Multipartite entanglement in 2 x 2 x n quantum systems

0.23 MB·

by  Akimasa Miyake

Multipartite entanglement in 2 2 n quantum systems
× ×
Akimasa Miyake
∗
Department of Physics, Graduate School of Science, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan
Quantum Computation and Information Project, ERATO, JST, Hongo 5-28-3, Bunkyo-ku, Tokyo 113-0033, Japan
Frank Verstraete
†
Max-Planck Institut fu¨r Quantenoptik, Hans-Kopfermann Str. 1, Garching, D-85748, Germany
(Dated: July 9, 2003)
We classify multipartite entangled states in the Hilbert space H = C2 ⊗C2⊗Cn (n ≥ 4), for
example the 4-qubit system distributed over 3 parties, under local filtering operations. We show
that there exist nine essentially different classes of states, giving rise to a five-graded partially
ordered structure, including the celebrated Greenberger-Horne-Zeilinger (GHZ) and W classes of
threequbits. Inparticular,all2×2×n-statescanbedeterministicallypreparedfromonemaximally
4
entangled state, and some applications like entanglement swapping are discussed.
0
0
PACSnumbers: 03.65.Ud,03.67.-a
2

n
a I. INTRODUCTION even probabilistically.
J
In this paper, we will generalize these results and

7
present one of the very few exact and complete results
Entanglement is the key ingredient of all applications
aboutmultipartite quantumsystems,by classifyingmul-
3 inthefieldofquantuminformation. Duetothenon-local
tipartite entanglement in the 2 2 n cases. Since
v character of the correlations that entanglement induces, × ×
7 it is expected that entanglement is especially valuable in this include the 4-qubit system distributed over 3 par-
6 ties, which is the case in e.g., entanglement swapping,
thecontextofmanyparties. Despite alotofeffortshow-
0 our results will clarify what kinds of essentially differ-
ever, it has been proven exceedingly hard to get insight
7 ent multipartite entanglement there exist in this sit-
into the structure of multipartite entanglement. Still,
0
uation, and give better understanding for multi-party
3 the motivation of our work is as follows. In the bipar-
LOCC protocols. More specifically, we will address the
0 tite (pure) setting, the entanglement present in a Bell-
stochasticLOCC(SLOCC)classificationofentanglement
/ Einstein-Podolsky-Rosen (Bell-EPR) state is essentially
h [1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], which is a coarse-
p unique;i.e.,wecanevaluateanybipartiteentangledstate grained classification under LOCC. Let us consider the
- by the number ofequivalentBellpairs,ineither a qubit-
t singlecopyofamultipartitepurestate Ψ ontheHilbert
n ora qudit- system, both inthe single-copyandmultiple- space =Ck1 Ckl (precisely, in| aibuse of the no-
a copies case. H ⊗···⊗
tation, we would denote a ray on its complex projective
u
q The situation is totally different in the multipartite space CPk1×···×kl−1 by Ψ ),
settinghowever,whereinterconvertibilityunderlocalop- | i
:
v erations and classical communication (LOCC) is not ex- k1−1,...,kl−1
Xi pected to hold [1]. Multipartite entanglement exhibits a |Ψi= ψi1...il|i1i⊗···⊗|ili, (1)
r much richer structure than bipartite entanglement. The i1,.X..,il=0
a first celebrated example thereof was the 3-qubit GHZ
where a set of i i constitutes the standard
state, called after Greenberger, Horne and Zeilinger [2]. | 1i⊗···⊗| li
computational basis and it often will be abbreviated to
This state was introduced because it allows to disprove
i i . In LOCC, we recognize two states Ψ and
the Einstein locality for quantum systems without in- | 1··· li | i
Ψ which are interconvertible deterministically, e.g., by
voking statistical arguments such as needed in the argu- | ′i
local unitary operations, as equivalent entangled states.
ments of Bell. Another interesting aspect of multipar-
Onthe otherhandinSLOCC,weidentify twostates Ψ
tite entanglement was discovered by Wootters et al. [3]. | i
and Ψ as equivalent if they are interconvertible prob-
They showed that a quantum state has only a limited | ′i
abilistically, i.e., with a nonvanishing probability, since
shareability for quantum correlations: the more bipar-
theyaresupposedtobeabletoperformthesametasksin
tite correlations in a state, the less genuine multipartite
quantum information processing but with different suc-
entanglement that can be present in the system. This
cess probabilities. Mathematically, Ψ and Ψ belong
led to the introduction of the so-called 3-qubit W state | i | ′i
to the same SLOCC entangled class if and only if they
[4], which was shown to be essentially different from the
can be converted to each other by invertible SLOCC op-
GHZ state as they are not interconvertible under LOCC
erations,
Ψ =M M Ψ , (2)
′ 1 l
| i ⊗···⊗ | i
∗Electronicaddress: [email protected] where Mi is any local operation having a nonzero de-
†Electronicaddress: [email protected] terminant on the i-th party [4], i.e., Mi is an element
2
of the general linear group GL(k ,C) (we do not care |000>+|011>+|102>+|113>
i
about the overall normalization and phase so that we |000>+ 1 (|011>+|101>)+|112>
2
can take its determinant 1, i.e., Mi ∈ SL(ki,C).). It dim |000>+|011>+|112>
can be also said that an invertible SLOCC operation is GHZ
15
a completely positive map followed by the postselection 14 W
13 8
of one successful outcome. Mathematically, the SLOCC 11
classification is equivalent to the classification of orbits 10 B1 dual
5
generated by a direct product of special linear groups
S
SL(k ,C) SL(k,C). Note that in the bipartite
1 ×···× l 8 B2 B3 6
l=2case,theSLOCCclassificationmeanstheclassifica-
tion just by the Schmidt rank (or equivalently, the rank
of a coefficient ”matrix” ψ in Eq. (1)). We will also
i1i2
address the question of noninvertible SLOCC operations 2x2x2
(atleastoneoftheranksofM inEq.(2)isnotfull). The
i
setofinvertibleandnoninvertibleSLOCCoperationsare 2x2x3
also called local filtering operations. Consider the bipar- 2x2x4
tite case as an example: SLOCC entangled classes are
found to be totally orderedin such a way that an entan- FIG. 1: The onion-like classification of multipartite en-
gled class of the larger Schmidt rank is more entangled tangled classes (SLOCC orbits) in the Hilbert space H =
C2⊗C2⊗Cn(n≥4). Therearenineclassesdividedby”onion
thanthatofthesmallerone,becausetheSchmidtrankis
skins”(theorbit closures). Thepicturesfor 2×2×n(n>4)
always decreasing under noninvertible local operations.
casesareessentiallysamealthoughthedimensionsofSLOCC
Thepaperisorganizedasfollows. InSec.II,weclassify
orbits are different. These classes merge into four classes, di-
multipartite 2 2 n pure states under SLOCC, so as
videdbytheskinsofthesolidline,inthe”bipartite”(AB)-C
× ×
to show that nine entangled classes are hierarchized in a picture. NotethatalthoughnoninvertibleSLOCCoperations
five-graded partial order. We discuss the characteristics generallycausetheconversionsinsidetheonionstructure,an
of multipartite entanglement in our situation in Sec. III, outer class can not necessarily convert into its neighboring
and extend the classification of multipartite pure states innerclass (cf. Fig. 2).
to mixed states in Sec. IV. The conclusion is given in
Sec. V.
containsthe classificationfor the 2 2 2 (3-qubit)case
II. CLASSIFICATION OF MULTIPARTITE × ×
[4,5,8]andthe2 2 3case[5]. WefindthatSLOCCor-
ENTANGLEMENT × ×
bitsareaddedoutsidethe onion-likepicture(Fig.1)and
the partially ordered structure (Fig. 2) becomes higher,
In this section, we give the complete SLOCC classifi-
as the third party Clare has her larger subsystem. Re-
cation of multipartite entanglement in 2 2 n cases.
markably, for the 2 2 n (n 4) cases, the generic
× ×
Moreover, we present a convenient criterion to distin- × × ≥
class is one ”maximally entangled” class located on the
guish inequivalent entangled classes by SLOCC invari-
top of the hierarchy. This is a clear contrast with the
ants.
situationofthe 2 2 2and2 2 3cases,wherethere
× × × ×
aretwo differententangledclassesonits top. It suggests
that,eveninthemultipartite situation,thereisaunique
A. Five-graded partial order of nine entangled
entangledclasswhichcanserveasresourcestocreateany
classes
entangledstate,ifthe Clare’ssubsystemis largeenough.
This will be proven in Sec. III.
Weshowthattherearenineentangledclassesandthey
We note that it is sufficient to consider the 2 2 4
constitute five-graded partially ordered structure under × ×
case in the proof of the theorem, since Clare can only
noninvertible SLOCC operations.
have support on a 4-dimensional subspace. This is an
Theorem 1 Consider pure states in the Hilbert space analogy with the bipartite k k′ (k < k′) case whose
×
=C2 C2 Cn (n 4), they are divided into nine en- SLOCC classification is equivalent to that of the k k
tHangled c⊗lasse⊗s, seen i≥n Fig. 1, under invertible SLOCC case, because the SLOCC-invariant Schmidt local r×ank
operations. These nine entangled classes constitute the takesatmostk. Inany2 2 n(n 4)case,thepartially
× × ≥
five-graded partially ordered structure of Fig. 2, where ordered structure of multipartite entanglement consists
noninvertible SLOCC operations degrade higher entan- of nine finite classes. Our result not only describes the
gled classes into lower entangled ones. situationthatonlyClarehastheabundantresources,but
also would be useful in analyzing entanglement of two-
Some remarksare givenbefore its proof. The theorem qubit mixed states attached with an environment (the
gives the complete classification of multipartite pure en- rest of the world), which could e.g. be used to analyze
tangled states in 2 2 n (n 4) cases. It naturally the power of an eavesdropper in quantum cryptography.
× × ≥
3
|000>+|011>+|102>+|113> ways. Readers who are interested just in applying our
(2,2,4) resultscanskiptoSec.IIC,whereaconvenientcriterion
for distinguishing nine classes is given.
Proof. We first give an algebraic proof, utilizing the
|000>+ 1 (|011>+|101>)+|112> |000>+|011>+|112>
2 matrixanalysis(cf. Ref.[6,7,8,9]). Anystateisparam-
(2,2,3) (2,2,3) eterizedbyathreeindextensorψ withi ,i 0,1
i1i2i3 1 2 ∈{ }
and i 0,1,2,3 . This tensor can be rewritten as a
3
∈ { }
4 4 matrix Ψ˜ =(ψ ) by concatenatingthe indices
|000>+|111> |001>+|010>+|100> × (i1i2)i3
(i ,i ). Next we define the matrix R as
(2,2,2) GHZ (2,2,2) W 1 2
R=TΨ˜, (3)
|001>+|010> |001>+|100> |010>+|100>
where T is defined as
(1,2,2) B1 (2,1,2) B2 (2,2,1) B3
1 0 0 1
1 0 i i 0
T = . (4)
|000> √20 1 1 0 
−
(1,1,1) S i 0 0 i
 − 
 
Letusobservethatboth2 2matricesM andM belong
1 2
FIG. 2: The five-graded partially ordered structure of nine toSL(2,C)ifandonlyifO×=T(M M )T SO(4,C)
entangled classes in the 2×2×n (n ≥ 4) case. Every class 1⊗ 2 † ∈
and det(M ) = det(M ) = 1, because of a consequence
is labeled by its representative, its set of local ranks, and its 1 2
of an accident in the Lie group theory: SL(2,C)
name. NoninvertibleSLOCCoperations,indicatedbydashed SL(2,C) SO(4,C)(cf. SU(2) SU(2) SO(4)). Ac⊗-
arrows,degradehigherentangledclassesintolowerentangled
≃ ⊗ ≃
ones. cordingly,weseethataSLOCCtransformationofEq.(2)
results in a transformation
R =ORMT. (5)
′ 3
Let us consider the situation where Alice and Bob are Thus, our problem is equivalent to finding appropriate
considered as one party (or, one of Alice and Bob comes normal forms for the complex 4 4 matrix R under left
×
to have two qubits) and call it the ”bipartite” (AB)-C multiplication with a complex orthogonal matrix O
picture. When two parties have two qubits for each, the SO(4,C)andrightmultiplicationwithanarbitrary4 ∈4
onion-like structure of Fig. 1 becomes coarser. The nine matrix MT SL(4,C). ×
3 ∈
entangled classes merge into four classes, and the struc- If the matrix R has full rank, it is enough to operate
turecoincideswiththatofthebipartite4 4case. Wesee M chosen to be T (R 1)T. As a result, the state Ψ˜ is
× 3 † −
thatwecanperformLOCCoperationsmorefreelyinthe (proportional to) the identity matrix 11, or
bipartite situation. Likewise, in the bipartite A-(BC) or
000 + 011 + 102 + 113 , (6)
B-(AC) pictures, the onion-like structure coincides with
| i | i | i | i
that of the 2 8 (i.e., 2 2) case so that just two en-
the representative of the highest class in the hierarchy.
× ×
tangled classes, divided by the onion skin of B or B
1 2 SupposehoweverthattherankofRisthree. Asafirst
respectively, remain.
step, R can always be multiplied left by a permutation
On the other hand, it can be said that the SLOCC- matrix and right byMT so as to yield an R of the form
3
invariantonionstructureofthe 2 2 4caseisacoarse-
grained one of the 4 qubits (2 ×2× 2 2) case (see 1 0 0 0
alsoRef.[9,11,14]),i.e.,theform×eris×emb×eddedintothe 0 1 0 0
R= . (7)
latterinthesamewayasthestructureofthebipartite4  0 0 1 0
4caseisembeddedintothatofthe2 2 4case. So,iftw×o  α β γ 0
× ×  
4-qubit states belong to different classes in the 2 2 4
Suppose α = i, then it can easily be checked that left
× ×
classification,thesestatesmustbealsodifferentinthe4- 6 ±
multiplication by the complex orthogonalmatrix
qubit classification. It would be interesting to note that
the 4-qubit entangled states are divided into infinitely 1/√α2+1 0 0 α/√α2+1
many classes [4, 5, 9], in comparison with finitely many 0 1 0 0
O =  (8)
classes of the 2 2 4 case. In other words, there are 0 0 1 0
× ×
infinitely many orbits in the 4 qubits case between some  α/√α2+1 0 0 1/√α2+1 
 − 
onion skins, while there exists one orbit in the 2 2 4  
× × and right multiplication with
case. This suggests that a drastic change occurs in the
structureofmultipartiteentanglementevenwhenaparty 1 αβ/(α2+1) αγ/(α2+1) 0
comes to have two qubits in hands [15]. 0 − 1 − 0 0
MT = (9)
Now,wegivethe proofofthe Theorem1intwodiffer- 3  0 0 1 0
ent, algebraic (in Sec. IIA) and geometric (in Sec. IIB), 0 0 0 1
 
 
4
yield a new R of the form Note that the last two cases cannot be transformed into
eachother due to the constraintthat O has determinant
1 0 0 0
+1. The corresponding representative states are easily
0 1 0 0
R= . (10) obtained by choosing symmetric ones:
 0 0 1 0 
 0 β′ γ′ 0  000 + 111 , (15)
  | i | i
Exactly the same can be done in the case where β,γ = 001 + 010 + 100 , (16)
6 | i | i | i
i,andthereforeweonlyhavetoconsiderthecasewhere 000 + 011 , (17)
± | i | i
α,β,γ 0,i, i . It can however be checked that in
000 + 101 . (18)
∈ { − }
the case that when 2 or 3 elements α,β,γ are not equal | i | i
to zero, a new R can be made where all α,β,γ become The first state is the celebrated Greenberger-Horne-
equal to zero: this can be done by first multiplying R Zeilinger(GHZ)state,thesecondonetheWstatenamed
with orthogonalmatrices of the kind in [4] for the 3-qubit case,and the remaining ones repre-
sent biseparable B (i = 1,2) states with only bipartite
1 0 0 0 i
entanglementbetweenBobandClare,orAliceandClare,
0 1 0 0
O= , (11) respectively.
0 0 1/√2 1/√2
− Asalastclass,wehavetoconsidertheonewhereRhas
 0 0 1/√2 1/√2  rank equal to 1. This leads to the following two possible
 
  normal forms for R:
and repeating the procedure outlined above. There re-
mains the case where exactly one of the elements is 1 0 0 0 1 0 0 0
equal to i. Without loss of generality, we assume that 0 0 0 0 0 0 0 0
(α,β,γ) ±= (i,0,0) (this is possible because one can do  0 0 0 0,  0 0 0 0. (19)
permutations(withsigns)byappropriateO SO(4)and 0 0 0 0 i 0 0 0
∈    
M3). This case is fundamentally different from the one    
where all α,β,γ are equal to zero as the corresponding The corresponding states are given by
matrixRTRhasrank2asopposedtorank3ofR. There
isnowayinwhichthisbehaviorcanbechangedbymulti- 000 + 110 , (20)
| i | i
plyingRleftandrightwithappropriatetransformations, 000 , (21)
and we therefore have identified a second class (which is | i
clearlyofmeasurezero: agenericrank3stateRwillalso which are the biseparable B state and the completely
3
yield a rank 3 RTR). separable S state, respectively. This ends the complete
It is now straightforwardto constructa representative classification.
stateofeachclass. Asarepresentativeofthemajorclass Itremainstobeproventhatanystatethatishigherin
in the rank 3 R, we choose the state thehierarchyofFig.2canbetransformedtoalltheother
ones that are strictly lower. The first step downwards is
1
000 + (011 + 101 )+ 112 . (12) evident from the fact that right multiplication of a rank
| i √2 | i | i | i 4 R with a rank deficient M can yield whatever R of
3
rank 3. In going from a rank 3 R of the major class to a
As a representative of the minor class in the rank 3 R,
rank2,thestate 000 +(011 + 101 )/√2+ 112 canbe
we choose the state
| i | i | i | i
transformedinto the GHZ state by a projection of Clare
000 + 011 + 112 , (13) on the subspace 0 , 2 and into the W state by Clare
| i | i | i {| i | i}
implementing the POVM element
asitmakesclearthatthestatesinthisclasscanbetrans-
formedtohave3termsinsomeproductbasis(asopposed 1 0 0 0
to the states in the major class that can be transformed 0 1 0 0
. (22)
to have 4 product terms).  0 i 0 0
The case where R has rank 2 can be solved in a com- 0 0 0 0
 
pletely analogousway. Exactly the same reasoningleads  
to the following four possible normal forms for R: From a rank 3 R of the minor class, the GHZ state
can easily be constructed by a projection of Clare on
1 0 0 0 1 0 0 0
her 1 , 2 subspace, while the W state is obtained by
0 1 0 0 0 1 0 0 {| i | i}
, , Clareprojectingonher 0 , 1 + 2 subspace. Finally,
 0 0 0 0 0 0 0 0  {| i | i | i}
theconversionoftheGHZandWstatestotheBellstate
0 0 0 0 i 0 0 0
    amongtwoparties(thebiseparablestate),aswellasthat
    (14)
1 0 0 0 1 0 0 0 of the Bell state to the completely separable state, is
0 1 0 0 0 1 0 0 straightforward. (cid:3)
, .
 0 i 0 0 0 i 0 0 The proof not only gives a constructive transforma-
−
i 0 0 0 i 0 0 0 tionstorepresentativesofnineentangledclasses,butalso
   
   
5
suggestsaverysimplewayofdeterminingtowhichclassa change any more for n 4. This is intuitively because
≥
givenstatebelongs. Onehastocalculatetherankr(.)of the subsystem of one party is too large, compared with
thematricesR(seeEq. (3)),ofRTR,andofthereduced the subsystems of the other parties. Remember that it
density matrix ρ . One gets the following classification: is again an analogy to the bipartite k k case (k <k ),
1 ′ ′
×
where there is no determinant but its onion structure
Class r(R) r(RTR) r(ρ ) remains unchanged from that of the k k case.
1 ×
In general, the hyperdeterminants can be defined for
000 + 011 + 102 + 113 4 4 2
| i | i | i | i =Ck1 Ckl, if and only if
000 + 1 (011 + 101 )+ 112 3 3 2 H ⊗···⊗
| i √2 | i | i | i
000 + 011 + 112 3 2 2 ki 1 (kj 1) i=1,...,l (24)
| i | i | i − ≤ − ∀
000 + 111 2 2 2 Xj6=i
| i | i
001 + 010 + 100 2 1 2 aresatisfied[5,16]. Ofcourse,inthebipartitecases,this
| i | i | i condition suggests that the determinants can be defined
000 + 101 2 0 2
| i | i just for square (k = k ) matrices as usual. Instead, in
000 + 011 2 0 1 1 2
| i | i the 2 2 4 case, the zero locus of the ordinary deter-
000 + 110 1 1 2 minan×t of×degree 4 for the ”flattened” matrix Ψ˜,
| i | i
000 1 0 1
| i ψ ψ ψ ψ
(23) 000 001 002 003
Note that the representative states in the GHZ-type (cid:12) ψ010 ψ011 ψ012 ψ013 (cid:12) =detΨ˜ , (25)
(cid:12) (cid:12)
classes were chosen to be the ones with maximal entan- (cid:12) ψ100 ψ101 ψ102 ψ103 (cid:12)
(cid:12) (cid:12) (cid:16) (cid:17)
glement: following [6], the states with maximal entan- (cid:12) ψ ψ ψ ψ (cid:12)
(cid:12) 110 111 112 113 (cid:12)
glement in a SLOCC class are the ones for which all lo- (cid:12) (cid:12)
givesthee(cid:12)quationofthelargestcl(cid:12)osedsubset. Notethat
cal density operators are proportional to the maximally (cid:12) (cid:12)
it is the SLOCC invariant for the bipartite 4 4 format
mixed state. This is in accordance with the intuition ×
as well as the tripartite 2 2 4 format. It means that
that the local disorder or entropy is proportional to the × ×
thelargestsubsetisdualto thesetB ofthebiseparable
entanglement present in the (pure) state. 3
states, i.e., the set of the separable states in the ”bipar-
tite” (AB)-C picture. We should stress that this duality
itself is valid in any 2 2 n(n 4) case, regardless of
B. Geometry of nine entangled classes × × ≥
the absence of the (hyper)determinant.
Next, let us show that the dual set of S is the second
We explore how the whole Hilbert space is geomet-
largest subset for the 2 2 4 case. In order to de-
rically divided into different nine classes, drawn in the × ×
cide the dual set of S, we seek for the state Ψ included
onion-like picture Fig. 1. This subsection can be seen | i
in the hyperplane (the orthogonal 1-codimensional sub-
as an alternative proof of the theorem in Sec. IIA by a
space) tangent at a completely separable state x (see
geometric way. | i
Ref. [5] in detail.). Mathematically speaking, we should
Weutilizeadualitybetweenthesetofseparablestates
decide the conditionfor Ψ suchthat a setof equations,
and the set of entangled states in order to classify mul- | i
tipartite entangled states under SLOCC [5]. The set S
1,1,3
of completely separable states is the smallestclosed sub-
F(Ψ,x)= ψ x(1)x(2)x(3) =0,
set, as seen in Fig. 1. In many cases (such as the l-qubit i1i2i3 i1 i2 i3
cthaeselsa)rgoefsitntceloressetdtsoubquseatnwtuhmichincfoornmsiasttsioonf,aitllsddeugaelnseertaties  ∂ F(Ψ,ix1,)iX2=,i30=0 j,i , (26)
entangled states, and is given by the zero hyperdeter- ∂x(j) ∀ j
mk inkanctasDe,ettΨhe=set0.SWisethreeasdmilayllseesetstuhbaste,tinoftthheebSicphamrtiidtet hasat leaisjt a nontrivial solution x = (x(1),x(2),x(3)) of
ra×nk 1, while its dual set is the largest subset where the every x(j) = 0. For simplicity, let us suppose that the
6
Schmidt rank is not full (i.e., detΨ=0). point of tangency is the completely separable state 000
However, the entangled states in = C2 C2 Cn (i.e., x(1) =x(2) =x(3) =1, others=0), the corresp|ondi-
H ⊗ ⊗ 0 0 0
(n 4)haveapeculiar structure froma geometricview- ing state Ψ should satisfy
poi≥nt. It is not the case here that the largest subset is | i
Ψ ψ =ψ =ψ =ψ =ψ =ψ =0 ,
dual to the smallest subset S. Indeed, the largestsubset 000 100 010 001 002 003
| i∈{ }
(27)
is dual to (the closure of) the set B of the biseparable
3
accordingto Eq. (26). We find that the state Ψ should
states,i.e.,thesecondsmallestclosedsubsetofdimension
| i
belong to the class of dimension 13, because any state,
6 in Fig. 1. The dual set of S is the second largest sub-
set of dimension 13. The reason will be explained later. Ψ =ψ 011 +ψ 012 +ψ 013 +ψ 101
011 012 013 101
Significantly, this suggests that for the 2 2 n (n 4) | i | i | i | i | i
cases, there are no hyperdeterminants in×the×Gelfan≥d et +ψ102|102i+ψ103|103i+ψ110|110i+ψ111|111i
+ψ 112 +ψ 113 , (28)
al.’s sense; in other word, the onion structure will not 112 113
| i | i
6
inEq.(27)canconverttoitsrepresentative 011 +102 + First we calculate a set (r ,r ,r ) of the SLOCC-
1 2 3
| i | i
113 under invertible SLOCC operations. invariant local ranks of the reduced density matrices.
| i
Inbrief,wefindthatthe 14dimensionallargestsubset (i) In the (2,2,4) case, we find that the state Ψ be-
is the dual set of the biseparable states B3, and the 13 longstothe genericclassofdimension15(the dim|enision
dimensional second largest subset is the dual set of the isindicatedforreaders’convenience,but itisthe one for
completely separable states S. Moreover,we notice that the 2 2 4 case.).
the inside of the largest subset, given by zero locus of × ×
(ii) In the (2,2,3) case, there are two possibilities.
Eq. (25), is equivalent to the structure of the 2 2 3
× × Changingthe localbasisfor Clare,wecanalwayschoose
case(sincethelocalrankforClareshouldbelessthanor
all new ψ = 0 (i 3). We evaluate the hyperde-
equal to 3), which has already been clarified in Ref. [5]. i1i2i3 3 ≥
terminant of degree 6 for the new, 2 2 3 formated
That is how we obtain the onion-like picture of Fig. 1. × ×
ψ ,
Ingeneral,wecantakeadvantageofallkindsofthedual i1i2i3
pairs for sets (typically, one is a large set and the other
is a small set), in order to distinguish inequivalent en-
ψ ψ ψ ψ ψ ψ
tangled classes. This strategy will be exploredelsewhere 000 001 002 010 011 012
[17]. DetΨ2 2 3 =(cid:12)ψ010 ψ011 ψ012 (cid:12)(cid:12) ψ100 ψ101 ψ102 (cid:12)
× × (cid:12) (cid:12)(cid:12) (cid:12)
(cid:12)ψ100 ψ101 ψ102 (cid:12)(cid:12) ψ110 ψ111 ψ112 (cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)
ψ (cid:12) ψ ψ ψ (cid:12)(cid:12) ψ ψ (cid:12)
C. Convenient criterion to distinguish nine 000(cid:12) 001 002 00(cid:12)0(cid:12) 001 002 (cid:12)
entangled classes −(cid:12)(cid:12) ψ010 ψ011 ψ012 (cid:12)(cid:12)(cid:12)(cid:12) ψ100 ψ101 ψ102 (cid:12)(cid:12). (31)
(cid:12) ψ110 ψ111 ψ112 (cid:12)(cid:12) ψ110 ψ111 ψ112 (cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)
We give a convenient criterion to distinguish nine en- (cid:12) (cid:12)(cid:12) (cid:12)
(cid:12) (cid:12)(cid:12) (cid:12)
tangled classes by a complete set of SLOCC invariants. If DetΨ = 0, then Ψ belongs to the major class
2 2 3
Letusdenotelocalranksofthereduceddensitymatrices of dimen×sio×n 16 4. Otherw|isie (i.e., DetΨ = 0), it
2 2 3
ρ1,ρ2, and ρ3 such as belongs to the minor class of dimension 13×. ×
(iii)Inthe(2,2,2)case,therearealsotwopossibilities.
ρ =tr (Ψ Ψ) i=1,2,3, (29)
i ∀j6=i | ih | Changingthe localbasisfor Clare,wecanalwayschoose
allnew ψ =0(i 2). We evaluatethe hyperdeter-
by the 3-tuples (r1,r2,r3). These local ranks are al- minant ofi1id2ei3gree 4 (3it≥s absolute value is also known as
ways useful SLOCC invariants. In the bipartite setting,
the 3-tangle [3]) for the 2 2 2 formated ψ ,
the 2-tuples (r1,r2) are enough to distinguish entangled × × i1i2i3
classes, for both r and r are indeed nothing but the
1 2
Schmidt rank. In the multipartite setting, however, we DetΨ
2 2 2
× ×
needmoreSLOCCinvariantsinadditiontothesetofthe =ψ2 ψ2 +ψ2 ψ2 +ψ2 ψ2 +ψ2 ψ2
local ranks. 000 111 001 110 010 101 100 011
2(ψ ψ ψ ψ +ψ ψ ψ ψ
The proof of the Theorem 1 in Sec. IIA has suggested − 000 001 110 111 000 010 101 111
thatacompletesetofSLOCCinvariantsistherankofR +ψ000ψ100ψ011ψ111+ψ001ψ010ψ101ψ110
in Eq. (3) (i.e., r3), rank of RTR, and r1 (alternatively, +ψ001ψ100ψ011ψ110+ψ010ψ100ψ011ψ101)
r ). Although we have successfully found the rank of
2 +4(ψ ψ ψ ψ +ψ ψ ψ ψ ). (32)
RTR as an additional SLOCC invariant, this is specific 000 011 101 110 001 010 100 111
to the substructure associated with 2 qubits, i.e., to a
homomorphism SL(2,C) SL(2,C) SO(4,C). Likewise,ifDetΨ2 2 2 =0,then Ψ belongstotheGHZ
In the following, we int⊗roduce anot≃her complete set of class of dimension×1×1. 6 Otherwise|, iit belongs to the W
SLOCC invariants, since it also gives an insight about class of dimension 10.
how entanglement measures, distinguishing entangled (iv) Inthe (1,2,2),(2,1,2),and(2,2,1)cases, Ψ be-
classes, are derived in general. The set consists of poly- longs to the biseparable B , B , and B class of|dimi en-
1 2 3
nomialinvariants(hyperdeterminants[5,16])adjustedto sion 8, 8, and 6, respectively.
smallerformats,aswellas3-tuples(r ,r ,r )ofthelocal
1 2 3 (v) In the (1,1,1) case, Ψ belongs to the completely
ranks. The criterion reflects the onion structure drawn | i
separable class S of dimension 5.
in Fig. 1, and suggests that we can utilize the results of
the SLOCC classification for smaller formats recursively Inthismanner,wecanimmediatelycheckwhichclassa
as if we were skinning the onion recursively. givenstate Ψ belongsto. Weremarkthattherepresen-
| i
Any pure state in =C2 C2 Cn is written in the tatives of nine entangled classes in previous subsections
form, H ⊗ ⊗ have been chosen with the help of hyperdeterminants;
the ”GHZ-like” representatives are chosen to maximize
1,1,n 1 the absolute value of (hyper)determinants in Eqs.(25),
−
Ψ = ψ i i i . (30) (31),and(32),whichareentanglementmonotonesunder
| i i1i2i3| 1i⊗| 2i⊗| 3i
general LOCC [5, 6] (cf. Ref. [18, 19]).
i1,iX2,i3=0
7
III. CHARACTERISTICS OF MULTIPARTITE B. Two Bell pairs create any state with certainty.
ENTANGLEMENT
We show that two Bell pairs are powerful enough to
A. LOCC protocols as noninvertible flows createanystatewith certaintyinour2 2 ncases. We
× ×
find that this is also the case when one of multiparties
The recent trend of experimental quantum optics has a half of the total Hilbert space.
reachesthe stagethatwe canmanipulate twoBellstates
collectively. LOCC protocols involving local collective Theorem 2 Consider pure states in the Hilbert space
operations over two Bell states are key procedures in , =C2 C2 Cn. Two Bell pairs, the representative of
H ⊗ ⊗
for example, entanglement swapping [20, 21] (a building thegenericclass,cancreateanystate Ψ withprobability
| i
blockofquantumcommunicationprotocolslikequantum 1 by means of a local POVM measurement M on Clare
i
teleportation[22]andthequantumrepeater[23])andthe followed by local unitary operations U (i) and U (i) on
A B
creation of multipartite GHZ and W states. Although Alice and Bob, respectively.
there appear 4 particles (qubits), these can be seen as
LOCC operations in 3 parties ( = C2 C2 C4) be- Proof. We prove that we can always choose a local
H ⊗ ⊗
cause the third party Clare has initially two particles, POVM Mi on Clare, local unitary operations UA(i) and
each of which is in a Bell state with another particle on UB(i) on Alice and Bob (depending on the outcome i of
Alice’s or Bob’s side respectively, and locally performs the POVM Mi), such that
collective operations on them.
Entanglement swapping is the LOCC protocol where Ψ =UA(i) UB(i) Mi(000 +011 +102 +113 ) i,
| i ⊗ ⊗ | i | i | i | i ∀
the initial state is prepared as two Bell pairs shared (35)
among Alice, Bob, and Clare in the manner described where iMi†Mi = 11. In terms of the ”flattened” ma-
above. We note that two Bell pairs are equivalent to the trix form Ψ˜ where the indices (i ,i ) are concatenated,
P 1 2
representativeofthegenericentangledclassofdimension Eq. (35) is rewritten as
15,
|2 Belli=(|00i+|11i)AC1 ⊗(|00i+|11i)BC2 Ψ˜ =[UA(i)⊗UB(i)]11MiT ∀i. (36)
= 00(00) + 01(01) + 10(10) + 11(11) . (33)
| i | i | i | iABC12 BychoosingMiT =(Mi∗)† =(UA(i)⊗UB(i))†Ψ˜,itshould
2 Bell isalsoequivalentto 3 Φ Φ ,where be satisfied that
| i i=0| iiAB⊗| iiC12
a set of Φ (i=0,1,2,3)is the standardBell basis. So,
i
this pro|tociol can create thePbiseparable B3 state which 11= (Mi∗)†Mi∗
containsmaximalentanglement(aBellpair)betweenAl- i
X
ice and Bob, = [U (i) U (i)] Ψ˜Ψ˜ [U (i) U (i)]. (37)
A B † † A B
⊗ ⊗
(|00i+|11i)AB⊗(|(00)i+|(11)i)C12, (34) Xi
byClare’slocalcollectiveBellmeasurement(any Φ Such a local POVM M always exists, because we can
i AB i
correspondingto the outcome i of her Bell measur|emient depolarize any Ψ˜Ψ˜† to the identity 11 by random local
is equivalent to (00 + 11 ) under LOCC). Thus, en- unitary operations U (i) U (i) on Alice and Bob [24,
AB A B
| i | i ⊗
tanglement swapping can be seen as a protocol creating 25]. Thisrandomizationcanbealternativelyachievedby
isolated(maximal)entanglementbetweenAlice andBob applyingasetof16localunitaryoperationsσµ σν with
fromgenericentanglement. Inotherwords,itisgivenby equalprobabilities,whereσµ andσν (µ,ν =0A,1⊗,2,B3)are
a downward flow in Fig. 2 from the generic class to the the Pauli matrices. This completes the proof. (cid:3)
biseparable class B . Now, we readily find that the en-
3
tanglement swapping protocol is (probabilistically) suc- Theorem 3 Consider l-partite pure states in the Hilbert
cessful even when we initially prepare other 4-qubit en- space =Ck1 Ck2 Ckl−1 Ck1×k2×···×kl−1, the
H ⊗ ⊗···⊗ ⊗
tangled states in the generic class. maximallyentangledstate,which is the(k k )
1 l 1
Ontheotherhand,twoBellpairscancreatetwodiffer- (k k ) identity matrix 11 in conc×at·e·n·×atin−g th×e
1 l 1
ent kinds of genuine 3-qubit entanglement, GHZ and W indi×ces··(·i×,...−,i ),cancreateanystatewithprobability
1 l 1
by Clare’s local collective operations. These LOCC pro- 1 by means of a−local POVM on the l-th party followed
tocols are given by the downward flow, in Fig. 2, from by local unitary operations on the rest of the parties.
the genericclassestotheGHZ andWclass,respectively.
Thatis howwesee thatimportantLOCC protocolsin Proof. Thegeneralizationoftheproofinthe2 2 n
quantum information are given as noninvertible (down- case is straightforward. × ×(cid:3)
ward) flows in the partially ordered structure, such as These theorems suggestthat when one of multiparties
Fig. 2, of multipartite entangled classes. So, we expect holds at least a half of the total Hilbert space, the situ-
that the SLOCC classification can give us an insight in ation is somehow analogous to the bipartite cases. The
looking for new novel LOCC protocols by means of sev- maximally entangled state, i.e., the representative of the
eral entangled states over multiparties. generic class, can create any state with certainty.
8
IV. EXTENSION TO MIXED STATES
|000>+|011>+|102>+|113>
In this section, we extend the onion-like SLOCC clas-
sification of pure states in Sec. II to mixed states. |000>+ 1 (|011>+|101>)+|112>
2
A multipartite mixed state ρ can be written as a con-
|000>+|011>+|112>
vexcombinationofprojectorsontopurestates(extremal
points), GHZ
W
ρ= p Ψ ( ) Ψ ( ), p >0, (38)
i i i i i i
| O ih O |
Xi B
where each pure state Ψ ( ) belongs to one of the
i i
| O i
SLOCCentangledclasses(i.e.,anSLOCCorbit i). Our S
O
idea is to discuss, in Eq. (38), how ρ needs at least an
outer entangled class , among the set , in the
max i
O {O }
onion structure of Fig. 1. That is, we are interested in
FIG. 3: The SLOCC classification of multipartite mixed
theminimumof forallpossibledecompositionofρ.
max statesinthe2×2×n(n≥4)cases. Mixedstatesintheclass,
O
Because the onion picture is divided by every SLOCC- labeledby|Ψ(Omax)i∈{|000i+|011i+|102i+|113i,...,S},
invariant closed subset(i.e., everySLOCCorbitclosure) areconvexcombinationsofpurestatesinsidethe”onionskin”
ofpure states, their convexcombinationinEq.(38) con- of |Ψ(Omax)i in Fig. 1. So the outer the class is, the more
stitutes the SLOCC-invariant closed convex subsets of kinds of multipartite entangled pure states the mixed states
mixedstates(seeFig.3.). Notethat,intheonionpicture contain. The edges of the ”fan” reflect the structure of ex-
ofthemultipartitepurecases,therecanbe”competitive” tremal points (pure states), and noninvertible SLOCC oper-
ations can neverupgrade an inner class to its outerclasses.
closed subsets which never contain nor are contained by
each other. An example is the closures of three bisepa-
rable classes B in Fig. 1. So, in the extension to mixed
i
states, we should assemble all subsets of mixed states
is lower bounded by
which require at most these biseparable classes B into
i
onebiseparableconvexsubsetbytheirconvexhull. (The
argumentis similarto the classificationof3-qubitmixed kA−Bk2 ≥ (σiA−σiB)2
states in Ref. [26].) s i
X
We find that these entangled classes constitute a to- A
tally ordered structure, seen in Fig. 3, where nonin- kA−Bk2 ≥ 2(1k+ kA2 ) (τiA−τiB)2
vertible SLOCC operations can never upgrade an in- k k2 s i
X
ner class to its outer classes. For instance, we see
that the closure of W3 class of mixed states (labeled by where we assumed that kAk2 ≥kBk2.
000 + 011 + 112 )isincludedintheclosureofGHZ -
3
c|lassi(la|beleid b|y 00i0 + 1 (011 + 101 )+ 112 ). This Proof. The first inequality can readily be proven using
| i √2 | i | i | i standard results of linear algebra [27]. The second in-
classification reflects a diversity of multipartite pure en-
equality can be proven as follows. Defined X = A B;
tangled states a mixed state ρ consists of: the outer the −
then
class of ρ is, the more kinds of resources it contains.
Needless tosay,itis verydifficult togivethe criterionto ATA BTB = XAT +AXT XTX
distinguish convex subsets, even to distinguish the sepa- k − k k − k
2 X A + X 2 (39)
rableconvexsubset(i.e.,theseparabilityproblem),since ≤ k kk k k k
wefacethetroubleevaluatingallpossibledecompositions
The left term of this inequality is bounded below by
in Eq. (38) for a given ρ. Let us however prove that the
convex combination of nine classes of pure states gives
rise to convex sets that are not of measure zero, in con- ATA BTB (τA τB)2. (40)
k − k≥ i − i
trast with the pure case (cf. Ref. [26]). This can easily s i
X
be established with the help of the following lemma:
The second inequality of the lemma can now be checked
Lemma 1 Given two matrices A,B with corresponding by making use of straightforwardalgebra. (cid:3)
ordered singular values σA,B . Denote the ordered sin- The fact that a structure of convex sets as depicted
{ i } in Fig. 3 is obtained, can now be proven by combin-
gular values of the matrices ATA and BTB as τA,B .
{ i } ing the previous lemma with the results of the table in
Then the Hilbert-Schmidt norm
Eq. (23): indeed, it can easily be shown that whenever
there exists a pure state in one class that is separated
kA−Bk2 = tr((A−B)†(A−B)) fromallpurestatesinanotherclasswithafinitenon-zero
q
9
Hilbert-Schmidt distance, then the corresponding class quantum information processing are given as noninvert-
for mixed states is absolutely separated from the other ible(downward)flowsbetweendifferententangledclasses
one. The previous lemma guarantees that the Hilbert- inthepartiallyorderedstructureofFig.2. Inparticular,
Schmidt norm will be non-zero for all states having a weshowthattwoBellpairsarepowerfulenoughtocreate
different rank for the matrices R or RTR (see the ta- anystatewith certaintyinoursituation. Basedonthese
ble in Eq. (23)). More specifically, all the W-classes are observations, we suggest that SLOCC classifications can
embeddedin the respectiveGHZ-classes,andthe convex be useful in looking for new prototypes of novel LOCC
structure as depicted in Fig. 3 is obtained. protocols.
V. CONCLUSION
Acknowledgments
In this paper, (i) we give the complete classification
of multipartite entangled states in the Hilbert space A.M. would like to express his sincere gratitude to
= C2 C2 Cn under stochastic local operations J.I. Cirac, M. Lewenstein, and the members of these
H ⊗ ⊗
andclassicalcommunication(SLOCC).Ourstudycanbe groups for stimulating discussions and warm hospital-
seen as the first example of the SLOCC classification of ity during his visit. He also thanks K. Matsumoto
multipartite entanglement where one of multiparties has for the discussions on Sec. IIB, as well as the mem-
more than one qubits. We show that nine classes consti- bers of both the University of Tokyo and the ERATO
tute the five-gradedpartiallyorderedstructure ofFig.2. Project for helpful comments. The work of A.M. is par-
Remarkably, a unique maximally entangled class lies on tially supported by the Grant-in-Aid for JSPS Fellows.
its top, in contrast with the l-qubit (l 3) cases. We The work of F.V. was supported in part by the E.C.
≥
also present a convenient criterion to distinguish these (projects RESQ and QUPRODIS) and the Kompeten-
classes by SLOCC-invariantentanglement measures. znetzwerk ”Quanteninformationsverarbeitung” der Bay-
(ii) We illustrate that important LOCC protocols in erischen Staatsregierung.
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