On Estimation of Fully Entangled Fraction

0.12 MB·English

by  Rui-Juan Gu

On Estimation of Fully Entangled Fraction
Rui-Juan Gu 1, Ming Li2, Shao-Ming Fei1 and Xianqing Li-Jost3,4
1 School of Mathematical Sciences, Capital Normal University, 100048 Beijing
2 College of Mathematics and Computational Science, China University of
0 Petroleum, 257061 Dongying
1
0 3 Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig
2
n 4 Department of Mathematics, Hainan Normal University, 571158 Haikou
a
J

6 Abstract
2

] We study the fully entangled fraction (FEF) of arbitrary mixed states.
h
p New upper bounds of FEF are derived. These upper bounds make
-
t
n complements on the estimation of the value of FEF. For weakly mixed
a
u quantum states, an upper bound is shown to be very tight to the exact
q
[
value of FEF.

1
v PACS numbers: 03.67.-a, 02.20.Hj, 03.65.-w
6
5
7
4 Quantum entanglement plays crucial roles in quantum information processing such as
.
1
quantum computation [1, 2], quantum teleportation [3, 4], dense coding [5], quantum cryp-
0
0 tographic schemes [6], entanglement swapping [7] and remote states preparation (RSP) [8–
1
: 10]. For instance in terms of a classical communication channel and a quantum resource
v
i (a nonlocal entangled state like an EPR-pair of particles), the teleportation protocol gives
X
r ways to transmit an unknown quantum state from a sender to a receiver that are spatially
a
separated. When the sender and receiver share a maximally entangled pure state, the state
can be perfectly teleported. However when the shared entangled state is an arbitrary mixed
state ρ, then the optimal fidelity of teleportation is given by [4, 11],
d (ρ) 1
f(ρ) = F + , (1)
d+1 d+1
which solely depends on the fully entangled fraction (FEF) (ρ) of ρ. Detailed discussion
F
about FEF can be found in Ref. [12].
In fact the quantity FEF plays essential roles in many other quantum information pro-
cessing such as dense coding, entanglement swapping and quantum cryptography (Bell in-
equalities). Thus it is very important to compute the FEF of general quantum states.
1
Unfortunately, precise formula of FEF has been only obtained for two qubits systems [13].
For high dimensional systems, it becomes quite difficult to derive an analytic formula for
FEF. In [14] we have derived an upper bound of FEF to give an estimation of the value
of FEF. In this paper, we derive more tight upper bounds for FEF. These bounds make
complements on the estimation of FEF.
Let be a d-dimensional complex vector space with computational basis i , i = 1,...,d.
H | i
The fully entangled fraction of a density matrix ρ is defined by
∈ H⊗H
(ρ) = max φ ρ φ , (2)
F φ ǫ h | | i
∈
where ǫ denotes the set of d d-dimensional maximally entangled pure states. (2) can be
×
also alternatively expressed as
(ρ) = max ψ (I U )ρ(I U) ψ , (3)
+ † +
F U h | ⊗ ⊗ | i
d
where the maximization is taken over all unitary transformations U, ψ = 1 ii is the
| +i √d | i
i=1
maximally entangled state and I is the corresponding identity matrix. P
Let h and g be d d matrices such that h j = (j + 1) mod d ,g j = ωj j , with
× | i | i | i | i
ω = exp 2iπ . We can introduce d2 linear-independent d d-matrices U = htgs, which
{−d } × st
satisfy
UstUs′t′ = ωst′−ts′Us′t′Ust,Tr(Ust) = dδs0δt0. (4)
U also satisfy the condition for bases of unitary operators in the sense of [15], i.e.
st
{ }
tr UstUs+′t′ = dδtt′δss′,
(5)
Us(cid:0)tUs+t = I(cid:1).
Ust form a complete basis of d d-matrices, namely, for any d d matrix W, W can be
{ } × ×
expressed as
1
W = tr(U+W)U . (6)
d st st
s,t
X
From U , we can introduce the generalized Bell-states,
st
{ }
1
Φ = (I U ) ψ = (U ) ij ,and Φ = ψ , (7)
| sti ⊗ s∗t | +i √d st ∗ij| i | 00i | +i
i,j
X
Φ are all maximally entangled states and form a complete orthogonal normalized basis
st
| i
of .
H⊗H
Theorem 1: For any quantum state ρ , the fully entangled fraction defined in
∈ H⊗H
(2) and (3) fulfills the following inequality:
(ρ) max λ , (8)
j
F ≤ j { }
2
T iT
where λ s are the eigenvalues of the real part of the matrix M = , T is a d2 d2
j
 iT T  ×
−
matrix with entries T = Φ ρ Φ andΦ is the maximally entangled basis states defined
n,m n m j
h | | i
in (7).
Proof: From (6), any d d unitary matrix U can be represented as U = d2 z U ,
× k=1 k k
where zk = d1Tr(Uk†U), Uk are the unitary matrices defined in (4). Define P
Re[z ],1 l d2; U ,1 l d2;
l ′ l
x = ≤ ≤ and U = ≤ ≤ (9)
l  Im[z ],d2 < l 2d2 l i U ,d2 < l 2d2.
 l ≤  ∗ l ≤
Then the unitary matrix U can be rewritten as U = 2d2 x U′. The necessary unitary
k=1 k k
condition of U, Tr(UU ) = d, requires that x2 = 1. Set
† k k P
P 2d2
F(ρ) ψ (I U )ρ(I U) ψ = x x M , (10)
+ † + m n mn
≡ h | ⊗ ⊗ | i
m,n=1
X
where M is the entry of the matrix M defined in theorem. From the hermiticity of ρ it is
mn
easily verified that
M = M . (11)
m∗n nm
To maximize F(ρ) under constraints we get the following equation
∂
F(ρ)+λ( x2 1) = 0. (12)
∂x { l − }
k
l
X
Taking into account (11) we obtain an eigenvalue equation,
2d2
Re[M ]x = λx . (13)
k,n n k
−
n=1
X
Therefore
(ρ) = maxF(ρ) max η , (14)
j
F U ≤ j { }
where η = λ is the corresponding eigenvalues of the real part of matrix M. (cid:3)
j j
−
The upper bound derived in [14] says that for any ρ , the fully entangled fraction
∈ H⊗H
(ρ) satisfies
F
1
(ρ) +4 NT(ρ)N(P ) , (15)
F ≤ d2 || + ||KF
where N(ρ) denotes the correlation matrix with entries n (ρ) given in the expression of ρ
ij
d2 1 d2 1 d2 1
1 1 − 1 − −
ρ = I I + r (ρ)λ I + s (ρ)I λ + n (ρ)λ λ , (16)
i i j j ij i j
d2 ⊗ d ⊗ d ⊗ ⊗
i=1 j=1 i,j=1
X X X
3
λ , i = 1,...,d2 1, are the generators of the SU(d) algebra with Tr λ λ = 2δ , r (ρ) =
i i j ij i
− { }
1Tr ρλ (1) I , s (ρ) = 1Tr ρI λ (2) , n (ρ) = 1Tr ρλ (1) λ (2) , P stands for
2 { i ⊗ } j 2 { ⊗ j } ij 4 { i ⊗ j } +
the projection operator to ψ , N(P ) is similarly defined to N(ρ), NT stands for the
+ +
| i
transpose of N, N = Tr√NN is the Ky Fan norm of N. This upper bound was used
KF †
|| ||
to improve the distillation protocol proposed in [16]. Here we show that the upper bound
in (8) is different from that in (15) by an example.
Example 1: We consider the bound entangled state [17]
a 0 0 0 a 0 0 0 a
 0 a 0 0 0 0 0 0 0 
 0 0 a 0 0 0 0 0 0 
 
 
 0 0 0 a 0 0 0 0 0 
 
1  
ρ(a) = a 0 0 0 a 0 0 0 a . (17)
8a+1  
 
 0 0 0 0 0 a 0 0 0 
 
 0 0 0 0 0 0 1+a 0 √1 a2 
−
 2 2 
 
 0 0 0 0 0 0 0 a 0 
 
a 0 0 0 a 0 √1−a2 0 1+a 
 2 2 
 
From Fig. 1 we see that for 0 a < 0.572, the upper bound in (8) is larger than that in
≤
(15). But for 0.572 < a < 1 the upper bound in (8) is always lower than that in (15), i.e.
the upper bound (8) is tighter than (15) in this case.
UpperBound
0.342
0.340
0.338
0.336
0.334
a
0.5 0.6 0.7 0.8 0.9 1.0
FIG. 1: Upper bound of (ρ) from (8) (solid line) and upper bound from (15) (dashed line).
F
By using the operator norm, we have further
4
Observation: For any ρ , the fully entangled fraction (ρ) satisfies
∈ H⊗H F
(ρ) max λ (18)
i
F ≤ i { }
where λ s are the eigenvalues of ρ.
i
Proof: For any quantum state ρ and unitary U, we have
ψ (I U )ρ(I U) ψ ψ (I U )ρ(I U) ψ
+ † + + † +
h | ⊗ ⊗ | i ≤ kh |kk ⊗ ⊗ | ik
(I U )ρ(I U) ψ 2 = ρ ψ 2 = ρ , (19)
† + +
≤ k ⊗ ⊗ kk| ik k kk| ik k k
where ρ stands for the operator norm, ρ = sup( ρ x : x = 1), x = x x .
k k k k k | ik k| ik k| ik h | i
We have used the Cauchy-Schwarz inequality to obtain the first inequality. The second
p
inequality is due to the basic property of operator norm. The followed equality follows from
the fact that unitary transformation does not change the operator norm.
From [18] ρ is an eigenvalue of ρ, actually, it is the maximal eigenvalue of ρ, i.e.
k k
ρ = max λ where λ s are the eigenvalues of ρ, which ends the proof. (cid:3)
i i i
k k { }
This bound can give rise to further
Corollary: Let ψ = a ij with ψ = 1 be the normalized eigenvector of ρ
| i ij ij| i ||| i||
with respect to the maximal eigenvalue λ . If the matrix A with elements A = √da
P max ij ij
are unitary, the upper bound derived in (18) becomes the exact value of FEF.
Proof: A simple computation shows that
1
(ρ) λ = ψ ρ ψ = √da ij ρ√da kl
F ≤ max h | | i d ∗ijh | kl| i
ij,kl
X
1
= ii I A ρI A kk = ψ I A ρI A ψ (ρ).
dh | ⊗ † ⊗ | i h +| ⊗ † ⊗ | +i ≤ F
i,k
X
Thus we have (ρ) = λ . (cid:3)
max
F
According to the corollary, we can find out when the upper bound derived in theorem 2
becomes the exact value of FEF.
Example 2: Consider the 3 3 state [19]
⊗
2 α 5 α
ρ = ψ ψ + σ + − σ , (20)
+ + +
7| ih | 7 7 −
where σ = 1( 01 01 + 12 12 + 20 20 ),σ = 1( 10 10 + 21 21 + 02 02 ). ρ is
+ 3 | ih | | ih | | ih | − 3 | ih | | ih | | ih |
entangled when 3 < α 5. The maximal eigenvalue of ρ is 2, with the corresponding
≤ 7
normalized eigenvector 1 ,0,0,0, 1 ,0,0,0, 1 . The matrix A related to this eigenvector
{√3 √3 √3}
is just the 3 3 identity matrix which is obviously unitary. Thus we have for the state (20),
×
(ρ) = 2.
F 7
The following upper bound of FEF gives a very tight estimation of FEF for weakly mixed
quantum states.
5
Theorem 2: For any bipartite quantum state ρ , the following inequality holds:
∈ H⊗H
1
(ρ) (Tr√ρ )2, (21)
A
F ≤ d
where ρ is the reduced matrix of ρ.
A
Proof: Note that the FEF for pure state ψ is given by [16]
| i
1
( ψ ) = (Tr ρ|ψi)2, (22)
F | i d A
q
ψ
where ρ| i is the reduced matrix of ψ ψ .
A | ih |
For mixed state ρ = p ρi, we have
i i
P
(ρ) = max ψ (I U )ρ(I U) ψ
+ † +
F U h | ⊗ ⊗ | i
p max ψ (I U )ρi(I U) ψ
i + † +
≤ U h | ⊗ ⊗ | i
i
X
1 1
= p (Tr ρi )2 = (Tr p ρi )2. (23)
d i A d i A
i q i q
X X
Let λ be the real and nonnegative eigenvalues of the matrix p ρi . Recall that for any
ij i A
1
function F = i( j x2ij)2 subjected to the constraints zj = ixij with xij being real and
nonnegative, the inequality z2 F2 holds. It follows that
P P j j ≤ P
P 2
2
1 1 1
(ρ) λ λ = (Tr√ρ )2, (24)
ij ij A
F ≤ d ≤ d   d
!
i j j s i
X Xp X X
 
(cid:3)
which ends the proof.
We now give an example to show that when the quantum state is weakly mixed, theorem
3 will be a very good estimation for the FEF.
Example 3: Consider the following 3 3 mixed state: ρ = 1 pI + p ψ ψ , where
⊗ −9 9 | ih |
ψ = x2+2 x,0,0,0,1,0,0,0,1 is a pure state with one parameter x. To show the
| i 3 { }
effectiveqness of (21), we compare it with the single fraction of entanglement F = ψ ρ ψ .
s + +
h | | i
As seen from the Fig. 2, for weakly mixed states (with larger parameter p), the bound
provides excellent estimation of the FEF.
We have studied the fully entangled fraction that has tight relations with many quantum
information processing. New upper bounds for FEF have been derived. They make comple-
ments on estimation of the value of FEF. The conditions for the bounds to be exact or to be
more tight have been analyzed. These bounds provide a better estimation of FEF and can
be use in related information processing, e.g. to detect the entanglement of the non-local
source used in quantum teleportation. Our results also give rise to good estimations for the
conditional min-entropy H (A B) and q according to relation in [21].
min corr(AB)
| |
6
1 1
1
0.95
0.9
0.9
0.9
d 0.8 d d0.85
n n n
u u 0.8 u
o o o
B B B 0.8
0.7
0.75
0.7
0.7
0.6
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
p=0.80 p=0.90 p=0.99
FIG. 2: Upper bound of (ρ) from (21) (solid line) and single fraction of entanglement F
s
F
Acknowledgments This work is supported by NSFC 10675086, 10875081, 10871227,
KZ200810028013 and NKBRPC(2004CB318000).
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