Classification of Mixed Three-Qubit States
A.Acín,1D.Bruß,2M.Lewenstein,2and A.Sanpera 2
1
Departament d’Estructura i Constituents de la Matèria,Universitat de Barcelona,08028Barcelona,Spain
2
Institut für Theoretische Physik,Universität Hannover,30167Hannover,Germany
(Received 8March 2001;published 3July 2001)We introduce a classification of mixed three-qubit states,in which we define the class of parable,biparable,W ,and Greenberger-Horne-Zeilinger states.The class are successively embedded into each other.We show that contrary to pure W -type states,the mixed W class is not of measure zero.We construct witness operators that detect the class of a mixed state.We discuss the conjecture that all entangled states with positive partial transpo (PPTES)belong to the W class.Finally,we prent a new family of PPTES “edge”states with maximal ranks.
DOI:10.1103/PhysRevLett.87.040401
PACS numbers:03.65.Ud,03.65.Ca,03.67.Hk
The rapidly increasing interest in quantum information processing has motivated the detailed study of entangle-ment.Whereas entanglement of pure bipartite systems is well understood,the classification of mixed states accord-ing to the degree and character of their entanglement is still a matter of intensive rearch (e [1]).It was soon real-ized that the entanglement of pure tripartite quantum states is not a trivial extension of the entanglement of bipartite systems [2,3].Recently,the first results concerning the en-tanglement of pure tripartite systems have been achieved [4–6].There,the main goal has been to generalize the con-cept of the Schmidt decomposition to three-party systems [4,5],and to distinguish class of locally inequivalent states [6].The knowledge of mixed tripartite entangle-ment is much less advanced (e,however,[7–9]).
In this Letter we introduce a classification of the whole space of mixed three-qubit states into different entangle-ment class.We provide a method to determine to which class a given state belongs (tripartite witness).We also discuss the characterization of entangled states that are posi-tive under partial transposition (PPTES).Finally,we in-troduce a new family of PPTES for mixed tripartite qubits.Our proposal to classify mixed tripartite-qubit states is done by specifying compact convex subts of the space of all states,which are embedded into each other.This idea vaguely rembles the classification of bipartite systems by their Schmidt number [9–11].However,as shown later our classification does not follow the Schmidt number [9].Also in this respect,entanglement of tripartite systems dif-fers genuinely from the one of bipartite quantum systems.Before prenting our results concerning mixed states,we briefly review some of the recent results on pure three-qubit states.Any three-qubit vector (pure state)can be written as
j c GHZ ͘l 0j 000͘1l 1e i u j 100͘1l 2j 101͘1l 3j 110͘1l 4j 111͘,
(1)where l i $0,P i l 2
i 1,u [͓0,p ͔,and ͕j 0͘,j 1͖͘de-notes an orthonormal basis in Alice ’s,Bob ’s,and Charlie ’s spaces,respectively [4].Apart from parable and bipa-rable pure states,there exist also two differ
ent types of locally inequivalent entangled vectors:the so-called Greenberger-Horne-Zeilinger (GHZ)type [2]and W type [6].V ectors belonging to GHZ and W types cannot be transformed into each other by local invertible and,in particular,local unitary [12]operations.Generically,a vector described by Eq.(1)is of the GHZ type,while W vectors can be written as
j c W ͘l 0j 000͘1l 1j 100͘1l 2j 101͘1l 3j 110͘.(2)W vectors form a t of measure zero among all pure states [6].Also,given a W vector one can always find a GHZ vector as clo to it as desired by adding an in finitesimal l 4term to the right-hand side of Eq.(2).Furthermore,the so-called tangle,t ,introduced in [13],can be ud to detect the type,since t ͑j c W ͒͘0[6].
Mixed states of three-qubit systems can be classi fied generalizing the classi fication of pure states.To this aim we de fine the following (e Fig.1):(i)the class S of pa-rable ,tho that can be expresd as a convex sum of projectors onto product vectors;(ii)the class B of biparable ,tho that can be expresd as a con-vex sum of projectors onto product and bipartite entangled vectors (A -BC ,B -AC ,and C -AB );(iii)the class W of W ,tho that can be expresd as a convex sum
FIG.1.Schematic structure of the t of all three-qubit states.S :parable class;B :biparable class (convex hull of bipa-rable states with respect to any partition);W class and GHZ class.
of projectors onto product,biparable,and W -type vec-tors;(iv)the class GHZ of GHZ ,the t of all physical states.All the ts are convex and compact,and satisfy S ,B ,W ,GHZ.States in S are not en-tangled.No genuine three-party entanglement is needed to prepare entangled states in the subt B n S .The forma-tion of entangled states in W n B requires W -type vectors with three-party entanglement,but zero tangle,which is an entanglement monotone decreasing under local opera-tions and classical communication [6].Finally,the class GHZ contains all types of entanglement,and,in particu-lar,GHZ-type vectors are needed to prepare states from GHZ n W .The introduced class are invariant under lo-cal unitary or invertible nonunitary operations,while local POVM ’s can transform states only from a “higher ”to a “lower ”class.
Notice that since GHZ vectors can be expresd as the sum of only two product ,j GHZ ͑͘j 000͘1j 111͒͘͞p 2,whereas the minimum number of product terms forming a W vector is three [4,6],as in the state j W ͑͘j 100͘1j 010͘1j 001͒͘͞p our scheme may em somehow counterintuitive.Indeed,for bipartite systems,states with lower Schmidt ,lower number of product terms in the Schmidt decomposition,are embedded into the t of states with higher Schmidt number [10].One is tempted to extend this classi fication to tripartite systems as S ,B ,GHZ ,W ,where now W is the t of all states.However,such generalization is evidently wrong,becau the t of GHZ states in such classi fication cannot be clod.
Having established the structure of the t of mixed three-qubit states,we show how to determine to which class a given state r belongs.To this aim,we u the approach developed previously in the construction and op-timization of witness operators [11,14,15].
We denote the range of r by R ͑r ͒,its rank by r ͑r ͒,its kernel by K ͑r ͒,and the dimension of K ͑r ͒by k ͑r ͒.Fol-lowing the approach of the best parable approximation [16],one can decompo any state r as a convex combi-nation of a W -class state and a remainder d ,
r l W r W 1͑12l W ͒d ,
占地面积英文(3)
where 0#l W #1,and R ͑d ͒does not contain any W vec-tor.Maximization of l W leads to the best W approxima-tion of r .Notice that only for r belonging to the GHZ n W class,this decomposition is ,l W fi1.Also,r ͑d ͒1,since any subspace spanned by two linearly in-dependent GHZ vectors contains at least one pure state with zero tangle.Therefore,any W approximation must have the form
r l W r W 1͑12l W ͒j c GHZ ͗͘c GHZ j .
(4)
Similarly,one can express r in the best biparable ap-proximation as
r l B r B 1͑12l B ͒d ,
(5)
where now R ͑d ͒must not contain any biparable ,r ͑d ͒,4,since any N -dimensional subspace of the 23N space contains at least one product vector [17].We u the above decompositions to construct opera-tors that detect the desired subt (e [15]).In analogy to entanglement witness and Schmidt witness we term the operators tripartite witness.The existence of wit-ness operators is a conquence of the Hahn-Banach theo-rem,which states that a point outside a convex compact t is parated from that t by a hyperplane.The equa-tion Tr ͑W r ͒0describes such a hyperplane,and one calls W a witness operator.For example,in our tting,a W witness is an operator W W such that Tr ͑W W r B ͒$0holds ;r B [B ,but for which there exists a r W [W n B such that Tr ͑W W r W ͒,0.
Any GHZ witness (W witness)has the canonical form W Q 2e ',where Q is a positive operator whic
h has no W -type (B -type)vectors in its kernel;thus k ͑Q ͒1[k ͑Q ͒,4][11,15].An example of a GHZ witness is
W GHZ
3
4
'2P GHZ ,(6)
where P GHZ is the projector onto j GHZ ͘.The value 3͞4corresponds to the maximal squared overlap between j GHZ ͘and a W vector.This construction guarantees that Tr ͑W GHZ r W ͒$0for any W state,and since Tr ͑W GHZ P GHZ ͒,0,there is a GHZ n W state which is detected by W GHZ .The maximal overlap is obtained as follows:due to the symmetry of j GHZ ͘we need to consider only W vectors that are symmetric under the exchange of any of the three qubits [18].Therefore,we have to consider all local trilateral rotations of j c W ͘k 0j 000͘1k 1͑j 100͘1j 010͘1j 001͒͘,where k 0,k 1are
real and k 2013k 2
11.Becau of the symmetry,such rotations can be parametrized for all parties as j 0͘!a j 0͘1b j 1͘,j 1͘!b ءj 0͘2a ءj 1͘,with j a 2j 1j b 2j 1.Thus,the overlap ͗GHZ j c W ͘is a function of six parameters with two constraints,and can be maximized using Lagrange multipliers.An optimal choice of parame-ters is k 00,k 11͞p ,and b 2a 1͞p
万人迷英文
This
leads to j ͗GHZ j c W ͘j 2
应试教育英文max 3͞4.
Analogously,we can construct a W witness as
W W 1
2
3
'2P W ,(7)
where P W is now the projector onto a vector j W ͘,and 2͞3corresponds to the maximal squared overlap between j W ͘and a B vector.Another example of a W witness is
W W 2
1非限制定语从句
2
'2P GHZ ,(8)
where now 1͞2is the maximal squared overlap between j GHZ ͘and a B type vector [19].The W vector that has maximal overlap with j GHZ ͘is detected by W W 2.
The tripartite witness W W 2allows one to prove that the class of mixed W n B states is not of measure zero:consider the family of states in C 2≠C 2≠C 2given by the convex
sum of the identity and a projector onto a W -type vector that maximizes the squared overlap with j GHZ ͘,
r
12p
8
'1pP W .(9)
Obviously,the states (9)belong at most to W .The range for the parameter p ,in which W W 2detects r ,i.e.,Tr ͑W W 2r ͒,0,is found to be 3͞5,p #1,and is bigger than the one found by using W W 1.Taking any p which has a finite distance to the border of this ,p 23͞5.D and 12p .D ,it is always possible to find a finite region around r which still belongs to the W n B class.This can be en by considering
˜r ͑12e ͒∑12p
8'1pP W ∏1es ,(10)
where s is an arbitrary density matrix,which covers all directions of possible deviations from r in the operator space.In the worst ca s is orthogonal to P GHZ ,so that Tr ͑P GHZ s ͒0,and therefore Tr ͑W W 2˜r ͒͑12e ͒Tr ͑W W 2r ͒1e ͞2.As long as the relation e ,͑5p 23͒͑͞5p 11͒holds,the corresponding state ˜r is still detected by W W 2.Moreover,one can also find
a finite e 0such that if e ,e 0,then ˜r
is in the W class.The bound e 0
is obtained,for instance,by demanding that ͑12e 0͒͑12p ͒'͞81e 0s is biparable.The interction of the two intervals gives a finite range for e where the state ˜r is in the W n B class.This proves that the t of mixed W n B states contains a ,is not of measure zero.squirrels
We discuss now some possible conquences of our re-sults for PPTES of three qubits,for which the partial trans-pos r T A ,r T B ,and r T C are positive.Any of the states can be decompod as
r l S r S 1͑12l S ͒d ,
(11)
where r S is a parable state and d is an edge state [20].
We conjecture that PPTES cannot belong to the GHZ n W class ;i.e.,they are at most in the W class.This conjecture is true for states that have edge states with low ranks in the above decomposition.It was shown in [17]that for bipartite systems in C 2≠C N ,the rank of PPTES must be
larger than N ,and if r ͑r ͒#N and r T A $0,then the state r is parable.Thus,any PPTES of three qubits with r ͑r ͒#4is biparable with respect to any partition;examples of such states are the UPB states from Ref.[7].For the ca of higher ranks we can give some support only for our conjecture.We proceed as in [11],and obrve first that it suf fices to prove the conjecture for the edge states.For the states,the sum of ranks satis fies r ͑d ͒1r ͑d T A ͒1r ͑d T B ͒1r ͑d T C ͒#28[20].Any PPT entangled state can be detected only by a nondecompos-able entanglement witness,which in the ca of tripartite systems has the canonical form W nd W d 2e 'where W d P 1P Q T X
X is a decomposable operator with P ,Q X $0,R ͑P ͒K ͑d ͒,R ͑Q X ͒K ͑d T X ͒for some edge state d ,and X A ,B ,C [20].We restrict ourlves to edge states with the maximal sum of ,states d with ͓r ͑d ͒,r ͑d T A ͒,r ͑d T B ͒,r ͑d T C ͔͓͒8,8,7,5͔,͓8,8,6,6͔,͓8,7,7,6͔,͓7,7,7,7͔and permuta-tions.Indeed,if the conjecture is true for the states,it will be true for all edge states,and thus for all PPTES,since the edge states with maximal sum of ranks are den in the t of all edge states [11].We conjecture that for the ca of edge states with maximal sum of ranks it is always possible to find a pure W -type vector,j f W ͘,such that for any nondecomposable witness W nd of d ,͗f W jW d j f W ͘#0,so that ͗f W jW nd j f W ͘,0.That means W nd cannot be a GHZ witness,so the edge state d belongs to the W class.If this holds for any d it implies that all PPTES belong to the W class.
Any W vector can be obtained by local invertible opera-tions applied to j W ͘,i.e.,can be written as
j f W ͘a A j e 2,f 1,g 1͘1a B j e 1,f 2,g 1͘1a C j e 1,f 1,g 2͘.
(12)
We denote
j F A ͘j e ء
2
,f 1,g 1͘,j C A ͘a B j e ء1,f 2,g 1͘1a C j e ء
1,f 1,g 2͘,j F B ͘j e 1,f ء
2
,g 1͘,j C B ͘a A j e 2,f ء1,g 1͘1a C j e 1,f ء
1,g 2͘,(13)
重听
j F C ͘j e 1,f 1,g ء2͘,
j C C ͘a A j e 2,f 1,g ء1
͘1a B j e 1,f 2,g ء
1͘.In order to ful fill the condition ͗f W jW d j f W ͘#0we demand that Q X j F X ͘0;P j f W ͘0,and Q X j C X ͘0for X A ,B ,C .The latter four conditions form four linear homogeneous equations for the a X ’s,who solu-tions exist if two 333determinants vanish.Together with the first three conditions this gives at most five equations in the ca r ͑d ͒,8,and six equations in the worst ca r ͑d ͒8,for the six complex parameters characterizing j e i ͘,j f i ͘,and j g i ͘,with i 1,2.For r ͑d ͒,8[r ͑d ͒8]one expects here a one complex parameter (finite,but large)family of solutions.At the same time ͗f W jW d j f W ͘2Re P X a X ͗C ءX X j Q T X X j F ءX X ͘,(where j F ءX ͘denotes partial complex conjugation with respect to X ),i.e.,is a Hermitian form of a X ’s,who diagonal elements vanish,since j C X ͘does not depend on a X .Employing the freedom of choosing the solu-tions from the family,one expects to find at least one with ͗f W jW d j f W ͘#0.In this way we obtain the W vector we were looking for.For the cas ͓6,8,8,6͔and ͓5,8,8,7͔,a similar argument indeed shows that there should exist a biparable state,j c B ͘,such that ͗c B jW nd j c B ͘,0.Note that the above method of arch-ing j c W ͘(j c B ͘)for a given d ,if successful,provides a suf ficient condition for d to belong to the W class (B class).
Finally,we prent an example for a PPTES entangled edge state with ranks͓7,7,7,7͔.We introduce
r1
n
phenomB B新年快乐的英语
B B
B B
B B
B B
B B
B@
10000001轮廓的意思
0a000000
00b00000
000c0000
00001000
000001b00
0000001a0
10000001
1
C C
C C
C C
C C
C C
C C
C A
(14)
with a,b,c.0and n21a11͞a1b11͞b1
c11͞c.The basis is͕000,001,010,011,100,101,110, 111͖.This density matrix has a positive partial trans-
po with respect to each subsystem.One es imme-
diately that r͑r͒r͑r T A͒r͑r T B͒r͑r T AB͒7.In order to check that r is a PPT entangled edge state,one
has to prove that it is impossible tofind a product vector j f͘[R͑r͒,such that at the same time j fءX͘[R͑r T X͒for XA,B,C.This,indeed,is not possible,as one read-ily concludes by looking at the kernels directly:one can-notfind a product vector j f͘that is orthogonal to j000͘2 j111͘,whereas at the same time
j fءA͘Ќj011͘2c j100͘, j fءB͘Ќj010͘2b j101͘,and j fءC͘Ќj001͘2a j110͘,un-less the condition abc is fulfilled.Thus,for generic a,b,c we have found a family of bound PPT entangled edge states of three qubits with maximal sum of ranks.By direct inspection we obrve that r fulfills our conjecture, and is biparable with respect to any partition.It can be ,as a sum of parable projectors and a B state acting in the232subspace spanned by Alice’s space and the vectors j00͘and j11͘in Bob’s-Charlie’s space.
To summarize,we show that the t of density matrices
for three qubits has an“onion”structure(e Fig.1)and contains convex compact subts of states belonging to the parable S,biparable B,W and GHZ class,respec-tively.We provide the canonical form of witness operators for the GHZ and W class,and give thefirst examples of such witness.The study of the family of tripartite states given in Eq.(9)allows us to prove that the W class is not of measure zero.We conjecture and give some evi-dence that all PPTES of three-qubit systems do not require GHZ-type pure states for their formation.We formulate a sufficient condition which allows us to check construc-tively if a state belongs to the W class(B class).Finally, we prent a family of PPT entangled edge states of three qubits with maximal sum of ranks.
This work has been supported by DFG(SFB407 and Schwerpunkt“Quanteninformationsverarbeitung”), the ESF-Programme PESC,and the EU IST-Programme EQUIP. A.A.thanks the University of Hannover for hospitality,E.Janéfor uful comments,and the Spanish MEC(AP-98)forfinancial
support.
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