Method for measuring the entanglement of formation for arbitrary-dimensional pure states

0.1 MB·English

by  Ming Li

Method for measuring the entanglement of formation for arbitrary-dimensional pure
states
Ming Li1 and Shao-Ming Fei2,3
1Department of Mathematics, School of Science, China University of Petroleum, Qingdao 266555, China
2School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
3Max-Planck-Institute for Mathematics in the Sciences, Leipzig 04103, Germany
Entanglement of formation is an important measure of quantum entanglement. We present an
experimental way to measure the entanglement of formation for arbitrary dimensional pure states.
The measurement only evolves local quantummechanical observables.
PACSnumbers: 03.67.-a,02.20.Hj,03.65.-w
2
1
0
Quantum entangled states have become the most im- quantum state on is generally of the form,
2 HA⊗HB
portant physical resources in quantum communication,
n information processing and quantum computing. One m n
a of the most difficult and fundamental problems in en- |ψi=XXaij|iji, aij ∈C (1)
J tanglement theory is to quantify the quantum entangle- i=1j=1
0
ment. A number of entanglement measures such as the
3 with normalization
entanglementofformationanddistillation[1,2],negativ-

] ity [3] and concurrence [4] have been proposed. Among m n
h theseentanglementmeasures,theentanglementofforma- XXaija∗ij =1. (2)
p tion,whichquantifiestherequiredminimallyphysicalre- i=1j=1
-
t sourcestoprepareaquantumstate,playsimportantroles
n The entanglement of formation of ψ is defined as the
inquantumphasetransitionforvariousinteractingquan- | i
a partial entropy with respect to the subsystems [1],
u tum many-body systems [5] and may significantly affect
q macroscopic properties of solids [6]. Thus the quantita- E(ψ )= Tr(ρAlog ρA)= Tr(ρBlog ρB), (3)
[ tive evaluation of entanglement of formation is of great | i − 2 − 2

significance. where ρA (resp. ρB) is the reduced density matrix ob-
1
v tained by tracing ψ ψ over the space B (resp. A).
| ih | H H
2 Comparingwith the concurrence,entanglementoffor- This definition can be extendedto mixed states ρ by the
9 mationismoredifficulttodealwith,andlessresultshave convex roof,
2
been derived. The experimentalmeasurementof concur-
6
rence has been proposed in [7] for pure two-qubit sys- E(ρ) min p E(ψ ), (4)
1. temsbyusingtwocopiesoftheunknownquantumstate. ≡{pi,|ψii}Xi i | ii
0 In [8] the authors presented an approach of measuring
2 where the minimization goes over all possible ensemble
concurrence for arbitrary dimensional pure multipartite
1 realizations of ρ,
systems in terms of only one copy of the unknown quan-
:
v tum state. However, due to the complicated expression
Xi of entanglement of formation, there is no experimental ρ=Xpi|ψiihψi|, pi ≥0, Xpi =1. (5)
way yet to determine the entanglement of formation for i i
r
a an unknown quantum pure state, except for the case of A bipartite quantum state ψ can be written in the
two-qubit systems for which the concurrence and entan- m | i
g[2le].ment of formation have a simple monotonic relations Schmidt form |ψi = iP=1√λi|iAi|iBi, λi ≥ 0, Piλi = 1,
under suitable basis i and i . λ , i =
A A B B i
1,...,m, are also the|eigie∈nvHalues of ρ|A.i ∈E(Hψ ) can be
In this brief report we present an experimental de- further expressed as | i
termination of the entanglement of formation for arbi-
trary dimensional pure quantum states. The measure- m
E(ψ )=S(ρA)= λ logλ . (6)
ment onlyevolveslocalquantummechanicalobservables | i −X i i
and the entanglement of formation can be obtained ac- i=1
cording to the mean values of these observables.
For two-qubit case, m = n = 2, ψ = a 00 +
11
| i | i
a 01 +a 10 +a 11 , a 2+ a 2+ a 2+ a 2 =
12 21 22 11 12 21 22
The entanglement of formation is defined for bipartite 1. (|3)ican be| wiritten|asi[2|], | | | | | | |
systems. Let and be m and n (m n) dimen-
A B
H H ≤
sionalcomplexHilbertspaceswithorthonormalbasis i , 1+√1 C2
i = 1,...,m, and j , i = 1,...,n respectively. A pu|rie E(ψ )=h( − ), (7)
| i | i 2
2
where h(x) = xlog x (1 x)log (1 x), C = where ψ = λ δ ac λ δ ad λ δ bc +
− 2 − − 2 − | iαβ b bd| i − b bc| i − a ad| i
2a a a a isthe concurrence. Inthis specialcase λ δ bd .
11 22 12 21 a ac
E|(ψ ) is−just a m|onotonically increasing function of the We|noiw compute the eigenvalues of ρ′A = Tr (ρ′ )
| i αβ B αβ
concurrenceC. Howeverform 3,thereisnosuchrela- according to the values of a,b,c and d:
≥
tionslike(7)betweentheentanglementofformationand i). a=b=c=d. We have ψ =0.
αβ
concurrence in general. Since for the case m=2, due to ii). b6>a6=c6<dandb=d.| Wie get ψ =√λ bd .
αβ a
the normalization condition, λ1 +λ2 = 1, only one free The eigenvalue of ρ′A cor6responding t|o tihis case is|λi.
parameter is left in the formula (6). For general high αβ a
Asa=ccanbechosentobe1,2, ,m 2,banddhave
dimensional case, E(ψ ) depends on more free param- ··· −
eters. Nevertheless, i|f ρiA has only two non-zero eigen- only m−k and m−k−1 (corresponding to a = c = k,
k =1,2, ,m 2)kindsofchoices. Altogetherwehave
values (each of which may be degenerate), the maximal (m k)(·m·· k− 1) eigenvalues of ρ′A to be λ in this
non-zero diagonal determinant D of ρA is a generalized − − − αβ k
case, with k =1,2, ,m 2.
concurrence, namely, the corresponding entanglement of ··· −
iii). a < b = d > c and a = c. We have ψ =
formationisagainamonotonicallyincreasingfunctionof √λ ac . The eigenvalue of ρ′A6is λ . In this ca|seibαβ= d
D[9]. Theconstructionofsuchkindofstatesispresented b| i αβ b
can be 3,4, ,m. Then a and c have only k 1 and
in [10]. In [11], the results are generalized to more gen-
··· −
eralcase: relationslike(7)holdsforstateswithρAhaving k 2(correspondingtob=d=k,k =3,4, ,m)kinds
− ···
of choices. Hence we have (k 1)(k 2) eigenvalues of
morenon-zeroeigenvaluessuchthatalltheseeigenvalues
are functions of two independent parameters. ρ′αAβ to be λk in this case, k =−3,4,···−,m.
To measure the quantity (6) experimentally, we first iv). b > a = c < d and b = d. We obtain ψ αβ =
rewritethe expression(6)accordingto the entanglement √λ ac +√λ bd . The eigenvalues of ρ′A are|λi and
of formation of some “two-qubit” states. Let LA and λ ,ba|ndia = cac|anibe 1,2, ,m 1. Thαeβn b = ad can
α b
LB be the generators of special unitary groups SO(m) be k+1,k+2, ,m (corr·e·s·pond−ing to a = c = k,k =
β
and SO(n), with the m(m 1)/2 generators LA given 1,2, ,m 1).··W· e have (m 1) eigenvalues of ρ′A to
− α ··· − − αβ
by i j j i , 1 i < j m, and the n(n 1)/2 be λ , k =1,2, ,m.
k
gen{e|raithor|s−L|Bihgi|v}en b≤y k l≤ l k , 1 k <−l n, v). a < b = c··<· d. We have ψ = √λ ad . The
respectively.βThe matrix{|opiehr|a−tor|sihLA|}(res≤p. LB) ≤have eigenvalue of ρ′A is λ . b = c|caiαnβbe −2,3, b| ,mi 1.
α β αβ b ··· −
m 2(resp. n 2)rowsandm 2(resp. n 2)columns Then a and d have only k 1 and m k (corresponding
− − − − − −
that are identically zero. to b = c = k, k = 2,3, ,m 1) kinds of choices.
Let ρ = ψ ψ be the density matrix with respect to We have (k 1)(m k) ·e·ig·enva−lues of ρ′A to be λ ,
the pure sta|teihψ|. We define k =2,3, ,−m 1. − αβ k
| i ··· −
vi). c < d = a < b. We have ψ = √λ bc .
ραβ = LLAαA⊗LLBβBρρ((LLAαA))††⊗((LLBβB))†† , (8) The eigenvalue of ρ′αAβ is λa. In th|isiαcβase a−= da|cain
|| α ⊗ β α ⊗ β || be 2,3, ,m 1. c and b have only k 1 and m k
··· − − −
(corresponding to b = c = k, k = 2,3, ,m 1) kinds
where α = 1,2, ,m(m−1);β = 1,2, ,n(n−1), and of choices. Therefore we have (k 1)(m··· k) e−igenvalues
X = Tr(XX··†·) is th2e trace norm o·f··matri2x X. As of ρ′A to be λ , with k =2,3, −,m 1−.
t|t|hheat|m|aarterpixidLenAαt⊗icaLllBβyhzaesrom,nρ−4hroawssaatnmdomstn−4 4c4olu=m1n6s Leαtβλiαβ staknd for the eigen·v·a·lues−of ρAαβ. From the
αβ × analysis of cases i)-vi) and formula (6), we get
nonzero elements and is called “two-qubit” state. ρ is
αβ
still a normalized pure state. 1
E(ψ )= λi log(λi ). (12)
Theorem 1 For any m n (m n) pure quantum state | i −(m 1)2 XX αβ αβ
⊗ ≤ − αβ i=1
ψ ,
A B
| i∈H ⊗H
Since
1 E(ρ )+log(C )
αβ αβ
E(|ψi)= (m 1)2 X C , (9) ρ′
− αβ αβ ρ = αβ =C ρ′ , (13)
αβ Tr ρ′ αβ αβ
where C =1/Tr LA LB ψ ψ (LA)† (LB)† . { αβ}
αβ { α ⊗ β| ih | α ⊗ β }
we have λi C =1 for any α and β. Therefore
bParooafn.dTLoBca=lculcatedE(ραdβ)cwefodrecnoonteveLnAαien=ce|,awihhb|er−e Pi αβ αβ
| ih | β | ih | − | ih | E(ρ ) = C λi log(C λi )
1 a<b m and 1 c<d m. Set αβ −X αβ αβ αβ αβ
≤ ≤ ≤ ≤
i=1
ρ′αβ =LAα ⊗LBβ|ψihψ|(LAα)†⊗(LBβ)†. (10) =−XCαβλiαβlog(Cαβ)−XCαβλiαβlog(λiαβ)
It is direct to verify that i=1 i=1
= log(C ) C λi log(λi ).
ρ′ = ψ ψ , (11) − αβ − αβX αβ αβ
αβ | iαβh | i=1
3
That is The quantity C(ρ ) can be measured experimentally
αβ
intermsofthemethodintroducedin[8],withafewmod-
m
λi log(λi )= E(ραβ)+log(Cαβ). (14) ifications of the measurement operators. Corresponding
−X αβ αβ C to the caseofL = i j j i andL = k l l k ,
αβ α β
i=1 | ih |−| ih | | ih |−| ih |
we define m m matrix operators Σ , s=0,1,2,3,such
s
×
Substituting (14) into (12), we obtain that that (Σ0)pq = δpiδqi +δpjδqj, (Σ1)pq = δpiδqj +δpjδqi,
(Σ ) = Iδ δ Iδ δ , (Σ ) = δ δ δ δ ,
2 pq pi qj pj qi 3 pq pi qi pj qj
− −
1 E(ραβ)+log(Cαβ) p,q = 1,...,m. Similarly we define n n matrix op-
E(ψ )= , (15) ×
| i (m 1)2 X Cαβ erators Γ0,Γ1,Γ2 and Γ3 by replacing the indices i,j
− αβ in Σ ,Σ ,Σ and Σ with k,l respectively, and setting
0 1 2 3
p,q =1,...,n. ItisstraightforwardtoderivethatC(ρ )
which proves the theorem. αβ
can be expressed as the mean values of the above local
The theorem shows that one can derive the entangle-
observables,
ment of formation of a pure quantum state by measur-
ing the values of the entanglement of formation of all 1 C2
the states ραβ and values of Cαβ. Here if |ψiαβ = 0, C2(ραβ) = 2 + 2αβ (cid:0)hΣ3⊗Γ3i2−hΣ3⊗Γ0i2
then C goes to infinity and this term does not con-
αβ
tribute to the summation in (9). Hence the summation Σ0 Γ3 2 Σ0 Γ1 2+ Σ3 Γ1 2
−h ⊗ i −h ⊗ i h ⊗ i
in (9) simply goes over all the terms such that Σ Γ 2+ Σ Γ 2 . (16)
Pαβ −h 0⊗ 2i h 3⊗ 2i (cid:1)
Tr LA LB ψ ψ (LA)† (LB)† =0.
{ α ⊗ β| ih | α ⊗ β }6 In summary we have presented an experimental way
With formula (9), we now show how to get the value
to measure the entanglement of formation for arbitrary
of E(ψ ) experimentally by measuring the quantities on
| i dimensional pure states, by measuring some local quan-
the right hand side of (9).
′ tum mechanical observables. We reduced the difficult
The quantity C =1/Tr ρ canbe determined by
αβ { αβ} problem to find the concurrence of “two-qubit” states
Tr ρ′ . Since Tr ρ′ = ψ (LA)†LA (LB)†LB ψ ,
{ αβ} { αβ} h | α α ⊗ β β| i for which many results have been already derived. Re-
one can obtain C by measuring the local Hermitian
αβ cently high dimensional bipartite systems like in NMR
operator (LA)†LA (LB)†LB associated with the state
α α ⊗ β β and nitrogen-vacancy defect center have been success-
ψ . fully used for quantum computation and simulation ex-
| i
TomeasureE(ραβ),wefirstnotethatalthoughραβ are periments [18]. Our results present a plausible way to
m n bipartite quantumstates, they are basically“two- measure the entanglement of formation in these systems
⊗
qubit” ones. For given Lα = i j j i and Lβ = and to investigate the roles played by the entanglement
| ih |− | ih |
k l l k , i= j, k = l, the non-zero elements of ραβ of formation in these quantum information processing.
| ih |−| ih | 6 6
are located at the i (m 1)+kth, i (m 1)+lth,
So far experimental measurement on entanglement of
∗ − ∗ −
j (m 1)+kth, and j (m 1)+lth rows and the
formationandconcurrenceconcernsonlypurestates. For
∗ − ∗ −
i (m 1)+kth, i (m 1)+lth, j (m 1)+kth,
mixedstates,lessisknownexceptforexperimentaldeter-
∗ − ∗ − ∗ −
and j (m 1)+lth columns. They constitute a 4 4
mination of separability, both sufficiently and necessary,
∗ − ×
matrix,
for two-qubit [19] and qubit-qutrit systems [20]. Gener-
ally(4)hasonlyanalyticalresultsforsomespecialstates
ρ ρ ρ ρ
ik,ik ik,il ik,jk ik,jl [21]andanalyticallowerbounds[22]whicharenotexper-
σ′ = ρil,ik ρil,il ρil,jk ρil,jl . imentallymeasurable. Recentlyin[23]wehavepresented
αβ ρ ρ ρ ρ
 jk,ik jk,il jk,jk jk,jl  a measurable lowerbound of entanglementof formation.
 ρ ρ ρ ρ 
jl,ik jl,il jl,jk jl,jl The formula (9) for pure state may also help to study
measurable lower bounds of entanglement of formation
′ ′
Set σ = σ /Tr σ . Obviously E(ρ ) = E(σ ).
αβ αβ { αβ} αβ αβ for mixed states.
But σ are actually two-qubit pure states. According
αβ
to the formula (7), E(σ ) is determined by the concur-
αβ
Acknowledgments This work is supported by the
renceC(σ )=C(ρ ). Thereforeifwecanmeasurethe
αβ αβ
NSFC 10875081and PHR201007107.
quantity C(ρ ), we can obtain E(ρ ).
αβ αβ
[1] C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. [2] W. K. Wootters, Phys.Rev.Lett. 80, 2245 (1998).
Wootters, Phys. Rev.A 54, 3824 (1996). [3] A. Peres, Phys.Rev.Lett. 77 1413 (1996).
C.H. Bennett, H.J. Bernstein, S. Popescu, and B. Schu- K. z˙yczkowski and P. Horodecki, Phys. Rev. A 58, 883
macher, Phys.Rev. A 53, 2046 (1996). (1998).
V. Vedral, M.B. Plenio, M.A. Rippin, and P.L. Knight, [4] A. Uhlmann,Phys. Rev.A 62, 032307 (2000).
Phys.Rev.Lett. 78, 2275 (1997). P. Rungta, V. Buˇzek, C.M. Caves, M. Hillery, and G.J.
4
Milburn, Phys.Rev.A 64, 042315 (2001). 95, 210501(2005).
S. Albeverio and S. M. Fei, J. Opt. B: Quantum Semi- [14] B. M. Terhal and K. H. Vollbrecht, Phys. Rev. Lett. 85,
class. Opt.3, 223(2001). 2625(2000).
[5] A.Osterloh et al., Nature416 , 608 (2002). [15] F. Mintert and A. Buchleitner, Phys. Rev. Lett. 98,
L. A. Wu, M. S. Sarandy, and D. A. Lidar, Phys. Rev. 140505 (2007).
Lett.93, 250404 (2004). [16] C. J. Zhang, Y. X. Gong, Y. S. Zhang, and G. C. Guo,
[6] S. Ghosh, T. F. Rosenbaum, G. Aeppli, S. N. Copper- Phys. Rev.A 78, 042308(2008).
smith, Nature425, 48 (2003). [17] L. Aolita, A. Buchleitner, and F. Mintert, Phys. Rev. A
V.Vedral, Nature425, 28 (2003). 78, 022308(2008).
[7] F.Mintert,M.Ku´sandA.Buchleitner,Phys.Rev.Lett. [18] F. Shiet al, Phys. Rev.Lett. 105, 040504 (2010).
95, 260502 (2005). J. F. Du et al, Phys. Rev.Lett. 104, 030502 (2010).
S. P. Walborn, P. H. Souto Ribeiro, L. Davidovich, F. [19] S. Yu, J. W. Pan, Z. B. Chen and Y. D. Zhang, Phys.
Mintert and A.Buchleitner, Nature. 440, 20(2006). Rev. Lett.91, 21(2003).
[8] S. M. Fei, M. J. Zhao, K. Chen and Z. X. Wang, Phys. [20] S. M. Fei, M. J. Zhao, K. Chen and Z. X. Wang, Phys.
Rev.A 80, 032320(2009). Rev. A 80, 032320(2009).
[9] S. M. Fei, J. Jost, X. Q. Li-Jost and G. F. Wang, Phys. [21] B.M.TerhalandKarlG.H.Vollbrecht,Phys.Rev.Lett.
Lett.A 310, 333(2003). 85, 2625 (2000).
[10] S. M. Fei and X. Q. Li-Jost, Rep. Math. Phys. 53, K. G.H.Vollbrecht andR.F. Werner,Phys.Rev.A 64,
195(2004). 062307(2001).
[11] S. M. Fei, Z. X. Wang and H. Zhao, Phys. Lett. A 329, [22] K. Chen, S. Albeverio and S. M. Fei, Phys. Rev. Lett.
414(2004). 95, 210501(2005).
[12] E. Gerjuoy Phys.Rev. A 67, 052308 (2003). [23] M. Li and S. M. Fei, Phys. Rev.A 82, 044303(2010).
[13] K. Chen, S. Albeverio, and S. M. Fei, Phys. Rev. Lett.