Candidates for Universal Measures of Multipartite Entanglement

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by  Samuel R. Hedemann

Candidates for Universal Measures of Multipartite Entanglement
SamuelR.Hedemann
P.O.Box 72, Freeland, MD 21053, USA
(Dated: January 17, 2017)
We propose and examine several candidates for universal multipartite entanglement measures.
The most promising candidate for applications needing entanglement in the full Hilbert space is
the ent-concurrence, which detects all nonlocal correlations while distinguishing between different
typesofdistinctlymultipartiteentanglement,andsimplifiestotheconcurrencefortwo-qubitmixed
states. For applications where subsystems need internal entanglement, we develop the absolute
ent-concurrence which detects the entanglement in the reduced states as well as the full state.
PACSnumbers: 03.67.Mn,03.65.Ud
7
1 I. INTRODUCTION
0 GM-entangled
2
Entanglement[1,2],inthesimplestcaseofpurequan-
n tumstates,iswhenastatesuchas|ψ(cid:105)=a|1(cid:105)⊗|1(cid:105)+b|1(cid:105)⊗ 2-separable
a
J |2(cid:105)+c|2(cid:105)⊗|1(cid:105)+d|2(cid:105)⊗|2(cid:105),where|a|2+|b|2+|c|2+|d|2 =1, .
.
cannotbefactoredasatensorproductofpurestatessuch .
2
1 as|ψ(cid:105)=(w|1(cid:105)+x|2(cid:105))⊗(y|1(cid:105)+z|2(cid:105)), where|w|2+|x|2 =1
and |y|2+|z|2 = 1, where we have expressed each qubit
(N −1)-separable
] inagenericbasis{|1(cid:105),|2(cid:105)}(ourconventioninthispaper,
h
and these kets do not imply Fock states [3]). 2-separable
p
t- p Q∈u[a0n,1tu],m(cid:80)mipxed=s1t,ataensdaarell ρ|ψ≡(cid:105)(cid:80)arjeppj|uψrje(cid:105).(cid:104)ψFjo|,r wbihpaerre- (N −1)N-s-espeapraarbalbele
n j j j j
a tite systems,thosecomposedoftwosubsystems(modes),
u mixed states ς ≡ς(1,2) are separable if and only iff (iff)
q
[ ς(1,2) =(cid:88) p ς(1)⊗ς(2), (1)
j j j
1 j FIG. 1: (color online) Relationships of k-separabilities [4].
v Each k-separability implies all lower-k-separabilities, and is
2 where parenthetical superscripts are mode labels, and necessary for all higher-k-separabilities. Thus, N-separable
8 each ς(m) is pure. Each mode-m reduced state states are also (N − 1)-separable, all the way down to 2-
j
7 ˇς(m)≡tr (ς), where m means “not m” (see App.A), separable, butsome2-separablestatesarenot3-separableor
3 admits amdecomposition of the form ˇς(m) ≡(cid:80) p ˇς(m) higher. The “1-separable” states are “genuinely multipartite
j j j
0 =(cid:80) p ς(m) as proved in App.B, so if we only knew re- (GM) entangled,” also defined as all states that are not 2-
1. ductjionjsjˇς(m), we could search decompositions of each separable. Thus,theGM-entangledregionisstrictlycrescent-
0 one to find the pair with matching sets {p } such that shaped here, while the k-separable regions are each ellipse-
j
7 ς(1,2)=(cid:80) p ˇς(1)⊗ˇς(2). Therefore knowledge of the re- shaped and coinciding with parts of all lower-k-separabilities.
1 j j j j (Theshapesarearbitrary,merelyrepresentingrelationships.)
ductions allows reconstruction of the parent state ς.
:
v For N-partite (N-mode) systems, separability can oc-
i cur in more than one way. For example, two different By definition, k-separability of pure states is when
X
any member of the set of all possible k-partitions is k-
3-qubit pure states could have separable bipartitions as
ar ρ=ˇρ(1)⊗ˇρ(2,3) and (cid:37)=ˇ(cid:37)(1,2)⊗ˇ(cid:37)(3), so we call both of separable,forafixedvalueofk. Thus,forapureρ(1,2,3),
if only ρ(2|1,3) is 2-separable, but not ρ(1|2,3) or ρ(3|1,2),
them biseparable or 2-separable, even though the mode
then that is sufficient for ρ(1,2,3) to be 2-separable.
groups that are separable for each state are different.
Mixed states are k-separable if a decomposition exists
These different mode-groupings are called partitions,
for which all pure decomposition states are at least k-
which are definitions of new modes composed of (but
separable, with one being exactly k-separable [4] (since
not subdividing) the original modes m , as explained in
k
App.C. For example, a tripartite state like ρ(1,2,3) can higher-than-k-separabilities are also k-separable, we can
havethreeuniquebipartitionsρ(1|2,3),ρ(2|1,3),ρ(3|1,2) and just say that all decomposition states need to be k-
oneuniquetripartitionρ(1|2|3) =ρ(1,2,3), showingthatin separable). For example, the state
theabsenceofpartitions,thecommasare thepartitions. ρ=(cid:80)p ρ(1)⊗ρ(2,3)+(cid:80)q ρ(2)⊗ρ(1,3)+(cid:80)r ρ(3)⊗ρ(1,2),
To handle the general N-partite phenomenon of sep- j j j k k k l l l
j k l
arability of a given partitioning having the potential to (2)
(cid:80) (cid:80) (cid:80)
occur in different ways, the notion of k-separability was where p ,q ,r ∈[0,1], ( p )+( q )+( r )=1,
j k l j j k k l l
developed [4–17], as depicted in Fig.1. with pure entangled bipartite states ρ(2,3),ρ(1,3),ρ(1,2)
j k l
2
andpureρ(1),ρ(2),ρ(3),is2-separable,eventhougheach Yet, 2-separability alone does not detect the strong
j k l
group of pure decomposition states is separable over dif- nonlocal correlations contained within the Bell states of
ferent bipartitions [4]. ρ in(4). Thus,whileN-entanglementcandetectthe
|ΦBP(cid:105)
By definition, for k > 1, the absence of k-separability presence of all nonlocal correlations but cannot distin-
is k-entanglement. For example, the absence of 2- guish k-separabilities, lone sub-N k-entanglement mea-
separability (biseparability) is 2-entanglement, known as sures are not sufficient to detect the presence of all non-
“genuinely multipartite (GM) entanglement.” Figure 2 local correlations, but can verify k-separability.
shows how all the k-entanglements are related. Therefore our main goal here is to define a few can-
didate universal entanglement measures that distinguish
between types of multipartite entanglement without dis-
2-entangled carding information about nonlocal correlations, which
individual k-entanglement measures cannot do alone.
.
. The building-block of our candidate measures is the
.
N-entanglement measure the ent [21], given by
(N −1)-entangled(N−1)-entangled Υ(ρ)≡Υ(ρ,n)≡ 1 (cid:32)1− 1 (cid:88)N nmP(ˇρ(m))−1(cid:33),
N-entangled M(L ) N n −1
∗ m
m=1
(5)
N-entangled
for pure states ρ of an N-mode n-level system where
modemhasn levelsandn≡(n ,...,n ),sothatn=
m 1 N
N-separable n ···n = det(diag(n)), N = dim(n), P(σ) ≡ tr(σ2) is
1 N
the purity of σ, and ˇρ(m) is the n -level single-mode re-
m
ductionofρformodem(seeApp.A). Thenormalization
factor M(L ) ≡ M(L ,n) is given in App.E. Basically,
∗ ∗
theentmeasureshowsimultaneouslymixedtheˇρ(m) are.
FIG.2: (coloronline)Relationshipsofk-entanglements.Here,
We will also use the partitional ent Υ(m(T))(ρ)≡
a given k-entanglement implies all higher-k-entanglements,
Υ(ˇρ(m),n(m(T))),allowingustorepartitionρ’sreduction
while being itself necessary for all lower-k-entanglements.
Thus, each k-entanglement region is strictly crescent-shaped ˇρ(m) (including the nonreduction ˇρ(N) ≡ˇρ(1,...,N) =ρ)
and coinciding with parts of all higher-k-entanglements, with into new mode groups m(T) ≡(m(1)|...|m(T)) of levels
N-entanglement being the thickest crescent, and all lower- n(m(T)) ≡(n ,...,n )(seeApp.C)tomeasurethe
m(1) m(T)
k-entanglements having progressively thinner crescents. The T-entanglement of any T mode groups (see [21] for de-
entire region that is not N-separable is N-entangled. tails). Finally, when ρ or ˇρ(m) aremixed,weuseconvex-
roofextensionsasΥˆ andΥˆ(m(T)) (see[21,App.J]),which
are minimum average ents over all decompositions. The
On the other hand, N-partite states are entangled iff
main sections of this paper are:
they are N-partite entangled, as we prove in App.D.
This means that the complete absence of nonlocal cor-
relations can only occur in N-partite states that are N- I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1
II. CandidatePure-StateEntanglementMeasures 3
separable,meaningstateswithanoptimaldecomposition,
III. Tests of Candidate Pure-State Measures. . . 3
IV. Candidate Mixed-State Measures . . . . . . . . 6
ς(1,...,N) =(cid:88) p ⊗N ς(m) =(cid:88) p ς(1)⊗···⊗ς(N), V. Tests of Candidate Mixed-State Measures . 6
j j j j j
j m=1 j VI. Ent-Concurrence. . . . . . . . . . . . . . . . . . . . . 8
(3)
VII. Absolute Ent-Concurrence . . . . . . . . . . . . . 9
wheretheς(m) arepure. N-partitestatesthatcannot be
j VIII.Conclusions. . . . . . . . . . . . . . . . . . . . . . . . .11
expanded as (3) are N-partite entangled (N-entangled). App. Appendices . . . . . . . . . . . . . . . . . . . . . . . . .12
However,N-entanglementdoesnotdistinguishbetween A. Brief Review of Reduced States . . . . . . . . .12
different types of k-entanglement. For example, given B. N-Separability of N-Partite States Implies
Reconstructability by Smallest Reductions.12
ρ and ρ ≡ρ ⊗ρ , (4) C. Definition of Partitions . . . . . . . . . . . . . . . .12
|ΦGHZ(cid:105) |ΦBP(cid:105) |Φ+(cid:105) |Φ+(cid:105) D. ProofthatN-PartiteStatesareEntangledIf
where|Φ (cid:105)≡√1 (|1,1,1,1(cid:105)+|2,2,2,2(cid:105))isa4-qubitGHZ and Only If they are N-Partite Entangled. .13
state [18–G2H0Z] wher2e ρ ≡|A(cid:105)(cid:104)A|, and |Φ+(cid:105)≡√1 (|1,1(cid:105)+ E. Normalization Factor of the Ent . . . . . . . . .14
|A(cid:105) 2 F. True-Generalized X (TGX) States . . . . . . . 15
|2,2(cid:105)) is a 2-qubit Bell state so that ρ is a “Bell-
|ΦBP(cid:105) G. Full Set of 4-Qubit Maximally N-Entangled
product state,” since both ρ and ρ have
|ΦGHZ(cid:105) |ΦBP(cid:105) TGX States Involving |1(cid:105). . . . . . . . . . . . . . .15
maximal mixing in all single-mode reductions, an N-
H. QuantumMixedStatesCannotBeMadefrom
entanglement measure would report both states as be- Statistical Mixtures of Pure States . . . . . . .15
ing equally entangled, despite ρ being 2-entangled
|ΦGHZ(cid:105)
while ρ is 2-separable. andwhereverpossible,detailsareputintheAppendices.
|ΦBP(cid:105)
3
II. CANDIDATE PURE-STATE C. Full Distinguishably Multipartite (FDM) Ent:
ENTANGLEMENT MEASURES The Ent-Concurrence
Here we introduce the candidate entanglement mea-
The FDM ent (or the ent-concurrence) for pure ρ is
suresunderconsideration. Eachwillusetheentfrom(5)
as well as its various different forms due to partitioning. N
1 (cid:88)
See [21] for full explanations. We start with pure-state Υ (ρ)≡ Υ (ρ), (10)
FDM M DMk
measures, and discuss mixed input after initial tests. FDM k=2
whereM isanormalizationfactor,andwedefinethe
FDM
distinguishably multipartite k-ent (DM ent) as
A. Full Genuinely Multipartite (FGM) Ent k
The FGM ent for pure ρ is ΥDMk(ρ)≡ MD1Mk (cid:88)h{N=k}1(cid:113)Υ(N(hk))(ρ), (11)
N where M is a normalization factor, {N} are Stirling
1 (cid:88) DMk k
ΥFGM(ρ)≡ M ΥGMk(ρ), (6) numbers of the second kind as in (9), and {Υ(Nh(k))(ρ)}
FGM k=2 is the set of all N-mode k-partitional ents.
TheFDMentΥ isameasureofsimultaneousDM
where M is a normalization factor, and FDM k
FGM
ents Υ , and the Υ measure not only the combi-
DMk DMk
ΥGMk(ρ)≡min({Υ(N(hk))(ρ)}), (7) nhoawtioenquoaflalylldpiosstsriibbuleteNd-tmhoedyeakre-p,arrattiitniognsatlaetnestsf,obruwthailcsho
whichistheGMk ent,where{Υ(N(hk))(ρ)}isthesetofall all N-mode k-partitional ents have the highest combina-
N-mode k-partitional ents, each labeled by h. Note that tion and are the most equal and numerous as having the
[21]definedtheGM ent asΥGM(ρ)≡ΥGM2(ρ), whichis highest DMk ent. √
the “ent-version” of GM concurrence CGM(ρ) [5]. Thisisbasedonpseudonorm |x|1/2 ≡(cid:80)k xk; xk ≥0,
SinceΥFGMsumsallGMkents,itisameasureofsimul- which obeys |x|1/2 = 0 ⇒ x = 0 and the triangle in-
taneous (GM)-k-entanglements; that is, it rates states equality, and although |ax| (cid:54)=|a|·|x| , that does not
1/2 1/2
for which the combination of all their k-entanglements is matter here, since our “vectors” are really just lists of
maximal as being “maximally FGM-entangled.” scalars. The main reason we use this pseudonorm is be-
cause it rates an equally distributed 1-norm-normalized
vector such as x = (0.25,0.25,0.25,0.25) as having the
B. Full Simultaneously Multipartite (FSM) Ent highest “1-norm” value out of all vectors of the same
2
1-norm, such as y=(0,0,0.5,0.5) or z=(0,1,0,0).
The FSM ent for pure ρ is For two qubits (N = 2), the FDM ent is the concur-
rence C [22, 23], since Υ=C2 as proven in [21], so that
N
1 (cid:88)
Υ (ρ)≡ Υ (ρ), (8) (cid:112)
FSM MFSM SMk ΥFDM(ρ)=ΥDMN(ρ)= Υ(ρ)=C(ρ), (12)
k=2
(extendible to mixed states, as shown later). Therefore,
whereM isanormalizationfactor, andwedefinethe
FSM if we let the N-mode k-partitional ent-concurrence be
simultaneously multipartite k-ent (SM ent) as
k
(cid:113)
ΥSMk(ρ)≡ MS1Mk (cid:88){hN=k}1Υ(N(hk))(ρ), (9) where{Υ(N(hk))(CρΥ()N}(hiks))t(hρe)s≡etofΥa(lNl(hNk)-)m(ρo),dek-partitio(1n3a)l
wbehresreof{tNkh}e≡seck1o!n(cid:80)dkjk=in0(d−,1(cid:0))kjk(cid:1)−≡j(cid:0)kjj(cid:1)!(jkkN−!j)a!r,e{ΥSt(iNrlh(ikn)g)(ρn)u}mis- e{nCtΥ(sN,h(tkh))e(nρ)t}hefoDrMakgivenent Υk.DMTkheirsefsoimrep,ltyheaΥ1-FnDoMrmino(f1a0l)l
the set of all N-mode k-partitional ents, and MSMk is canalsobecalledtheent-concurrence CΥforpurestates.
a normalization factor. Υ (ρ) detects the presence of
SMk
any nonlocal correlation among all possible k-partitions
of ρ. Thus, it cannot ignore entanglement within par- III. TESTS OF CANDIDATE PURE-STATE
MEASURES
ticular k-partitions just because a different k-partition is
separable as Υ (ρ) can.
GMk For simple tests, we use 4-qubit (2×2×2×2) states,
TheFSMentΥ isameasureofsimultaneous SM
FSM k
ents; which means it is a measure of the combination |ΦGHZ(cid:105) ≡ √1 (|1,1,1,1(cid:105)+|2,2,2,2(cid:105))
2
of all N-mode partitional ents, so it is a sum of all 2- |ΦBP(cid:105) ≡ √1 (|1,1,1,1(cid:105)+|1,1,2,2(cid:105)+|2,2,1,1(cid:105)+|2,2,2,2(cid:105))
partitionalentsofρ,all3-partitionalentsofρ,alltheway 4
uptotheN-partitionalent(theentitself). Thus,ifthere |ΦF(cid:105) ≡ √14(|1,1,1,1(cid:105)+|1,1,2,2(cid:105)+|2,2,1,2(cid:105)+|2,2,2,1(cid:105))
is any nonlocal correlation between any mode groups of |ψW(cid:105) ≡ √1 (|1,1,1,2(cid:105)+|1,1,2,1(cid:105)+|1,2,1,1(cid:105)+|2,1,1,1(cid:105))
4
ρ,theFSMwilldetectit,andstatesthatmaximizeitare |ψ (cid:105)≡ (cid:80)16 a |k(cid:105); (cid:80)16 |a |2 =1,
Rand k=1 k k=1 k
“maximally FSM-entangled.” (14)
4
where |Φ (cid:105)≡|Φ+(cid:105)⊗|Φ+(cid:105) is the Bell product from values, and actually outperforms |Φ (cid:105) in Υ , Υ ,
BP BP GM2 GM3
(4), |ψW(cid:105) is the W state [24], and |ψRand(cid:105) is a ran- and Υ(cid:101)FGM, despite its lower ΥGMN.
dom 16-level pure state, where here and throughout Interestingly, |ψRand(cid:105) reached a higher Υ(cid:101)FGM than all
we use basis abbreviation {|1(cid:105),|2(cid:105),...,|n(cid:105)} ≡ {|1,1,1,1(cid:105), other test states, since its Υ and Υ are higher
||Φ1,F1,(cid:105)1a,2r(cid:105)e,t.a.k.e,n|2f,2ro,2m,2t(cid:105)h}.esTethme asxtaimteasll|yΦNGH-eZn(cid:105),ta|nΦgBlePd(cid:105),trauned- tshliagnhtltyholosweeorftthhaeno1t.hTerhisstaptreGosvM,e2swbhyileexiatsmGMΥpl3GeMthNatisthsetrilel
generalized X (TGX) states (see App.F, [21, App.D], are nonmaximally-N-entangled FGM-entangled states
and [25]), chosen from the subset including |1(cid:105), gener- with higher Υ(cid:101)FGM than maximally N-entangled states.
atedbythe13-stepalgorithmof[21]. Thus,(14)provides
threemaximally N-entangled statesandtwoother kinds
of states for comparison.
Thesubset{|Φ (cid:105),|Φ (cid:105),|Φ (cid:105)}waschosenfromthe
GHZ BP F B. Tests and Analysis of FSM Ent
full set of N-entangled TGX states since they produced
distinct results for the measures under test and included
Here,Fig.4appliesthesametestsasinFig.3,butthis
|Φ (cid:105). See App.G for the full set initially used.
GHZ
time for the FSM ent of (8).
As an example showing the partitional ents involved,
unnormalized expansion of the FGM ent in (6) is

13.44
Υ(cid:101)FGM(ρ)=ΥGM2 +ΥGM3 +ΥGMN, (15) 12.76
where Υ =min{Υ(1|2,3,4),Υ(2|1,3,4),Υ(3|1,2,4),Υ(4|1,2,3), Υ(cid:101)FSM
ΥΥ((11|,32||23,,44))G,,ΥΥM((121|,43||22,,34)),,ΥΥ((21|,34||12,,43)),}Υ,(a2|n4d|1,Υ3)G,ΥM(33|4=|1,m2)}in,a{nΥd(1Υ|2G|3M,4N), ΥΥΥ(cid:101)(cid:101)(cid:101)SSSMMM23N
=min{Υ(1|2|3|4)}=Υ, all using N-mode partitional ents. max
6.667 R
5.778 maxR
max2
A. Tests and Analysis of FGM Ent max3
maxN
Figure3explorestheFGMentoftheteststatesin(14), 1.000
0.000
withonlytheresultsforthe|ψRand(cid:105)thathadthehighest |ΦGHZ(cid:105) |ΦBP(cid:105) |ΦF(cid:105) |ψW(cid:105) |ψRand(cid:105)
unnormalizedFGMentΥ(cid:101)FGMoutof1000randomstates.
FIG. 4: (color online) Unnormalized FSM ent Υ(cid:101)FSM of (8)
2.691 for the test states of (14), with only the |ψRand(cid:105) that max-
2.556 imizes it over 1000 random pure states. The unnormal-
Υ(cid:101)FGM ized SMk ents Υ(cid:101)SMk of (9) are also shown, and the maxi-
ΥΥGGMM23 mmuaxmRo≡femacahx(oΥ(cid:101)veFrSMal)latpesptlisetsattoesailslnmoanxrkan≡dommaxte(sΥ(cid:101)tSsMtakt)e,sw,ahnilde
ΥGMN maxR ≡max(Υ(cid:101)FSM(ρ|ψRand(cid:105))).
max
R
maxR
1.000
00..981029 mmaaxx23 asH|Φere,w(cid:105),ebsueettthhaattb|ΦotBhP|(cid:105)Φhast(cid:105)haensdam|Φe Υ(cid:101)(cid:105)SuMnkdaenrpdeΥr(cid:101)foFrSmM
maxN |ΦF(cid:105)GinHtZermsofΥ(cid:101)SM2,Υ(cid:101)SMG3H,aZndΥ(cid:101)FSMB,Pdespiteallthree
states being maximally N-entangled. This time, |ψ (cid:105)
W
0.000
underperforms all other test states in every area, while
|ΦGHZ(cid:105) |ΦBP(cid:105) |ΦF(cid:105) |ψW(cid:105) |ψRand(cid:105)
|ψ (cid:105)outperforms |Φ (cid:105)and|Φ (cid:105),whilestillunder-
Rand GHZ BP
FIG. 3: (color online) Unnormalized FGM ent Υ(cid:101)FGM of (6) performing |ΦF(cid:105), suggesting that |ΦF(cid:105) may be maximal
for the test states of (14), with only the |ψ (cid:105) that maxi-
Rand in all quantities being measured.
mizesitafter1000randompurestatesweretested. Thenor-
Thus, this proves by example that some maximally
malizedGM entsΥ of(7)arealsoshown,andthemaxi-
mumofeachkoverallGtMesktstatesismax ≡max(Υ ),while N-entangled states are more FSM-entangled than oth-
maxR≡max(Υ(cid:101)FGM)appliestoallnonrkandomtestGsMtaktes,and ers, even |ΦGHZ(cid:105), and suggests that maximally FSM-
maxR ≡max(Υ(cid:101)FGM(ρ|ψRand(cid:105))). entangled states may also be maximal for all ΥSMk.
As Fig.3 shows, the maximally N-entangled (N = 4)
states|Φ (cid:105)and|Φ (cid:105)haveidenticalresultsforallΥ
GHZ F GMk C. Tests and Analysis of FDM Ent
and Υ(cid:101)FGM, but the maximally N-entangled |ΦBP(cid:105) has
ΥGM2 =0 and therefore a lower Υ(cid:101)FGM, despite match-
ing |Φ (cid:105) and |Φ (cid:105) for Υ . As expected, |ψ (cid:105) is Here we apply the same tests as in Sec.IIIA and
GHZ F GMN W
not maximal in any quantity, but still has fairly high Sec.IIIB, this time to the FDM ent of (10).
5
13.70 fore the presence of true entanglement. Stated another
13.48
way; the presence of separability between all partitions is
Υ(cid:101)FDM necessary and sufficient for the absence of all nonlocal
Υ(cid:101)DM2 correlations, and thus the absence of true entanglement.
Υ(cid:101)DM3
Υ(cid:101)DMN This theorem reflects the fact that although a state
max may be separable over one particular k-partition, that
6.816 R
5.886 maxR is not sufficient to conclude the absence of all k-partite
max2 nonlocal correlations, so k-separability is not a sufficient
max3 measure of the absence of true k-entanglement.
maxN
Thelikelyreasonthatk-entanglementwasdefinedwith
1.000
the min function is that the presence of separability in
0.000

|ΦGHZ(cid:105) |ΦBP(cid:105) |ΦF(cid:105) |ψW(cid:105) |ψRand(cid:105) any one bipartition is sufficient to claim the absence of
nonlocal correlations for bipartite systems, since there is
FIG. 5: (color online) Unnormalized FDM ent Υ(cid:101)FDM (or only one bipartition. So if separability were our only cri-
ent-concurrence) of (10) for the test states of (14), with
terion for multimode systems, then GM k-entanglement
only the |ψ (cid:105) that maximizes it over 1000 random pure
Rand
wouldbecorrectlydefinedbecauseifastateisinanyway
sshtaotwesn., Tanhde utnhneomrmaaxliimzeudmDoMfkeaenchtsoΥ(cid:101)vDerMkalolft(e1st1)satareteaslsios k-separable,thenitsGMk-entanglementyields0. (After
maxk ≡max(Υ(cid:101)DMk), while maxR≡max(Υ(cid:101)FDM) applies to all, GM measures do correctly indicate whether a pure
allnonrandomteststates,andmaxR≡max(Υ(cid:101)FDM(ρ|ψRand(cid:105))). stateisaproductoveratleastoneparticulark-partition.)
But since we have just seen examples that k-separability
isnotsufficienttoconcludetheabsenceofallk-partitenon-
Figure 5 shows different results for |Φ (cid:105) and |Φ (cid:105),
GHZ BP localcorrelations,andsincenonlocalcorrelationsarewhat
which is mainly due to the one biseparability of |Φ (cid:105),
BP entanglementreallymeans,thenGMk-“entanglement”is
makingtheFDMenttheonlymeasureofthesethreethat
not reallyameasureofk-partiteentanglement.
distinguishesbetweentheseparabilityofthesestates,and
Unfortunately, there is now quite a lot of literature
yet does not neglect the other bipartite nonlocal correla-
(including the present work by necessity) that uses the
tionsin|Φ (cid:105). Again,|Φ (cid:105)seemstooutperformallother
BP F
terminology k-“entangled” to describe a condition that
states in every measure, as suggested by the fact that
is insufficient to determine the presence of nonlocal cor-
|ψ (cid:105) seems to approach its performance, and |ψ (cid:105)
Rand W
relations over all k-partitions. However, a simple way
slightly underperforms in every area.
to handle this without generating confusion is to always
puttheentanglement partofsuchtermsinquotessuchas
k-“entanglement” or k-“entangled.” That acknowledges
D. Comparison of FGM, FSM, and FDM Ents
that such measures are not true entanglement measures
without having to define new terminology. We will not
The most important difference between Υ and
FGM usethisconventionhere,sinceweneededtousethefamil-
both Υ and Υ is that Υ (ρ )=0 while
FSM FDM GM2 |ΦBP(cid:105) iarpreviousdefinitionstointroducethepresentconcepts.
Υ(cid:101)SM2(ρ|ΦBP(cid:105))(cid:54)=0 and Υ(cid:101)DM2(ρ|ΦBP(cid:105))(cid:54)=0. The fact that However, the true-entanglement theorem alone is not
not all bipartitions of |Φ (cid:105) are separable means that
BP sufficient to distinguish between states like |Φ (cid:105) and
there are some bipartitions that have nonlocal correla- GHZ
|Φ (cid:105). One missing concept is that the states that are
tions, and the sum over all 2-partitional quantities in BP
mostentangledhavethemostsimultaneous nonlocalcor-
Υ(cid:101)SM2 and Υ(cid:101)DM2 is why they detect these correlations, relationsoverallpossiblepartitions. Sincethisisexactly
while the minimum over all 2-partitions in Υ is why
GM2 what the FSM ent measures, then it is both necessary
it misses those nonlocal correlations, reporting “zero.”
and sufficient to detect all nonlocal correlations, and it
Therefore Υ is not sufficient to detect all nonlocal
GM2 measurestheirsimultaneouspresence, whichprovidesan
correlations of 2-partitions of a state, and thus Υ is
FGM ordering for multipartite entangled states.
not sufficient to detect all nonlocal correlations.
Thus, we must make an important new distinction; Yet the FSM’s ordering is still not sufficient to reveal
presence of separability in a particular k-partition is not the difference between states like |ΦGHZ(cid:105) and |ΦBP(cid:105), as
sufficient to claim absence of nonlocal correlations over seeninFig.4. Therefore,byalsorequiringthatthesimul-
all k-partitions for a given k. Sincenonlocalcorrelations taneous nonlocal correlations be as evenly and as widely
are what the word entanglement really means, we must distributed as possible, we attain ordering criteria that
requirethe necessary and sufficient detection of the pres- distinguish |ΦGHZ(cid:105) and |ΦBP(cid:105) without sacrificing infor-
ence of any nonlocal correlations as our prime criterion mation about nonlocal correlations. The FDM ent (or
for what constitutes an entanglement measure. We state ent-concurrence) achieves this as seen in Fig.5.
all of this in the following theorem. However, the true-entanglement theorem only applies
True-EntanglementTheorem: Theabsenceofsep- to applications of entanglement in the full Hilbert space
arabilitybetweenanypartitions isnecessaryandsufficient of ρ. For applications of entanglement in the reductions,
for the presence of any nonlocal correlations, and there- other principles are involved, as discussed in Sec.VII.
6
IV. CANDIDATE MIXED-STATE MEASURES Weincludeρ andρ becausetheyaremixtures
GHZ+1 F+1
of highly nonlocally correlated states with a basis state
Here we list the mixed-state entanglement-measure they already include, to see how the candidate measures
candidates that we test in Sec.V. In all cases, Eˆ(ρ) rate this lowering of nonlocal correlation.
means the convex-roof extension of a pure-state measure
Totestastatelike(2),ρ hasadecompositioninto
2-sep
E(ρ) to handle mixed-state input (see [21, App.J]).
pure states that are each 2-separable in different ways,
whereeachparthasstronginternalnonlocalcorrelations.
1. TheFGMentofformation(usingΥ (ρ)from(6)):
FGM
The state ρ , when viewed as a 2×8 system is a
Υˆ (ρ)≡ min (cid:16)(cid:88) p Υ (ρ )(cid:17). (16) mixed state wiMthMtEhe same entanglement as a maximally
FGM j FGM j
{pj,ρj}|ρ=(cid:80)jpjρj j entangled pure state. Rediscovered in the present work,
this phenomenon was originally discovered in [26, 27],
2. The strict FGM (SFGM) ent of formation: and called “mixed maximally entangled (MME) states.”
It is easy to prove that all decompositions of such states
1 (cid:88)N consistofmaximallyentangledpuredecompositionstates,
Υ (ρ)≡ Υ (ρ), (17)
SFGM M SGMk yielding an entanglement of 1 by any unit-normalized
SFGM
k=2 convex-roof (or nearest-separable-state [28]) measure.
whereMSFGM isanormalizationfactor,andthestrict Weusepure stateρ|ΦF(cid:105)asareferencesinceithadnear-
GM (SGM ) ent of formation is highest values in the GM measures of Fig.3, and it may
k k
havethehighestvaluesfortheSMmeasuresinFig.4and
ΥSGMk(ρ)≡min({Υˆ(N(hk))(ρ)}), (18) the DM measures in Fig.5.
where{Υˆ(Nh(k))(ρ)}isthesetofconvex-roofextensions
of all N-mode k-partitional ents. Here, the minimum
over all convex-roof-extensions of a given kind of k- A. Tests and Analysis of Mixed-Input FGM Ents
partitional ent ensures that if a state achieves strict
k-separability, all of its optimal-decomposition pure Figures 6–7 show similar results but differ in subtle
states are k-separable over the same partitions. ways briefly explained after each.
3. The FSM ent of formation (using Υ (ρ) from (8)):
FSM

Υˆ (ρ)≡ min (cid:16)(cid:88) p Υ (ρ )(cid:17). (19) 2.556
FSM j FSM j
{pj,ρj}|ρ=(cid:80)jpjρj j Υ(cid:101)ˆFGM
4. TΥˆheFD(ρM)≡entoffomrimnation(cid:16)((cid:88)withpΥΥFDM(ρ()ρfr)(cid:17)om. (1(02)0)): 1.480 ΥΥΥmˆˆˆGGGaxMMMF23N
FDM {pj,ρj}|ρ=(cid:80)jpjρj j j FDM j 01..808090 mmaaxx2F
0.667 max3
We do not use a “hat” if a convex-roof extension (CRE) maxN
has been applied already within the function.
0.000

ρGHZ+1 ρ2-sep ρF+1 ρMME ρ|ΦF(cid:105)
V. TESTS OF CANDIDATE MIXED-STATE
FIG. 6: (color online) Unnormalized FGM ent of forma-
MEASURES
tion Υ(cid:101)ˆFGM of (16) for the test states of (21). The (nor-
malized) GM ents of formation Υˆ (CREs of (7)) are
Limiting ourselves to rank-2 mixed states (since that k GMk
also shown, and the maximum of each over all test states
is computationally practical) we use test states,
ρGHZ+1 ≡ 12(|ΦGHZ(cid:105)(cid:104)ΦGHZ|+|1,1,1,1(cid:105)(cid:104)1,1,1,1|) iaCslRlmnEaosxnkuρs|≡ΦeFdm(cid:105)9ta0ex0s(tΥdˆsetGcaMotmeks)p,,oawsnihdtiilomenamsx,aFaxs≡Fin≡m[a2mx1a(].xΥ(cid:101)ˆ(FΥ(cid:101)ˆGFMG(Mρ)|ΦaFp(cid:105))p)l.ieTshtoe
ρ ≡ 1(ρ +ρ(1)⊗|Φ (cid:105)(cid:104)Φ |)
2-sep 2 |Φ+(cid:105)⊗|Φ+(cid:105) |1(cid:105) GHZ3 GHZ3
ρ ≡ 1(|Φ (cid:105)(cid:104)Φ |+|1,1,1,1(cid:105)(cid:104)1,1,1,1|)
F+1 2 F F
ρ ≡ 1(ρ +ρ ) The main items of interest in Fig.6 are the fact that
MME 2 √12(|1,1,1,1(cid:105)+|2,2,2,1(cid:105)) √12(|1,1,1,2(cid:105)+|2,2,2,2(cid:105)) both ρ2-sep and ρMME have ΥˆGM2 =0, and while we ex-
ρ ≡|Φ (cid:105)(cid:104)Φ |, pectthistobetruefromthewaytheFGMentminimizes
|ΦF(cid:105) F F
(21) over all bipartitions for each decomposition state within
where ρ ≡|A(cid:105)(cid:104)A|, and |Φ (cid:105), |Φ+(cid:105)⊗|Φ+(cid:105)≡|Φ (cid:105), the larger minimization of the CRE, it illustrates that
|A(cid:105) GHZ BP
and |Φ (cid:105) are from (14), ρ(1) is the first computa- GM measures ignore nonlocal correlations, since ρ
F |1(cid:105) MME
tional basis state for the mode-1 qubit, and |Φ (cid:105)≡ in particular has the maximal entanglement of a pure
GHZ3
√1 (|1,1,1(cid:105)(cid:104)1,1,1|+|2,2,2(cid:105)(cid:104)2,2,2|)isa3-qubitGHZstate. state for the (1|2,3,4) bipartition, as mentioned earlier.
2
7
2.556 correlations in both ρ2-sep and ρMME are detected since
Υ(cid:101)SFGM Υ(cid:101)ˆSM2 (cid:54)=0 for each.
ΥSGM2
1.590 ΥΥSSGGMM3N C. Tests and(TAhneaElynsti-sCoofnMcuirxreedn-cIne)put FDM Ent
max
F
maxF
1.000
0.889 max2
0.667 max3
13.70
maxN
11.60 Υ(cid:101)ˆFDM
0.000 ρGHZ+1 ρ2-sep ρF+1 ρMME ρ|ΦF(cid:105) ΥΥ(cid:101)(cid:101)ˆˆDDMM23
ˆ
FIG. 7: (color online) Unnormalized SFGM ent of forma- Υ(cid:101)DMN
max
tmioanlizΥ(cid:101)edS)FGSMGMof (1e7n)tsfoorf tfhoremtaetsitonstaΥtes of (o2f1)(.18)Thaere(naolsro- 56..888166 maxFF
shown, and thke maximum of each oSvGeMrkall test states is max2
max3
maxk ≡max(ΥSGMk),whilemaxF≡max(Υ(cid:101)SFGM)appliesto maxN
all nonρ|ΦF(cid:105) test states, and maxF ≡max(Υ(cid:101)SFGM(ρ|ΦF(cid:105))).
CREs used 900 decompositions. 1.000
0.000

ρGHZ+1 ρ2-sep ρF+1 ρMME ρ|ΦF(cid:105)
Thestrict version, theSFGMentofformationΥ(cid:101)SFGM FIG. 9: (color online) Unnormalized FDM ent of forma-
in Fig.7 does slightly better than the measure in Fig.6 tion (ent-concurrence of formation) Υ(cid:101)ˆFDM of (20) for the
because it correctly detects that no bipartitions of ρ test states of (21). The unnormalized DM ents of forma-
2-sep k
aictorersWrtweiellliatdcthioisoomcunutpsilnnseotfeρunlMrlyotMhcigaEenlr,ocifrosoesrrusrewetlshahwteicioihmtnhaΥsxGiSinGmMcMeam2litb=esiapΥs0auS.rGrteiMste2ren(cid:54)=goan0r,ldobicnuagtl tmmmiouaanxxmFFΥ(cid:101)ˆo≡≡DfMemmakcaahx(xC((oΥΥ(cid:101)ˆ(cid:101)ˆvRFFeEDrDsMMal()olρfta|eΦp(sF1pt(cid:105)1l)si)e)t).astaCterosReiaEsalsllmsnuoaosxenskdhρo≡|9Φw0Fmn0(cid:105),adtxaee(sncΥ(cid:101)tˆodDmsMttpahkote)es,simt,wioaahnxnilside-.
states like ρ and (2) in App.H.
2-sep
Here in Fig.9, we see the ratings of Υ(cid:101)ˆFDM are similar
tothoseofΥ(cid:101)ˆFSM inFig.8,exceptthathere,thevaluesfor
ρ are much closer to the values for ρ (though
B. Tests and Analysis of Mixed-Input FSM Ent GHZ+1 F+1
stilllessthantheyare). Thus,thistestdoesnotshowany
apparent problems, and exhibits the necessary features
that neither ρ nor ρ can be considered free of
2-sep MME
13.44 2-partite nonlocal correlations (see App.H).
ˆ
Υ(cid:101)FSM
ˆ
10.08 Υ(cid:101)ˆSM2 D. Comparison of all Mixed-Input Candidates
Υ(cid:101)SM3
ˆ
Υ(cid:101)SMN
max The main difference between the GM measures (from
6.667 F
5.778 maxF (16) and (17)) and the SM and DM measures of (19)
max2 and (20) is that the GM measures tend to undervalue
max3 the amount of entanglement, which is a consequence of
maxN their interpretation of separability as the prime criterion
1.000 for lack of entanglement. The SM and DM measures
0.000
take a more global approach, checking every possible k-
ρGHZ+1 ρ2-sep ρF+1 ρMME ρ|ΦF(cid:105) partition for the presence of nonlocal correlations, and
FIG. 8: (color online) Unnormalized FSM ent of formation as such, they correctly report entanglement of every k-
Υ(cid:101)ˆFSM of (19) for the test states of (21). The unnormal- partitional type, in particular correctly not ignoring the
sizheodwnS,MakndenttsheofmfaoxrmimautmionoΥf(cid:101)ˆSeMakch(CovReErsalolf t(e9s)t)satraeteaslsios maximal bipartite entanglement in ρMME.
In the pure-input case, the FDM ent Υ was the
mnoanxρk|Φ≡F(cid:105)mtaexst(Υ(cid:101)ˆsStMatke)s,,wahnidlemmaaxxFF≡≡mmaxa(xΥ(cid:101)ˆ(FΥ(cid:101)ˆSFMSM(ρ)|ΦaFp(cid:105)p))li.esCtoREalsl o|Φnly(cid:105)mewaisthuoreutthsaatcrciofiuclidngdidsteitnegcutiioshnboeftwnoenenlFoD|cΦMaGl HcoZr(cid:105)raenlad-
used 900 decompositions. BP
tions, and it equals the concurrence C for two qubits.
Since C for mixed states is a convex-roof extension
Figure 8 tests unnormalized FSM ent of formation (CRE), and since Υˆ is also a CRE, then Υˆ =C
FDM FDM
from (19), and we see that the known bipartite nonlocal for mixed two-qubit states, as well.
8
VI. ENT-CONCURRENCE Hierarchy of Maximally N-Entangled TGX States
The ability of Υˆ to detect and distinguish multi-
FDM The ent-concurrence identifies a hierarchy among the
partite nonlocal correlations and its link to C both sug-
maximally N-entangled states, which is easiest to see by
gestthatweadoptitasauniversalmeasureofmultipar-
examining its values for the subset of N-entangled TGX
tite entanglement, called the ent-concurrence,
states from App.G, as in Table I.
Cˆ (ρ)≡Υˆ (ρ), (22)
Υ FDM
where Υˆ is defined in (20), so that TABLEI:Normalizedent-concurrenceCΥ andnormalizedk-
FDM
ent-concurrencesC foreachofthemaximallyN-entangled
4-qubitTGXstatesΥ|kΦ[L∗](cid:105) involving the first computational
CˆΥ(ρ)≡{pj,ρj}m|ρi=n(cid:80)jpjρj(cid:32)(cid:80)j pjk(cid:80)N=2M1k h{(cid:80)N=k}1(cid:113)Υ(N(hk))(ρj)(cid:33), bbaraeersiosovfsetrlaettvheeel|ss1e(cid:105)ws≡ittah|t1en,s1oa,n1lo,zj1ne(cid:105)re,o,farsontmadtmeAacpoype.nffiGoct,ibewnehttesh.reeNnLoo∗rrmmisaatllhiizzeaanttiiuoomnnss-
(23) over all states. The test states of (14) are |Φ (cid:105)≡|Φ[2](cid:105),
wonhethreosMekof≡(1M0)FaDnMdM(1D1M),kainsda{nNo}rmaraeliSzatitriloinngfancutmorbbearsseodf |ΦBP(cid:105)≡|Φ[14](cid:105), and |ΦF(cid:105)≡|Φ[24](cid:105), in the first thGrHeeZrows. 1
k
thesecondkindasin(9),whichisthenumberofdifferent
N-mode k-partitional ents Υ(Nh(k))(ρ). |Φ[jL∗](cid:105) CΥ CΥ2 CΥ3 CΥN
While CˆΥ detects all entanglement and distinguishes |Φ[12](cid:105) 0.957 0.946 0.961 1.000
between different types of k-entanglement, it may also
|Φ[4](cid:105) 0.922 0.880 0.957 1.000
be useful to have a partition-specific view of how much 1
entanglement exists between particular mode groups. |Φ[4](cid:105) 1.000 1.000 1.000 1.000
2
Therefore, in the notation of [21], we also define the N- |Φ[4](cid:105) 0.922 0.880 0.957 1.000
3
mode partitional ent-concurrence vector as
|Φ[4](cid:105) 1.000 1.000 1.000 1.000
4
 {Cˆ(N(h2))(ρ)}  |Φ[54](cid:105) 0.922 0.880 0.957 1.000
Υ
Ξ(CNΥ)(ρ)≡{Cˆ(N(hN...))(ρ)}, (24) ||ΦΦ[6[744]](cid:105)(cid:105) 11..000000 11..000000 11..000000 11..000000
Υ |Φ[4](cid:105) 1.000 1.000 1.000 1.000
8
where{Cˆ(Nh(k))(ρ)}isthesetofallN-modek-partitional |Φ[4](cid:105) 1.000 1.000 1.000 1.000
Υ 9
ent-concurrencesofformation,thepure-inputversionsof
|Φ[6](cid:105) 0.957 0.946 0.961 1.000
which are in (13). For example, in a 4-mode system, 1
|Φ[6](cid:105) 0.957 0.946 0.961 1.000
2
Ξ(4) ≡Ξ(1,2,3,4)= |Φ[6](cid:105) 0.957 0.946 0.961 1.000
CΥ CΥ 3
Cˆ(1|2,3,4)Cˆ(2|1,3,4)Cˆ(3|1,2,4)Cˆ(4|1,2,3)Cˆ(1,2|3,4)Cˆ(1,3|2,4)Cˆ(1,4|2,3) |Φ[8](cid:105) 0.957 0.946 0.961 1.000
Υ Υ Υ Υ Υ Υ Υ 1
 Cˆ(1|2|3,4)Cˆ(1|3|2,4)Cˆ(1|4|2,3)Cˆ(2|3|1,4)Cˆ(2|4|1,3)Cˆ(3|4|1,2) ,
 Υ Υ Υ Υ Υ Υ 
Cˆ(1|2|3|4)
Υ
(25)
where, for instance, In Table I, the highest-rated maximally N-entangled
TGX states, having (C ,C ,C ,C )=(1,1,1,1), are
Cˆ(2|1,3,4) = min (cid:18)(cid:88) p (cid:113)Υ(2|1,3,4)(ρ )(cid:19). the tier-1 N-entangled sΥtateΥs2, Υ3 ΥN
Υ j j
{pj,ρj}|ρ=(cid:80)jpjρj j
(26)
Thus, the top row of Ξ(CNΥ) lists all 2-partitional entan- |Φ[24](cid:105)=√14(|1111(cid:105)+|1122(cid:105)+|2212(cid:105)+|2221(cid:105))
glement, and so-on until the lowest row gives the N- |Φ[4](cid:105)=√1 (|1111(cid:105)+|1212(cid:105)+|2122(cid:105)+|2221(cid:105))
partitional entanglement, yielding a fine-grained view of 4 4
the entanglement between each possible mode group. |Φ[64](cid:105)=√14(|1111(cid:105)+|1221(cid:105)+|2122(cid:105)+|2212(cid:105)) (28)
For an intermediate view of entanglement, we can de- |Φ[4](cid:105)=√1 (|1111(cid:105)+|1222(cid:105)+|2112(cid:105)+|2221(cid:105))
7 4
fine the N-mode k-ent-concurrences of formation as |Φ[4](cid:105)=√1 (|1111(cid:105)+|1222(cid:105)+|2121(cid:105)+|2212(cid:105))
8 4
CˆΥk(ρ)≡ΥˆDMk(ρ), (27) |Φ[94](cid:105)=√14(|1111(cid:105)+|1222(cid:105)+|2122(cid:105)+|2211(cid:105)),
whichisaCREofa1-normoverallCˆ(N(hk))(ρ)foragiven
Υ
k, as seen from the definition of Υ (ρ) in (11). where |abcd(cid:105) ≡ |a,b,c,d(cid:105). The tier-2 N-entangled states,
DMk
9
with (C ,C ,C ,C )≈(0.957,0.946,0.961,1), are example, the 0 in (33) shows that the mode groups
Υ Υ2 Υ3 ΥN
defined by the partitioning (1,2|3,4) are separable in
|Φ[2](cid:105)=√1 (|1111(cid:105)+|2222(cid:105)) ρ , which is true since that is the Bell product, but
1 2 |ΦBP(cid:105)
|Φ[6](cid:105)=√1 (|1111(cid:105)+|1122(cid:105)+|1212(cid:105)+|2121(cid:105)+|2211(cid:105)+|2222(cid:105)) (33) also shows that the entanglement is maximal be-
1 6 tween all other bipartitions of the state (seen in its top
|Φ[6](cid:105)=√1 (|1111(cid:105)+|1122(cid:105)+|1221(cid:105)+|2112(cid:105)+|2211(cid:105)+|2222(cid:105))
2 6 row). Thus, the ent-concurrence does not ignore all of
|Φ[36](cid:105)=√16(|1111(cid:105)+|1212(cid:105)+|1221(cid:105)+|2112(cid:105)+|2121(cid:105)+|2222(cid:105)) these bipartite nonlocal correlations just because one of
|Φ[8](cid:105)=√1 (|1111(cid:105)+|1122(cid:105)+|1212(cid:105)+|1221(cid:105) them is zero, as the GM measures do.
1 8
+|2112(cid:105)+|2121(cid:105)+|2211(cid:105)+|2222(cid:105)),
(29)
VII. ABSOLUTE ENT-CONCURRENCE
andthelowest-ratedgroup,thetier-3N-entangledstates,
with (C ,C ,C ,C )≈(0.922,0.880,0.957,1), are
Υ Υ2 Υ3 ΥN While the ent-concurrence (and its accompanying no-
|Φ[4](cid:105)=√1 (|1111(cid:105)+|1122(cid:105)+|2211(cid:105)+|2222(cid:105)) tions of N-mode partitional ent-concurrence vector and
1 4 k-ent-concurrence) evaluates the multipartite entangle-
|Φ[4](cid:105)=√1 (|1111(cid:105)+|1212(cid:105)+|2121(cid:105)+|2222(cid:105)) (30) ment resources of the entire input state in its full space,
3 4
|Φ[4](cid:105)=√1 (|1111(cid:105)+|1221(cid:105)+|2112(cid:105)+|2222(cid:105)). anotherdimensionofdetailscanbegleanedbyevaluating
5 4 the entanglement within reductions of the input state.
AllofthesestatesaremaximallyN-entangled,asseenin Thus for mode group m, the S-mode partitional ent-
Table I, and furthermore, these are only a small portion concurrence vector is
of the “phaseless” maximally N-entangled TGX states,
sincesimilarsetscanbegeneratedbyspecifyingadiffer-  {Cˆ(mh(2))(ρ)} 
Υ
entWchoemthmeornor“sntoatrtoitnhgerlesvtealt”estheaxnist|1t(cid:105)h=at|h1a1v1e1(cid:105)h.igherent- Ξ(CmΥ)(ρ)≡ ... , (34)
concurrence than the tier-1 states is still unknown, but {Cˆ(mh(S))(ρ)}
Υ
none were found in the present numerical tests.
To see the most fine-grained view, from (25) the N- where m ≡ (m1,...,mS); S ∈ 2,...,N are the modes
mode ent-concurrence vectors of |Φ (cid:105)≡|Φ[4](cid:105) (tier 1), to which ρ ≡ ρ(1,...,N) is being reduced before being
|ΦGHZ(cid:105)≡|Φ1[2](cid:105) (tier 2), and |ΦBP(cid:105)≡F|Φ1[4](cid:105) (2tier 3) are ppaarrttiittiioonnaedl,enant-dco{nCcˆuΥ(mrrh(eTn)c)}esisofthaegsievtenofraeldluSct-imonodˇρe(mT)-
 (cid:113) 
1 1 1 1 2 1 1 where T ∈ 2,...,S, where for a particular partitioning
Ξ(CNΥ)(ρ|ΦF(cid:105))=(cid:113)89 1 1 1 13 (cid:113)89 , (31) labeled by h, the partitional e(cid:18)nt-concu(cid:113)rrence is (cid:19)
1 CˆΥ(m(hT))(ρ)≡ min (cid:88) pj Υ(m(hT))(ρj) ,
{pj,ρj}|ρ=(cid:80)jpjρj j
(35)
 1 1 1 1 (cid:113)2 (cid:113)2 (cid:113)2  where Υ(m(hT)) is the partitional ent (of h-labeled parti-
3 3 3
Ξ(CNΥ)(ρ|ΦGHZ(cid:105))=(cid:113)89 (cid:113)89 (cid:113)89 (cid:113)89 (cid:113)89 (cid:113)89 , t[2io1n].mThh(uTs),)rmowenTt−io1neodfΞa(CfmtΥe)rli(s5t)saalnldpodsesifibnleedT-ipnadrteittaioilnianl
1 ent-concurrences of the mode-m reduction of ρ.
Since a partitional ent-concurrence vector exists for
(32)
each reduction m, we can define the ent-concurrence ar-
 1 1 1 1 0 1 1  ray as the matrix ∇(cid:101)CΥ (not a gradient) whose elements
(cid:113) (cid:113) are partitional ent-concurrence vectors,
Ξ(N)(ρ )= 2 1 1 1 1 2 . (33)
CΥ |ΦBP(cid:105)  3 3 
1 (∇(cid:101)CΥ)k,l(ρ)≡ΞC((ΥnCk(c,k))l,···), (36)
The sum of all elements in each of (31–33) is the unnor- where c≡(1,...,N), k ∈2,...,N, and l∈1,...,(cid:0)N(cid:1)
malizedent-concurrence,yielding13.70,13.11,and12.63, where (cid:0)N(cid:1) ≡ N! , and nCk(v,k) is the vectorizked
k k!(N−k)!
respectively, which are the first three Υ(cid:101)FDM values (in n-choose-k function yielding the matrix whose rows are
a different order) in Fig.5. In contrast, the square of each unique combinations of the elements of v chosen
all the elements in (31–33) (since these are pure states), k at a time, and A is the lth row of matrix A. For
l,···
yields the N-mode ent vectors of [21], the sums of which example in N =4, (suppressing input arguments ρ)
yield13.44,12.33,and12.33,respectively,whichexplains
 
itwnohFaybigtlhe.e4t,ovsadhluioswetsiinnoggfuΥti(cid:101)shhFaSttMhtehfsoeerFt|wSΦMoGHsmtZa(cid:105)etaeassn.udre|sΦwBPer(cid:105)eanreoteqaubalel ∇(cid:101)CΥ(ρ)=ΞC(1Υ,2Ξ)C(1ΥΞ,2,C(31Υ),3)Ξ(CΞ1Υ,C(21Υ,,44)) ΞΞC((C12ΥΥ,,33,)4)ΞΞ(C2C(Υ,24Υ,)3,4Ξ)(C3Υ,4),
The worth of Ξ(N) is that it shows us between which Ξ(1,2,3,4)
CΥ CΥ
mode groups entanglement and separability occur. For (37)
10
where the 2-mode partitional ent-concurrence vectors while for |Φ (cid:105), we have
GHZ
have just one element, such as
∇(cid:101)CΥ(ρ|ΦGHZ(cid:105))= 
(0) (0) (0) (0) (0) (0)
Ξ(2,3) =Cˆ(2|3) =Cˆ(2,3), (38) (cid:32) (cid:33)(cid:32) (cid:33)(cid:32) (cid:33)(cid:32) (cid:33)
CΥ Υ Υ  0 0 0 0 0 0 0 0 0 0 0 0 
 
 0 0 0 0 
and 3-mode partitional ent-concurrence vectors look like   (42)
  (cid:113) (cid:113) (cid:113)  ,
 1 1 1 1 2 2 2 
 3 3 3 
(cid:32) (cid:33)  (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) 
Ξ(1,2,4)= CˆΥ(1|2,4) CˆΥ(2|1,4) CˆΥ(4|1,2) , (39)   89 89 89 89 89 89 
CΥ Cˆ(1|2|4)
Υ 1
which shows that |ψ (cid:105) has many more sites of nonlocal
W
and the 4-mode partitional ent-concurrence vector
correlations than |Φ (cid:105), as shown also by their unnor-
Ξ(1,2,3,4) is given by (25). GHZ
CΥ malized absolute ent-concurrences, C(cid:101)Υabs(ρ|ψW(cid:105))≈26.19
Then, to create a universal multipartite entanglement and C(cid:101)Υabs(ρ|ΦGHZ(cid:105))≈13.11, although |ψW(cid:105) contains no
measurethatcandetectallpossible nonlocalcorrelations mode groups that are maximally entangled (since none
of a state including those of all of its reductions, we can get up to 1, since each element is normalized), while
define the absolute ent-concurrence as |Φ (cid:105) contains five maximally entangled mode groups.
GHZ
However, as pointed out in [21], it is important to
keep in mind that these nonlocal correlations may not
CΥabs(ρ)≡ ma|x|(∇(cid:101)||C∇(cid:101)ΥC(Υρ)(|ρ|)1||1), (40) aenllt-bceonsicmururletnanceeoaursrlayyavsahiolawbsleuassarlelsopuortceenst.iaRl aetnhtearn,gtlhee-
ment resources a state has to offer. Therefore, whether
which is the 1-norm over all partitional ent-concurrences we consider |ψ (cid:105) to be more or less “entangled” than
W
normalized to its maximum value over all input states. |Φ (cid:105) depends on our specific application, but the ent-
GHZ
Thus, by theorem 1 from App.D, C captures a concurrence array gives us a tool for assessing this.
measurementofallpossiblewaysinwhichΥaabsstatecanbe For comparison, for |ΦF(cid:105) of (14),
nonlocally correlated. The main drawback of the abso-
lute ent-concurrence of (40) is that it generally requires ∇(cid:101)CΥ(ρ|ΦF(cid:105))= 
(0) (0) (0) (0) (0) (0.0541)
ceovennvewx-hreonoftheextinenpsuitonissp(uCrRe,Essin)cienreadllucetlieomnesnotfstohfe∇(cid:101)pCuΥre, (cid:32)1 1 0.0541(cid:33)(cid:32)1 1 0.0541(cid:33)(cid:32)0 1 1(cid:33)(cid:32)0 1 1(cid:33)
mdeecaonmsptohsaittioitnisstautseusaρlljyocfomρpauretagtieonnearlallylyinmtriaxcetda.blTehtios  0.817  0.817(cid:113) (cid:113)23 (cid:113)23 , (43)
calculate C , even for pure ρ.  1111 211 
Υabs  (cid:113) 3(cid:113)  
For states like |ψW(cid:105), where one of its prime charac-   81111 8 
  9 9 
teristics is that it retains entanglement after the removal
1
of a particle (tracing away a mode) [24], finding the en-
teannt-gcloenmceunrtreonfciet,sarneddutchtieoennsti-scopnocsusirbrelencweitahrrtahyeoafb(s3o6lu)ties so that C(cid:101)Υabs(ρ|ΦF(cid:105))≈25.13, and for |ΦBP(cid:105) of (14),
an excellent tool for a high-resolution picture of all pos- ∇(cid:101)CΥ(ρ|ΦBP(cid:105))= 
sible nonlocal correlations of the state. These measures (1) (0) (0) (0) (0) (1)
would certainly show us exactly how |ψW(cid:105) differs from (cid:32)1 1 0(cid:33)(cid:32)1 1 0(cid:33)(cid:32)0 1 1(cid:33)(cid:32)0 1 1(cid:33)
 
|cΦleGs.HZF(cid:105)o,rwexhaicmhpilse,stehpeareanbt-lecoanfctuerrrernemceovaarrlaoyfoafn|yψWpa(cid:105)ritsi-  (cid:113)23 (cid:113)23 (cid:113)23 (cid:113)23 , (44)
   
 1111011 
 (cid:113) (cid:113) 
∇(cid:101)CΥ(ρ|ψW(cid:105))=   231111 23 
 (1) (1) (1) (1) (1) (1)  1
2 2 2 2 2 2
(cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) 
 12(cid:113)12 12 12(cid:113)12 12 12(cid:113)12 12 12(cid:113)12 12 hsoastthhaetmC(cid:101)oΥsatbso(cρc|uΦrBrPe(cid:105)n)c≈es25o.f90m.axiTmhaullsy,-e|nΦtBanPg(cid:105)leadctmuaoldlye
 1 1 1 1 
 2  (cid:113)3(cid:113)32(cid:113)3(cid:113)3(cid:113)2(cid:113)2 2(cid:113)2 2 , gnruomubpesrwaitth1921ofotfhtehmem. ,Twhhesileet|wΦoF(cid:105)sthaatessthalesonecxotnthaiignherset-
 4 4 4 4 3 3 3  ductions that are MME states as introduced in the text
 (cid:113) (cid:113) (cid:113) (cid:113) (cid:113) (cid:113)  
  13 13 13 13 13 13  after(21), and|Φ (cid:105)and|Φ (cid:105)alsohavesomerank-4re-
  18 18 18 18 18 18  F BP
  (cid:113)   ductionsthatwereluckilydiagonalandthereforesepara-
3
4 blebyanymeasure. Thus,CΥabs ordersstatesdifferently
(41) than C due to its inclusion of the reductions.
Υ