Entanglement witnesses and concurrence for multi-qubit states

0.11 MB·English

by  Hoshang Heydari

Entanglement witnesses and concurrence for
multi-qubit states
Hoshang Heydari
Institute of Quantum Science, Nihon University,
1-8 Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8308, Japan
7 Email: [email protected]
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February 1, 2008
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a
Abstract
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We establish a relation between concurrence and entanglement wit-
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1 nesses. Inparticular,weconstructentanglementwitnessesforthree-qubit
W and GHZ states in terms of concurrence and different set of opera-

1 tors that generate it. Wealso generalize our construction for multi-qubit
v states.
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0 1 Introduction
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Entanglement witnesses are a practical way to detect entangled states [1]. One
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recent example of detecting four-photon entangled state by entanglement wit-
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/ nesses has been reported in [2]. Entanglement witnesses has a rich geometrical
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structure andtheir constructionis a consequenceofthe Hahn-Banachtheorem.
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On other hand, concurrence is one of well-known measure of entanglement.
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t One can also find different construction and definition of concurrence for both
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a pure and mixed bipartite and multipartite states [3, 4]. Recently, we have
u also constructed generalized concurrence for pure general multipartite states
q
based on the complement of a positive operator valued measure (POVM) on
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v quantum phase [5]. In particular, by rewriting orthogonal complement of a
i POVM on quantum phase as sums and taking the expectation value of each of
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these operators, we were able to construct a general formula for concurrence.
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a In this paper, we will establish a connection between entanglement witnesses
and concurrence. In section2 we will give shortintroductionto constructionof
entanglement witnesses with some example for three-qubit W and GHZ state.
In section 3 we will introduce our construction of concurrence based on the
complement of POVM on quantum phase. And finally, in section 4 we will
show how we can construct entanglement witnesses for three-qubit states in
terms of concurrence and it’s generating operators. Our construction suggests
a systematicwayofconstructingentanglementwitnesses formulti-qubit states.
We will considera generalmultipartite quantumsystemwith m subsystems
which we denote as Q = Q Q ···Q , and denoting its general state as |Ψi =
1 2 m
N1 ··· Nm α |l ,l ,...,l i∈H =H ⊗H ⊗···⊗H ,where
Pl1=1 Plm=1 l1,l2,...,lm 1 2 m Q Q1 Q2 Qm
the dimension of the jth Hilbert space is given by N = dim(H ). Moreover,
let ρ = N p |Ψ ihΨ |, for all 0≤p ≤1 and Nj p =1, deQnjote a density
Q Pi=1 i i i i Pi=1 i
1
operator acting on the Hilbert space H . The most well know examples of
Q
multi-qubit states are |Ψ i and |Ψ i state. These quantum states are
GHZm Wm
defined by |ΨGHZmi= √12(|1,...,1i+|2,...,2i)and|ΨWmi= √1m(|m−1,2i),
where |m−1,2i denotes the totally symmetric state including m−1 ones and
1 twos. In the following section, we will call our local operators based on these
classes of states.
2 Entanglement witnesses
In this section we will give a short introduction to entanglement witnesses for
general multipartite state. Let ρ be a density operator acting on H . Then,
Q Q
the density operator ρ is said to be fully separable, which we will denote by
ρsep, with respect to tQhe Hilbert space decomposition, if it can be written as
ρQsep = N p m ρk , N p =1 for some positive integer N, where p
arQe posPitivk=e1reaklNnujm=1beQrsjanPdkρ=k1 dkenotes a density operator on Hilbert spacke
H . If ρp represents a pure staQtje, then the quantum system is fully separable
ifQρpj canQbe written as ρsep = m ρ , where ρ is the density operator on
H Q. If a state is not sepQarableN, tjh=e1n iQtjis said toQbje an entangled state.
Qj
Now, for every entangled state ρ there exist a Hermitian operator W that
satisfies the following conditions: i)QTr(Wρ ) < 0, and ii) Tr(Wρsep) ≥ 0, for
allseparablestateρsep. TheW operatorusuQallycalledentanglemenQtwitnesses.
By construction thiQs operator have a positive expectation value on the set of
all separable states. Thus, if the measurement of W on a quantum system
representedbyρ producenegativevalue,thenthisquantumρ isanentangled
Q Q
state. Moreover, the existence of this operator is guaranteed by the Hahn-
Banachtheorem: ForacompactandconvexsetS,ifρ isnotbelongtoS,then
thereexistsahyper-planethatseparateρ fromS. NeQxt,wewillconstructtwo
Q
wellknownentanglementwitnessesforGHZandWclassstates. Forthree-qubit
GHZ and W states entanglement witnesses can be constructed as
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WGHZ3 = 4I−|ΨGHZ3ihΨGHZ3|, WW3 = 3I−|ΨW3ihΨW3|, (1)
where I is an identity matrix. In following section we will rewrite these entan-
glement witnesses in terms of concurrence.
3 Different classes of POVM for general multi-
partite states
Inthissection,wewillconstructconcurrenceforgeneralpuremultipartitestates
Qp(N ,...,N ),wheresuperscriptpindicatesthatweareonlyconsideringpure
1 m
multipartite states. In our construction, we will use linear operators that are
constructed by the orthogonal complement of POVM on quantum phase [5].
The POVM for each subsystem Q is defined by
j
Nj
∆Qj(ϕkj,lj)= X eiϕkj,lj|kjihlj|, (2)
lj,kj=1
2
where ϕ =−ϕ (1−δ ). Moreover,the orthogonalcomplement of our
kj,lj lj,kj kjlj
POVMisgivenby∆ (ϕ )=I −∆ (ϕ ),whereI isthe N -by-N
Qj kj,lj Nj Qj kj,lj Nj j j
identity matrix foresubsystem j. For m-partite quantum systemwe construct a
operator (matrix) by taking the tensor product of m subsystems as follows
∆ (ϕ ,...,ϕ ) = ∆ (ϕ )⊗···⊗∆ (ϕ ), (3)
Q k1,l1 km,lm Q1 k1,l1 Qm km,lm
e e e
where ∆ (ϕ ,...,ϕ ) has phases that are sums or differences of phases
Q k1,l1 km,lm
originateing from two and m subsystems. That is, in the latter case the phases
of ∆ (ϕ ,...,ϕ ) take the form (ϕ ± ϕ ± ... ± ϕ ) and
Q k1,l1 km,lm k1,l1 k2,l2 km,lm
idenetification of these joint phases makes our distinguishing possible. Thus, we
can define linear operators for the Wm class which are sums and differences of
phases of two subsystems, i.e., (ϕ ±ϕ ). That is, for the Wm class
kr1,lr1 kr2,lr2
we have
∆Wm (Λ ) = I ⊗···⊗∆ (ϕπ2 )⊗···⊗∆ (ϕπ2 )⊗···⊗I ,
eQr1r2 m N1 eQr1 kr1,lr1 eQr2 kr2,lr2 Nm
(4)
π
where 1 ≤ r < r ≤ m and the notation ∆ (ϕ2 ) means that we evaluate
1 2 Qj kj,lj
e
∆ (ϕ ) at ϕ = π/2 for all k ,l . In order to simplify our presentation,
Qj kj,lj kj,lj j j
wee have used (Λ ) = (k ,l ; ...;k ,l ) as an abstract multi-index notation.
m 1 1 m m
Next, we could write the linear operator ∆Wm (Λ ) as a direct sum of the
m
upper and lower anti-diagonal eQr1r2
∆Wm (Λ ) = U∆Wm (Λ )+L∆Wm (Λ ). (5)
m m m
eQr1r2 eQr1r2 eQr1r2
For the GHZm class, we define linear operators based on our POVM which are
sums and differences of phases of m-subsystems, i.e., (ϕ ±ϕ ±...±
kr1,lr1 kr2,lr2
ϕ ). That is, for the GHZm class we have
km,lm
∆GHZm(Λ ) = ∆ (ϕπ )⊗···⊗∆ (ϕπ2 ) (6)
eQr1r2 m eQ1 k1,l1 π eQr1 kr1,lr1
⊗···⊗∆ (ϕ2 )⊗···⊗∆ (ϕπ ),
Qr2 kr2,lr2 Qm km,lm
e e
where ∆ (ϕπ ) indicates that we evaluate ∆ (ϕ ) at ϕ = π for all
Qj kj,lj Qj kj,lj kj,lj
k ,l . Neote also that, in this case we get an oeperator which has the structure
j j
of the Pauli operator σ embedded in a higher-dimensional Hilbert space and
x
coincides with σx for a single-qubit. There are m(m2−1) linear operators for the
GHZm class. In our recent paper [5] we have construct concurrence for general
multipartite states based on these sets of operators which is given by
C(|Ψi) = (N { C(QWm)+ C(QGHZm)+...})1/2,
m X r1r2 X r1r2
1 r1<r2 m 1 r1<r2 m
≤ ≤ ≤ ≤
where N is a normalization constant. The definition of these terms can be
m
fund in the above mentioned paper. For three-qubit state it is given by
C(|Ψi) = (2|α α −α α |2+2|α α −α α |2
1,1,1 2,2,1 1,2,1 2,1,1 1,1,2 2,2,2 1,2,2 2,1,2
+2|α α −α α |2+2|α α −α α |2
1,1,1 2,1,2 1,1,2 2,1,1 1,2,1 2,2,2 1,2,2 2,2,1
+2|α α −α α |2+2|α α −α α |2
1,1,1 1,2,2 1,1,2 1,2,1 2,1,1 2,2,2 2,1,2 2,2,1
+|α α −α α |2+|α α −α α |2
1,1,1 2,2,2 1,1,2 2,2,1 1,1,1 2,2,2 1,2,1 2,1,2
+|α α −α α |2+|α α −α α |2
1,1,1 2,2,2 1,2,2 2,1,1 1,1,2 2,2,1 1,2,1 2,1,2
+|α1,1,2α2,2,1−α1,2,2α2,1,1|2+|α1,2,1α2,1,2−α1,2,2α2,1,1|2)21,
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where we have set N = 1/4. Next, we evaluate this measure for |ΨW3i and
|ΨGHZ3istates. Forthesetwowell-knownstateswehaveC2(|ΨW3i)=2·391 = 23
and C2(|ΨGHZ3i)= 43 respectively.
4 Entanglement witnesses based on concurrence
Now we will systematically construct entanglement witnesses for multipartite
states. Butbeforethatweneedtointroducesomenewnotations. Form-partite
quantum system we will denote by ∆±(ϕk1,l1,...,ϕkm,lm) the operator which
has only phases that either can be weriQtten as (+ϕ +ϕ +...+ϕ )
k1,l1 k2,l2 km,lm
or (−ϕ −ϕ −...−ϕ ). This means the sign in front of all these
k1,l1 k2,l2 km,lm
phasesiseitherpositiveornegative. Theelementsof∆+(ϕ ,...,ϕ )and
k1,l1 km,lm
∆−(ϕk1,l1,...,ϕkm,lm)areplacedoverandundermaienQdiagonalrespectively. So
weeQcan also rewrite our operators as
∆±(ϕk1,l1,...,ϕkm,lm)=∆+(ϕk1,l1,...,ϕkm,lm)+∆−(ϕk1,l1,...,ϕkm,lm). (7)
eQ eQ eQ
For example for three-qubit states we have two classes of these operators, that
is W class and GHZ class operators. For W class we have
∆eQW13,2±(Λ3) = (cid:16)∆eQ1(ϕ1π2,2)⊗∆eQ2(ϕ1π2,2)⊗I2(cid:17)±.
q∆euQWb1i3,t3±G(ΛH3Z) awnedh∆eavQWe23,3±(Λ3) are defined in the similar way. Moreover, for three-
∆eGQH1,2Z3± = (cid:16)∆eQ1(ϕ1π2,2)⊗∆eQ2(ϕ1π2,2)⊗∆eQ3(ϕπ1,2)(cid:17)±.
Now, we are in a position to rewrite entanglement witnesses for three-qubit
states interms of concurrenceand it’s operators. For example for athree-qubit
GHZ state the entanglement witnesses can be constructed as
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WGHZ3 = 4I−|ΨGHZ3ihΨGHZ3|=C2(|ΨGHZ3i)I−|ΨGHZ3ihΨGHZ3|. (8)
We can also go one step further by setting C2(|ΨGHZ3i) = Cg = Cg +1, and
diagonal matrix D = diag(C ,C ,...,C ,C ) then an entanglement witnesses
g g g g g
for three-qubit GHZ state can be written as
WGHZ3 = Dg−∆eGQH1,2Z3±(Λ3). (9)
For three-qubit W state entanglement witnesses can be constructed as
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WW3 = 3I−|ΨW3ihΨW3|=C2(|ΨW3i)I−|ΨW3ihΨW3| (10)
Now,letC2(|ΨW3i)=Cw =Cw+1,anddefinediagonalmatrixDw =diag(Cw,Cw,
C ,C ,C ,C ,C ,C ).Thenanentanglementwitnessesforthree-qubitWstate
w w w w w w
can be written as
WW3 = Dw−Xr<sU∆eQWr3,s±(Λ3). (11)
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Theseconstructionsuggestthatwecangeneralizeentanglementwitnessesforat
least multipartite W and GHZ states. For example the entanglement witnesses
for multi-qubit W states is given by
W = γI−|Ψ ihΨ |=C2(|Ψ i)I−|Ψ ihΨ | (12)
Wm Wm Wm Wm Wm Wm
= Dw−Xr<sU∆eWQrm,s±(Λm),
where D is a diagonalmatrix with elements C =C2(|Ψ i)−1 for nonzero
w w Wm
coefficients α 6= 0 of multi-qubit W state and C = C2(|Ψ i) when
l1,l2,...,lm w Wm
α = 0 for multi-qubit W state. We have already seen construction of
l1,l2,...,lm
this diagonal matrix for three qubit states.
Moreover,anentanglementwitnessesformulti-qubitGHZ statesisgivenby
W = γI−|Ψ ihΨ |=C2(|Ψ i)I−|Ψ ihΨ |
GHZm GHZm GHZm GHZm GHZm GHZm
= Dg−∆eGQH1,2Zm±(Λm), (13)
where D is defined in the same way as for W state, e.g., by replacing W state
g
by GHZ state. However, we need carefully chose the normalization constant
N in the expression for concurrence. We also can write a general formula for
m
entanglement witnesses of pure multi-qubit state in terms of concurrence
WΨ = γI−|ΨihΨ|=C2(|Ψi)I−|ΨihΨ|=D−Xr<s∆eΨQ±r,s(Λm), (14)
where D is a diagonalmatrix with elements C =C2(|Ψi)−1 for nonzero coeffi-
cients α 6=0 of the state |Ψi and C =C2(|Ψi) whenever α =0
l1,l2,...,lm l1,l2,...,lm
for given state |Ψi. For GHZ states we only need to consider r = 1 and s = 2
withoutanypartitions. Thus,itispossibletosystematicallyconstructentangle-
mentwitnessesformulti-qubit statesintermsofconcurrenceandthe operators
which generateit. The geometricalstructure of entanglementwitnesses usually
related to geometry of Hilbert space in functional analysis. On the other hand
the geometry of concurrence and completely separable set is givenby the Segre
variety which is defined to be the image of the Segre embedding. Thus these
connectionbetweenentanglementwitnesses andconcurrencealsosuggestsome,
not yet known, similarity between geometrical structure of these entanglement
toolsthatdistinguishbetweenseparableandentangledsetofmulti-qubitstates.
Acknowledgments: The author acknowledges the financial support of the
Japan Society for the Promotion of Science (JSPS).
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