquitesimpleinfeasibility characterization of the bipartite entanglement is partic-
certificates. InSec.IVBweuse the Positivstellensatzon ularly interesting in case the bipartite entanglement of
theonehandtogetcertificatesforwhentwogivenvalues a multipartite qubit state is considered via a bipartite
of e.g. the source and the accessible entanglement can- splitting.
notcorrespondtoaphysicalstateandontheotherhand
to show that the source entanglement together with its
generalizations characterizes few-qubit bipartite entan- B. Properties of the source and accessible
entanglement
glement. Whereas in the first case no functions g are
k
required, we will use them in the second. Note that by
restricting the overall degree of the left-hand side of Eq. Inthissubsectionwefirstshowthattheformulaofthe
(10) one can find solutions of this equation efficiently, as source entanglement can be simplified and then discuss
this problem can then be stated as a semidefinite pro- general properties of the source and accessible entangle-
gram as shown in [8] (see Sec. IVB). ment.
III. PROPERTIES OF BIPARTITE 1. Simplification of the source entanglement formula
ENTANGLEMENT
Here we want to show that by using the results from
In this section we present an operational characteri- [12]onecansimplifythegeneralformulaofthesourceen-
zation of bipartite pure state entanglement. Moreover, tanglementofbipartitepurestates(seeEq.(5))thatwas
5
introduced in [7]. This result is stated in the following a polynomial vanishes, if the degree of the numerator is
Lemma. smaller than the degree of the denominator, leads to
Lemma 2. The source entanglement of bipartite pure ( d−1 1p k σ )d−1
states (see [7]) given in Eq. (5) can be simplified to Es(ψ)=1− k=d1−k1(σk l=σ1 l) . (13)
σX∈Σd P k=1 kP− k+1
( d σ λ )d−1
1 k=1 k k . (11) Note that this formula dQoes not depend on p , as
−σX∈Σd Pdk−=11(σk−σk+1) dl=1σd = d(d2+1) is a constant and thus, this terdm in
thesumcancels,asthedividedsymmetrizationofitvan-
Q
Hence, the source entanglement is a homogeneous func- P
ishes (see above).
tion in λ of degree d-1.
Proof. The proof is based on a result presented in [12]
2. General properties of E and E
concerning the divided symmetrization of a polynomial s a
f(x ,...x ),i.e. f = σ f(x1,...,xd) . Forall
1 d h i σ∈Σd Qdk−=11(xk−xk+1) Beforeinvestigatinginwhichparameterregiontheval-
functionsf(x1,...xd)wPithdegree(cid:16)smallerd−1th(cid:17)edivided ues of Ea and Es are lying, we summarize here some
symmetrizationvanishes,i.e. f =0. Thiscanbeeasily properties of these entanglement measures.
seen by writing f = g, as thheiVandermonde determi- Let us first show that the entanglement measures E
h i ∆ a
nant ∆ = 1≤i<j≤d(xi−xj) is the common denomina- and Es are no entanglement monotones. An entangle-
tor of all the terms in the divided symmetrization. Note ment monotone for pure states [15] is a function that is
Q
thatg has to be anantisymmetric polynomial,asthe di- nonincreasing on average under LOCC, i.e.
videdsymmetrizationis bydefinition symmetricandthe
Vandermonde determinant ∆ is an antisymmetric poly- E (ψ) p E (ψ ), (14)
mon i mon i
nomial. Furthermore,∆istheantisymmetricpolynomial ≥
i
of the smallest degree, deg (∆)= d . This is due to the X
2
fact that any antisymmetric polynomial vanishes if two for any pure state ensemble p , ψ that is obtained
i i
(cid:0) (cid:1) { | i}
of the variables are equaland thus, must have x x as from ψ via LOCC. A widely used feature of entangle-
i− j | i
afactorforalli=j. Nowforanypolynomialf(x ,...x ) ment monotones for pure states is the fact that the con-
1 d
6
with deg (f(x ,...x ))<d 1 the degree of the numera- vexroofconstruction[16]leadstoentanglementmeasures
1 d
−
tor of f is smaller than the degree of the denominator. on mixed states. Note that for the source and accessi-
h i
Therefore,the degree ofthe antisymmetric polynomial g ble entanglement such a constructionis not necessary as
has to be smaller than the degree of the Vandermonde they are already defined for any quantum state, includ-
determinant, which, as explained above, is not possible. ingmixedstatesofanysystemsize. Moreover,these two
Hence, g has to be equal to zero and therefore, the di- measures are defined in an operational way and thus, it
vided symmetrization vanishes, i.e. f = 0. Using this is not surprising that instead of fulfilling condition (14)
h i
propertyofthedividedsymmetrizationtogetherwiththe they only fulfill the physical LOCC monotonicity condi-
binomial theorem for the numerator in the original for- tion. For multipartite quantum states of four or more
mula of E in Eq. (5) one obtains directly the simplified qubits it is easy to show that E and E are indeed in-
s s a
version in Eq. (11). creasing on average under LOCC, thus violating cond.
(14),bysimplyconsideringanisolatedstate ψ . Such
| iiso
As mentioned before the formula of the source entan- a state can neither be reached nor transformed into any
glement (also the simplified version in Eq. (11)) is valid other state via LOCC. However, the state ψ can be
for sorted Schmidt vectors λ↓. Investigating the proper- transformedwithanonvanishingprobabilit|yiinistooanon-
ties of Es is often easier if the ordering is automatically isolated state, ψ1 . Denoting the other states in the en-
| i
fixed. Thus, we might want to change the variables in semble ofstates into which ψ is transformedby ψ ,
| iiso | ii
Es for certain problems and write the ordered Schmidt we have 0=Es(ψiso)<pEs(ψ1)+ ipiEs(ψi). A simi-
vector in terms of the extreme points of the convex set larargumentholdsfortheaccessibleentanglement. Note,
of sorted vectors, i.e. however, that for e.g. pure three-quPbit states within the
W-class the source and accessible entanglement are in-
d deed entanglement monotones. That is, the measures
λ↓ = p e =Mp, (12)
i i do notincreaseonaverageunder LOCC forthe W-class.
Xi=1 Similarfunctions,i.e. entanglementmonotonesforstates
in the W-class, were also introduced in [18].
11/21/3··· 1/d
01/21/3··· 1/d Also for bipartite states one can easily construct a
with p=(p1,...,pd) and M =00.. 00.. 1/0..3 ... 11//..dd. Using cthoeunbtiepra-erxtiatmepmleeatsoursehso.wItnhaotrdcoerndt.o(1d4o)sios,vcioolnastiedderbya
. . . .
  transformationofthestateλ =(0.6,0.3,0.1)intoanen-
0 0 0 ... 1/d ψ
Eq.(12)andthe factthat the dividedsymmetrizationof semblecontainingthe two statescorrespondingto λ =
ψ1
6
(0.8,0.15,0.05) and λ = (0.57,0.32,0.11) we obtain hand any entangled mixed state ρ can be transformed
ψ2
for the source and accessible entanglement respectively into some other entangled state. In particular ρ can be
E (ψ)=0.63, E (ψ)=0.36andp E (ψ )+p E (ψ )= transformed into ρ′ = pρ+(1 p) a,b a,b, which is
s a 1 s 1 2 s 2
− | ih |
0.631, p E (ψ )+p E (ψ ) = 0.37. Thus, the bipartite entangledforcertainvaluesofpandproductstates a,b ,
1 a 1 2 a 2
| i
source and accessible entanglement are no entanglement as the set of separable states is closed.
monotones[28]. Note that also the Renyi-entropies, i.e.
S (ρ)= 1 log(tr(ρα), are no entanglement monotones
α 1−α
for α>1, α= [16], as they are not concave functions IV. PHYSICAL REGION OF Es, Ea AND
6 ∞ OTHER ENTANGLEMENT MEASURES
of the Schmidt coefficients. Hence, S cannot simply
α
be generalized to mixed states by the convex roof con-
struction (see e.g. [16]) for α > 1. However, the Renyi- In this section we investigate the possible values Es
entropies are Schur concave functions for all α and are and/or Ea can take in terms of other measures. For
thus valid entanglement measures for pure states. As measures that are given in terms of a polynomial in the
mentioned before, in contrast to the Renyi-entropies the Schmidt coefficients the Positivstellensatz (see Eq. (1))
source and accessible entanglement are also defined for gives the analytic solution to the problem. We investi-
mixed states. gate also the boundaries of these sets of states in terms
Let us now show that both the source and accessible of the entanglement measures and highlight in some fig-
entanglement are not additive on tensor products, i.e. ures the states in the source and accessible set in terms
ofthe measuresofsomerandomlychosenstate. Further-
Es(a)(ψ φ)=Es(a)(ψ)+Es(a)(φ). (15) more for certain polynomial and some non-polynomial
⊗ 6
measures we plot the values the measures can have for
The property of additivity is especially useful for entan-
low-dimensional bipartite states.
glementmeasuresifmorecopiesofastateareconsidered.
In [9] a related problem has been considered. There,
Thesourceandtheaccessibleentanglementarebothun-
the authors investigate the optimization of functions of
surprisinglynotadditive. The non-additivity inEq.(15)
the form
can be easily proven by choosing two copies of the same
2-qubit state λψ = (λ1,1−λ1). We get for the source Sf(ρ)= f(λi), (17)
entanglement
i
X
E (ψ) = 2(1 λ ), giventhevalueofsomeotherfunctionofthisform. Note
s 1
−
E4→2(ψ) = 4(1 λ )3, (16) that the functions Sf can be written as a sum over a
s − 1 functionofasingleSchmidtcoefficient. Examplesofsuch
E (ψ ψ) = 2(1 λ )2(1+2λ (1+6λ (2λ 1))),
s ⊗ − 1 1 1 1− measuresareentropymeasures,e.g. the entanglementof
formation given by [1]
where we chose the source entanglement of all 2 2 and
×
all 4 4 states that can reach the 2 2 state ψ for the
single×copycase,suchthatwecompa×realsothe| mieasures S(ρA)=Ef(ψ)=−tr(ρAlog(ρA))=− λilog(λi),
with the same dimension. However, for both measures Xi
(18)
thevalueofasinglecopyisnotproportionaltothevalue
with ρ = tr (ψ ψ ). The Schmidt vectors which
A B
of two copies of the same state ψ . For the accessible | ih |
| i aremaximizingandminimizing the functionSf(ρ) given
entanglement we obtain a similar result.
Sf′(ρ), respectively, have been shown in [9] to be
Another obviousproperty ofthe source and the acces-
sible entanglement is that neither of them coincide with 1 λ
1
λ = λ ,λ ,...,λ ,λ = − λ (19)
theentanglementofformationforpurestates. Asamat- { i}max { 1 0 0} 0 d 1 ≤ 1
ter of fact many known entanglement measures reduce −
λ = λ ,...,λ ,λ ,0,...,0 ,λ =1 kλ ,k = 1/λ .
i min 1 1 0 0 1 1
to the entanglement of formation for pure states. Other { } { } − ⌊ ⌋
examples of measuresthat do not have this property are Hence, both classes of states are parametrized by the
thenegativity[20]ortherobustnessofentanglement[21]. single Schmidt coefficient λ . Note that in terms of the
1
Let us end this section with a brief discussion on the parameters p (see Eq. (12)) one can easily see that the
i
faithfulness of E and E . A measure is faithful if it states in Eq. (19) correspond to convex combinations of
s a
vanishes only on separable states and is strictly positive exactly two extreme points of the convex set of sorted
forentangledstates. Inthemultipartitesettingitisclear, vectors. That is for λ and λ there are
i max i min
{ } { }
thatE andE cannotbefaithful,duetotheexistenceof all but exactly two p ‘s equal to zero, i.e. p =
s a i i max
{ }
isolatedstates. Anotherwellknownmultipartitemeasure (p ,0,...,0,p ) and p = 0,0,...,p ,p ,0,..,0
1 d i min k k−1
{ } { }
that is not faithful is the 3-tangle [22], which vanishes with k 1,..,d . It is important to note here that the
∈ { }
forall3-qubitstatesintheW-class. Inthebipartitecase resultsfrom[9]donotapplyforE andE asbothmea-
s a
bothE andE are,however,faithful. Thiscanbeeasily sures involve products of Schmidt coefficients and can
s a
shown as, on the one hand, for any mixed state, there thus not be written as in Eq. (17). In spite of that, the
exists a pure state that can reach this state, e.g. Φ+ states λ and λ play also important roles in
i max i min
| i { } { }
can be transformed into any state ρ, and on the other our investigations as shown in the following sections.
7
A. Relation between source and accessible
entanglement
Here we consider 3 3 and 4 4 bipartite quantum
× ×
states(weskip2 2states,asforthesestatesanymeasure
×
depends only on a single Schmidt coefficient) and show
plots for several polynomial measures. For the formulas
of the plotted measures see Appendix A. In the second
part of this section we consider similar plots but include
theregionofpairsofentanglementmeasuresreachableby
statesinthesourceandaccessibleset(seeEq.(3)). That
is, we consider a random state φ and illustrate which
| i
values of E ,E the states in the source and accessible FIG.2: Sourceentanglementof3×3statesversusthesource
s a
set of φ can have. entanglementofall4×4statesthatreachtheconsidered3×3
| i state|ψi. Notethatthesetwomeasuresuniquelycharacterize
3×3 bipartite entanglement [7].
1. Bipartite 3×3 states
λ = λ ,λ ,1 2λ . That is these states do
Letus firstconsidertwomeasures,thatuniquelychar- { c}min { 1 1 − 1}
not only optimize a function S (ρ) (see Eq. (17)) given
acterize the bipartite entanglement of 3 3 states as f
shown in [7], i.e. the source entanglemen×t E and the another function Sf′(ρ), but also Es given Es4→3. This
s
can also be easily seen using Fig. 1, as the thick dashed
source entanglement of all 4 4 states, that reach a
certain 3 3 state, i.e. E4→3×(see Appendix A). That lines that enclose the set of sorted Schmidt vectors, i.e.
is, given t×he value of E ,Es4→3 the state is (up to LUs) the boundary lines of the set of states excluding LU-
s s equivalent states, correspond to the boundary lines in
uniquely defined.
Fig. 2. As mentioned above these lines are parametrized
FirstnotethatE andE4→3 canleadtoadifferentor-
s s by the convex combination of two extreme points of the
derof3 3statesfor,i.e. fortwostates φ and φ we
× | 1i | 2i set of sorted vectors. This can also be seen in Fig. 1,
can have E (φ ) > E (φ ) but E4→3(φ ) < E4→3(φ ).
s 1 s 2 s 1 s 2 where the dashed lines connect the vertices given by the
This property is also observed with many other entan-
separable state (1,0,0), the maximally entangled 2 2
glement measures such as the negativity and the entan- ×
state (1/2,1/2,0) and the maximally entangled 3 3
glement of formation, as it is a well known fact that dif- ×
state (1/3,1/3,1/3). Thus, for example all states on the
ferent entanglement measures measure different aspects
pink line in Fig. 2 are given by the convex combination
of entanglement and thus, it is not surprising that the
p(1/3,1/3,1/3)+(1 p)(1,0,0), which is equivalent to
order of the states can change for different measures. −
the above defined λ . In the following we will see
However, here the source entanglement and its gener- { a}max
that when consideringdifferent measures the boundaries
alization are very closely related and therefore, measure
are no longer equivalent to the ones in Fig. 1.
similar aspects of entanglement, but still they impose a
The next measures we consider are the source and the
different order on the states. Furthermore, the range of
possible values for the pairs (E ,E4→3) is surprisingly accessibleentanglement ofa 3 3 state. Note that these
s s ×
two measures do not uniquely characterize 3 3 entan-
well confined, as can be seen in Fig. 2. For example, if
×
E (ψ)=0.4, E4→3(ψ) can only have values between ap- glementand the maindifferences to the abovecompared
s s
measures are on the one hand that the range of values
proximately 0.09 and 0.2. That is a state with some
value for E can only have values for E4→3 within a is now much broader and on the other hand that the
s s
boundariesarenolongersolelygivenbythestatesinEq.
small range. When comparing other measures we will
(19). In Fig. 3 the dotted green and the black boundary
see that the range can be much larger than in this case.
correspond to the same states as in Fig. 2. In contrast
We also investigate the boundaries in Fig. 2. Let us
to Fig. 2 the states that maximize the accessible entan-
first note that in all figures one can go on the bound-
glement for a fixed value of the source entanglement are
ary lines always from right to left via LOCC, i.e. from
parametrizedbyλ =1/3, 1/6 λ 1/3forthedotted
one random state on a boundary line we can obtain all 2 3
≤ ≤
otherstatestotheleftofthe boundarydeterministically. red line and λ2 = λ3(1 2λ3), 0 λ3 < 1/6 for the
− ≤
This can be easily checked for all boundaries, that we dotdashed orange line and furthermore, the pink line is
p
parametrized (see below). For all other boundaries we nolongeronthe boundary. Hence, the states inEq.(19)
also checked this behavior numerically. The boundaries arenotenoughtocompletelycharacterizetheboundaries
inFig.2areindeedgivenbythestatesinEq.(19). Inter- of Fig. 3 and the boundaries are thus also not equal to
estingly, the dashed pink line corresponds to the states the ones in Fig. 1.
withλ =λ ,i.e. λ = λ ,1−λ1,1−λ1 ,theblack Given the operationalmeaning of E and E it is now
2 3 { a}max { 1 2 2 } s a
line and the dotted green line are given by λ = 0 and appealing to investigate what the entanglement proper-
3
λ = λ , respectively, i.e. λ = λ ,1 λ ,0 , ties (in terms of all these measures) are for the states
1 2 b min 1 1
{ } { − }
8
However, there exist of course incomparable states. In-
terestingly, even though they uniquely characterize the
entanglement of the states, there exist also incompara-
ble states for which both E and E4→3 are smaller but
s s
the states are not reachable by φ . The boundary lines
| i
of the pink and yellow set in Fig. 4 are as the bound-
ary lines of the set of all states in Fig. 2 equivalent to
the ones in Fig. 1, in which the source and accessible set
of a state φ is shown in terms of the Schmidt coeffi-
| i
cients. Again this is only the case if we consider E and
s
itsgeneralizationsandwewillseeespeciallyforthemore
involved4 4states thatthe boundariesofthe pink and
×
FIG. 3: Source entanglement versus the accessible entangle- yellow sets are no longer simply parametrized by states
ment of a 3×3 state |ψi. with E (ψ) = E (φ) for some i if we compare different
i i
measures.
Let us now consider the same state φ and its source
that can be reached by and for the ones that can be andaccessiblesetrespectively,asafunc|tiionofE versus
a
transformed into one specific state φ . That is, we ana- E . AscanbeseeninFig.5thereoccurmoreboundaries.
| i s
lyze given a state φ with a certain value of Es and Ea, In Fig. 5 the green and dashed orange line are the same
| i
which pairs of values are in the source and accessible set
of this state φ . In order to illustrate that, we include
| i
in Fig. 4, 5 the source and accessible entanglementof all
stateswhichareinthesourceoraccessiblesetof φ . Let
| i
us first investigate the source entanglement and its gen-
eralization. In the Fig. 4 the connected pink (lower left)
FIG. 5: Es versus Ea including the source and the accessi-
ble set of a 3×3 state |φi with λ = (0.52,0.28,0.2) with
φ
boundaries.
as in Fig. 4. The dotted black line corresponds to states
FIG. 4: Es versus Es4→3 including the source and the acces- |ψi with λ2 = λ3. The main difference to Fig. 4 is that
an additional boundary, namely the dotted black line, is
sible set of a 3×3 state |φi with λ = (0.52,0.28,0.2) with
φ
boundaries. required. Asthesemeasuresdonotuniquelycharacterize
the entanglement, one point in Fig. 5 can correspond to
differentstates. Thisisforexamplethecaseforthepoint
area and the connected yellow (upper right) area corre-
inthe pink accessiblesetwherethe dottedblackandthe
spond to the entanglementof the states in the accessible
green line intersect. The state on the green line is of
and the source set of a state φ , respectively. Hence,
| i the form (λ ,λgreen,λgreen), whereas the state on the
these sets include all states that can either be reached φ1 2 3
dotted black line is given by (λblack,λblack,λblack).
or can reach a certain state φ . The boundaries of the 1 2 2
| i
two sets, i.e. the green and the dashed orange line, are Interestingly,it seems to be a generalfeature (see also
given by states with one of the two entanglement mono- the 4 4 case), that for E and its generalizations the
s
×
tones E (ψ),E (ψ) being equal to that of the state φ . boundaries are easy to determine, whereas for the plots
2 3
| i
More precisely, the green line corresponds to states with including other measures, this is not the case. In the
E (ψ) = E (φ) and the orange line is parametrized by firstcase,the states correspondingto the boundariesare
2 2
states fulfilling E (ψ)=E (φ). Moreover, as we can see either of the same form as in Eq. (19) or in case of the
3 3
inFig.4moststatesareLOCC-comparable,i.e. thestate boundaries of the pairs corresponding to states in M
s
φ can either reach or be reached by most of the states. and M , i.e. the pink and yellow sets, have fixed values
a
| i
We can conclude this from Fig. 4 as any point in the for the entanglement monotones E . We elaborate on
i
figure corresponds to a single state, due to the fact that that also in the next subsection, where 4 4 states are
E and E4→3 uniquely characterize the entanglement. considered. ×
s s
9
2. Bipartite 4×4 states
We consider here the case of 4 4 states. In addition
×
to the investigationsperformed for 3 3 states, we com-
×
pareherealsothevaluesobtainablefortwo2-qubitstates
(viewed as 4 4 states, see Fig. 8). Let us first consider
×
the measures, which uniquely characterize the entangle-
ment of 4-dimensional states, i.e. E ,E5→4,E6→4 (as
s s s
can be also seen with the help of the Positivstellensatz).
The range ofpossible values is againquite constrainedif
we consider E and its generalizations, whereas we get
s
a wider range for values of the pair (Es,Ea). Similar FIG. 7: Es versus Es5→4 versus Es6→4 of the source (yellow)
and accessible (pink) set of a state |φi=(0.4,0.35,0.2,0.05).
Note that we do not show all the states that are neither in
thesource nor theaccessible set of |φi.
that at least one of the monotones E is equal to the
i
corresponding monotone of the state φ and if only one
| i
fulfills this equality in addition two Schmidt coefficients
of the states are equal to each other. Thus, these states
are also similar to the ones given in Eq. (19).
Similar to the 3 3 case the situation gets less trans-
×
parent if we consider E versus E . There, again the
s a
boundaries are not all given by those described above.
Note again that E and E do not uniquely charac-
s a
terize the entanglement of 4 4 states and thus, one
×
FIG. 6: Es versusits generalizations Es5→4 and Es6→4. point in the plot below can correspond to several differ-
ent states. In the subsequent figure we also illustrate in
orangethe entanglement measures of 4 4 states, which
to the 3 3 case in Fig. 6 the boundaries are again ×
× are of the form Φ Ψ , where both Φ and Ψ are 2-
given by the states in Eq. (19). Thus, the pink line cor- | i⊗| i | i | i
responds to the state λ = λ ,1−λ1,1−λ1,1−λ1 qubitstates. Itisinterestingto observethatthese states
{ a}max { 1 3 3 3 } almost optimize E given E . Note, however,again that
and the dotdashed black, the dashed blue and the dot- a s
these measures do not uniquely define the states. The
ted greenline are givenby the states λ = λ ,1
b min 1
{ } { − valuesofpairs(E ,E )forwhichtherealsoexistsastate
λ ,0,0 , λ = λ ,λ ,1 2λ ,0 and λ = a s
1 c min 1 1 1 d min
} { } { − } { } ψ = Φ Ψ canbereadoffcombiningFig.8withthe
λ1,λ1,λ1,1 3λ1 ,respectively. Hence, the boundaries | i6 | i⊗| i
{ − } figurefor4 4stateswhichistheequivalenttoFig.19for
are again completely characterized by the states in Eq. ×
3 3states. Notethattheredlineisparametrizedbyall
(19), as it is for instance also the case considering only ×
statesthataregivenbytwocopiesofa2-qubitstate,i.e.
E and E5→4.
s s Ψ Ψ . The maindifference ofFig.8to the 3 3case
Let us now, as in the 3 3 case, investigatethe values | i⊗| i ×
×
of E and its generalizationsfor states in the source and
s
accessible set of a certain state φ . To illustrate the
| i
result more clearly we show in Fig. 7 only the values
of the three measures for all states in either the source
or accessible set of the state φ and the boundaries (in
| i
light blue) of the set of all 4 4 states. The boundaries
×
of the two sets are as in the 3 3 case given by states
×
for which one of the entanglement monotones E is fixed
i
by the value it takes for the state φ . More precisely,
| i
the states on the gray line are parametrized by E (ψ)=
4
E (φ)andλ =λ ,onthedottedbrownlinebyE (ψ)=
4 2 3 3
E (φ)andE (ψ)=E (φ),onthedotdashedbluelineby
3 4 4
E (ψ)=E (φ) andλ =λ ,onthe dashedmagentaline
4 4 3 4
by E (ψ)=E (φ) and E (ψ)=E (φ), on the greenline FIG. 8: Source entanglement versus the accessible entangle-
2 2 3 3
by E (ψ) = E (φ) and λ = 0, on the turquoise line by ment of all 4×4 states |ψi.
2 2 4
E (ψ)=E (φ)andλ =λ andonthedotdashedpurple
2 2 3 4
line by E (ψ) = E (φ) and λ = λ . To summarize the isthatnotallstatesinEq.(19)lieevenontheboundary
4 4 1 2
statesontheboundariesofthepinkandyellowsetsfulfill ofthefigure. ThatisthestatesonthepinklineinFig.6,
10
givenby λ = λ ,1−λ1,1−λ1,1−λ1 ,arenotopti- the problem of deciding whether a polynomial is SOS or
{ a}max { 1 3 3 3 }
mizingE (ψ)forafixedvalueofE (ψ). Theotherstates not as a SDP feasibility problem, i.e.
a s
from Eq. (19) on the dotdashed black, the dotted green
and the dashed blue line are still lying on the boundary find Q
(for a parametrization see above). The other boundary subject to tr(A Q)=c (21)
α α
couldofcoursebeeasilycomputedbymaximizingE (ψ)
a Q 0.
given E (ψ) numerically [29]. ≥
s
The linear equality constraints in this feasibility prob-
lem are derived from F(x) = zTQz by comparing the
B. Relation between source and accessible
coefficients in these polynomials. Note that any SDP
entanglement by the Positivstellensatz
feasibility problem possesses a dual problem which gives
awitness,provingthattheprimaryproblemhasnosolu-
In this section we review the idea of efficiently find- tion. In the case investigated here, the potential witness
ing certificates for when a system of polynomial equa- can be found by writing F(x) = zTQz = tr(zzTQ) and
tions and inequalities has no solution in R (see [8]). We replacingzzT byamatrixW,thatfulfillsthesamelinear
will show then how to use these certificates to find the relationsamongitsentriesaszzT (relaxingthecondition
possible region of two (or more) entanglement measures of having rank 1). As long as Q represents the original
which are given as polynomial functions of the Schmidt polynomial F(x) the tr(WQ) does not depend on the
coefficients. For instance, this method can be used to specific choice of Q. Then the dual problem is equal to
determine analytically all possible pairs (E ,E ), which
s a
are accessible by a state. find W
It has been shown in [8] that one can efficiently find
subject to tr(WQ)<0 (22)
these certificates by using the Positivstellensatz and fix-
W 0
ingtheoveralldegreeofEq.(10),asthentheproblemcan ≥
be written as a semidefinite program (SDP). Hence, by wij =wkl for (i,j),(k,l) I,
{ }∈
solvingtheseSDPsandobtainingcertificateswecancom-
pletely solve the problem of when certain values of the where I is chosen such that the W-matrix fulfills the
source entanglement and its generalizations correspond samelinearconditionsamongtheentriesaszzT [30]. For
to a physical state or not. Note that this would not be instance for z = [x21,x22,x1x2] the entries of W have to
possibleformeasuresSf(ρ)asinEq.(18),astheyareno fulfill w12 =w33.
polynomials. For each point in the figures lying outside Using these results on SOS polynomials we now re-
the boundaries of the blue sets in all previous figures we view the method of finding bounded-degree certificates
find a certificate that tells us, that there exists no state from the Positivstellensatz. In order to do so, the over-
having these values for the entanglement measures. all degree d0 of Eq. (10), i.e. deg(f + g2 + h) = d0
To explain the idea of obtaining these certificates by is fixed [8]. Then the polynomial g, which is gener-
solving a SDP, we first recall the definition of sum of ated by the multiplicative monoid of the set {gk}tk=1,
squares (SOS) polynomials and the fact that the exis- is either equal to 1 if t = 0 or it is given by g =
tence of a SOS decomposition for polynomials can be ti=1gim, with m chosen such that the degree of g2
decidedbysolvingaSDP feasibilityproblem(see [8]and is less than or equal to d0. The polynomial f in the
references therein). The computational tractability of Qcone of the set of inequalities {fi}si=1 is parametrizedby
the SOS polynomials together with fixing the overallde- f =s0+s1f1+...+ssfs+s12f1f2+...+s12...sf1...fs,with
gree of Eq. (10) leads then also to the relaxations in the s{i1,...,is} SOS polynomials of degree less than or equal
Positivstellensatz as explained below. to d0. Furthermore, the polynomial h in the ideal of the
noAmiSaOl oSfpeovleynnodmegiarleeF,(txh)a,txc∈anRbmeiswarirtteeanl-vianlutehdepfoorlym- speotly{nhojm}mji=a1lsitsieoqfuadlegtorehe a=gat1inh1le+ss..t.h+antmohrmeqwuiatlhtsoomd0e.
F(x) = f (x)2, with f (x) R[x]. Clearly not every The correspondingSDP feasibility problemis then given
nonnegativie ipolynomial isi SO∈S [23], e.g. the Motzkin by
formM(Px,y,z)=x4y2+x2y4+z6 3x2y2z2 isnonnega-
− find Q ,
tive but cannotbe written asa SOS. A polynomialF(x) {i1,...,is}
of degree 2d is SOS iff it can be written as a quadratic subject to f +g2+h=0 (23)
homogeneous polynomial in z, where the vector z con-
Q 0,
tains all monomials of x with degree less or equal to d, {i1,...,is} ≥
i.e. with s = zT Q z
{i1,...,is} {i1,...,is} {i1,...,is} {i1,...,is}
F(x) is SOS Q 0: F(x)=zTQz, (20) and the monomial vector z{i1,...,is} =
⇔∃ ≥ [1,x ,x ,...,x ,x x ,...,xk ], with k d /2. Here,
where z =[1,x ,x ,...,x ,x x ,...,xd ] andthe positive k is1ch2osen fmor e1ac2h z m , such ≤that0the overall
1 2 m 1 2 m {i1,...,is}
semidefinite m m matrix Q is constant. Using this degree of Eq.(10) is equalto d . Note that the equation
0
decomposition o×f SOS polynomials it is easy to restate f +g2+h = 0 leads to the equality constraints for the