Schrödinger uncertainty relation, Wigner-Yanase-Dyson skew information and metric adjusted correlation measure

0.16 MB·English

by  Shigeru Furuichi

Schr¨odinger uncertainty relation, Wigner-Yanase-Dyson skew
information and metric adjusted correlation measure
Shigeru Furuichi1∗and Kenjiro Yanagi2†
1Department of Computer Science and System Analysis,
2 College of Humanities and Sciences, Nihon University,
1
3-25-40,Sakurajyousui,Setagaya-ku,Tokyo, 156-8550,Japan
0
2 2Division of Applied Mathematical Science,
Graduate School of Science and Engineering, Yamaguchi University,
n
a 2-16-1,Tokiwadai, Ube City, 755-0811,Japan
J

5
1
Abstract. In this paper, we give Schro¨dinger-type uncertainty relation using the Wigner-

] Yanase-Dyson skew information. In addition, we give Schro¨dinger-type uncertainty relation by
h
p useof atwo-parameter extended correlation measure. We finallyshow thefurthergeneralization
-
ofSchro¨dinger-typeuncertaintyrelationbyuseofthemetricadjustedcorrelationmeasure. These
t
n
results generalize our previous result in [Phys. Rev. A, Vol.82(2010), 034101].
a
u
q Keywords : Trace inequality, Wigner-Yanase-Dyson skew information, Schro¨dinger un-
[ certainty relation and metric adjusted correlation measure

4
2000 Mathematics Subject Classification : 15A45, 47A63 and 94A17
v
2
9
3
1 Introduction
0
.
0
1 In quantum information theory, one of the most important results is the strong subadditivity
0 of von Neumann entropy [22]. This important property of von Neumann entropy can be proven
1
by the use of Lieb’s theorem [16] which gave a complete solution for the conjecture of the
:
v
convexity of Wigner-Yanase-Dyson skew information. In addition, the uncertainty relation has
i
X been widely studied in quantum information theory [21, 31, 29]. In particular, the relations
r betweenskewinformationanduncertaintyrelationhavebeenstudiedin[17,4,8,9,7]. Quantum
a
Fisher information is also called monotone metric which was introduced by Petz [23] and the
Wigner-Yanase-Dyson metric is connected to quantum Fisher information (monotone metric) as
a special case. Recently, Hansen gave a further development of the notion of monotone metric,
so-called metric adjusted skew information [12]. The Wigner-Yanase-Dyson skew information is
also connected to the metric adjusted skew information as a special case. That is, the metric
adjusted skew information gave a class including the Wigner-Yanase-Dyson skew information,
whilethemonotonemetricgave aclassincludingtheWigner-Yanase-Dyson metric. Inthepaper
[12], the metric adjusted correlation measure was also introduced as a generalization of the
quantum covariance and correlation measure defined in [17]. Therefore there is a significance to
givetherelation amongtheWigner-Yanase-Dyson skewinformation, metricadjustedcorrelation
measure and uncertainty relation for the fundamental studies on quantum information theory.
∗E-mail:[email protected]
†E-mail:[email protected]
1
We start from the Heisenberg uncertainty relation [13]:
1
V (A)V (B) Tr[ρ[A,B]]2 (1)
ρ ρ
≥ 4| |
for a quantum state (density operator) ρ and two observables (self-adjoint operators) A and B.
The further stronger result was given by Schro¨dinger in [27, 28]:
1
V (A)V (B) Re Cov (A,B) 2 Tr[ρ[A,B]]2, (2)
ρ ρ ρ
−| { }| ≥ 4| |
where the covariance is defined by Cov (A,B) Tr[ρ(A Tr[ρA]I)(B Tr[ρB]I)].
ρ
≡ − −
The Wigner-Yanase skew information represents a measure for non-commutativity between
a quantum state ρ and an observable H. Luo introduced the quantity U (H) representing a
ρ
quantum uncertainty excluding the classical mixture [18]:
U (H) V (H)2 (V (H) I (H))2, (3)
ρ ρ ρ ρ
≡ − −
q
with the Wigner-Yanase skew information [32]:
1
I (H) Tr (i[ρ1/2,H ])2 = Tr[ρH2] Tr[ρ1/2H ρ1/2H ], H H Tr[ρH]I
ρ ≡ 2 0 0 − 0 0 0 ≡ −
h i
and then he successfully showed a new Heisenberg-type uncertainty relation on U (H) in [18]:
ρ
1
U (A)U (B) Tr[ρ[A,B]]2. (4)
ρ ρ
≥ 4| |
As stated in [18], the physical meaning of the quantity U (H) can be interpreted as follows.
ρ
For a mixed state ρ, the variance V (H) has both classical mixture and quantum uncertainty.
ρ
Also, the Wigner-Yanase skew information I (H) represents a kind of quantum uncertainty
ρ
[19, 20]. Thus, the difference V (H) I (H) has a classical mixture so that we can regard that
ρ ρ
−
the quantity U (H) has a quantum uncertainty excluding a classical mixture. Therefore it is
ρ
meaningful and suitable to study an uncertainty relation for a mixed state by the use of the
quantity U (H).
ρ
Recently, a one-parameter extension of the inequality (4) was given in [33]:
U (A)U (B) α(1 α)Tr[ρ[A,B]]2, (5)
ρ,α ρ,α
≥ − | |
where
U (H) V (H)2 (V (H) I (H))2,
ρ,α ρ ρ ρ,α
≡ − −
q
with the Wigner-Yanase-Dyson skew information I (H) is defined by
ρ,α
1
I (H) Tr (i[ρα,H ])(i[ρ1−α,H ]) = Tr[ρH2] Tr[ραH ρ1−αH ],
ρ,α ≡ 2 0 0 0 − 0 0
(cid:2) (cid:3)
It is notable that the convexity of I (H) with respect to ρ was successfully proven by Lieb
ρ,α
in [16]. The further generalization of the Heisenberg-type uncertainty relation on U (H) has
ρ
been given in [34] using the generalized Wigner-Yanase-Dyson skew information introduced in
[3]. See also [1, 5, 7, 8] for the recent studies on skew informations and uncertainty relations.
Motivated by the fact that the Schro¨dinger uncertainty relation is a stronger result than the
Heisenberg uncertainty relation, a new Schro¨dinger-type uncertainty relation for mixed states
using Wigner-Yanase skew information was shown in [4]. That is, for a quantum state ρ and
two observables A and B, we have
1
U (A)U (B) Re Corr (A,B) 2 Tr[ρ[A,B]]2, (6)
ρ ρ ρ
−| { }| ≥ 4| |
2
where the correlation measure [17] is defined by
Corr (X,Y) Tr[ρX∗Y] Tr[ρ1/2X∗ρ1/2Y]
ρ
≡ −
for any operators X and Y. This result refined the Heisenberg-type uncertainty relation (4)
shownin[18]formixedstates(generalstates). Weeasilyfindthattheinequality(6)isequivalent
to the following inequality:
U (A)U (B) Corr (A,B)2. (7)
ρ ρ ρ
≥ | |
The main purpose of this paper is to give some extensions of the inequality (7) by using the
Wigner-Yanase-Dyson skew information I (H) and the metric adjusted correlation measure
ρ,α
introduced in [12].
2 Schr¨odinger uncertainty relation with Wigner-Yanase-Dyson
skew information
In this section, we give a generalization of the Schro¨dinger type uncertainty relation (7) by the
use of the quantity U (H) defined by the Wigner-Yanase-Dyson skew information I (H).
ρ,α ρ,α
Theorem 2.1 For α [1/2,1], a quantum state ρ and two observables A and B, we have
∈
U (A)U (B) 4α(1 α)Corr (A,B)2. (8)
ρ,α ρ,α ρ,α
≥ − | |
where the generalized correlation measure [14, 36] is defined by
Corr (X,Y) Tr[ρX∗Y] Tr[ραX∗ρ1−αY]
ρ,α
≡ −
for any operators X and Y.
To prove Theorem 2.1, we need the following lemmas.
∞
Lemma 2.2 ([33])Foraspectral decomposition ofρ= λ φ φ , puttingh φ H φ ,
j=1 j| jih j| ij ≡ h i| 0| ji
we have the following relations.
P
(i) For the Wigner-Yanase-Dyson skew information, we have
I (H) = λα λα λ1−α λ1−α h 2.
ρ,α i − j i − j | ij|
Xi<j (cid:0) (cid:1)(cid:16) (cid:17)
(ii) For the quantity associated to the Wigner-Yanase-Dyson skew information:
1
J (H) Tr ( ρα,H )( ρ1−α,H ) = Tr[ρH2]+Tr[ραH ρ1−αH ],
ρ,α ≡ 2 { 0} 0 0 0 0
(cid:2) (cid:8) (cid:9) (cid:3)
where X,Y XY +YX is an anti-commutator, we have
{ } ≡
J (H) λα+λα λ1−α+λ1−α h 2.
ρ,α ≥ i j i j | ij|
Xi<j (cid:0) (cid:1)(cid:16) (cid:17)
Lemma 2.3 ([2, 33]) For any t > 0 and α [0,1], we have
∈
(1 2α)2(t 1)2 (tα t1−α)2.
− − ≥ −
3
∞
Proof of Theorem 2.1: We take a spectral decomposition ρ = λ φ φ . If we put
j=1 j| jih j|
a = φ A φ and b = φ B φ , where A = A Tr[ρA]I and B = B Tr[ρB]I, then
ij i 0 j ji j 0 i 0 0
h | | i h | | i − P −
we have
Corr (A,B) = Tr[ρAB] Tr[ραAρ1−αB]
ρ,α
−
= Tr[ρA B ] Tr[ραA ρ1−αB ]
0 0 0 0
−
∞
= (λ λαλ1−α)a b
i − i j ij ji
i,j=1
X
= (λ λαλ1−α)a b
i − i j ij ji
i6=j
X
= (λ λαλ1−α)a b +(λ λαλ1−α)a b . (9)
i − i j ij ji j − j i ji ij
Xi<j n o
Thus we have
Corr (A,B) λ λαλ1−α a b + λ λαλ1−α a b .
| ρ,α | ≤ | i− i j || ij|| ji| | j − j i || ji|| ij|
Xi<j n o
Since a = a and b = b ,takingasquareofbothsidesandthenusingSchwarzinequality
ij ji ij ji
| | | | | | | |
and Lemma 2.2, we have
4α(1 α)Corr (A,B)2
ρ,α
− | |
2
4α(1 α) λ λαλ1−α + λ λαλ1−α a b
≤ −  | i − i j | | j − j i | | ij|| ji|
Xi<j n o 
2
 
= 2 α(1 α) λα+λα λ1−α λ1−α a b
 − i j | i − j || ij|| ji|
Xi<j p (cid:0) (cid:1) 
2
 
2 α(1 α) λ λ a b
i j ij ji
≤  − | − || || |
Xi<j p 
2
  1/2
λα λα λ1−α λ1−α λα+λα λ1−α+λ1−α a b
≤  i − j i − j i j i j | ij|| ji|
Xi<j n(cid:0) (cid:1)(cid:16) (cid:17)(cid:0) (cid:1)(cid:16) (cid:17)o 
 
λα λα λ1−α λ1−α a 2 λα+λα λ1−α+λ1−α b 2
≤  i − j i − j | ij|  i j i j | ij| 
Xi<j (cid:0) (cid:1)(cid:16) (cid:17) Xi<j (cid:0) (cid:1)(cid:16) (cid:17) 
I (A)J (B)
ρ,α ρ,α
≤   
In the above process, the inequality (xα+yα)x1−α y1−α x y for x,y 0 and α [1,1]
| − | ≤ | − | ≥ ∈ 2
and the inequality 4α(1 α)(x y)2 (xα yα) x1−α y1−α (xα+yα) x1−α+y1−α for
− − ≤ − −
x,y 0 and α [0,1], which can be proven by Lemma 2.3, were used. By the similar way, we
≥ ∈ (cid:0) (cid:1) (cid:0) (cid:1)
also have
4α(1 α)Corr (A,B)2 I (B)J (A).
ρ,α ρ,α ρ,α
− | | ≤
Thus for α 1 we have
≥ 2
4α(1 α)Corr (A,B)2 U (A)U (B). (10)
ρ,α ρ,α ρ,α
− | | ≤
Note that Theorem 2.1 recovers the inequality (7), if we take α = 1.
2
4
Remark 2.4 We take α = 0.1 and
1 1 0 2 2 i 2 i
ρ= ,A = − ,B = ,
3 0 2 2+i 1 i 1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) − (cid:19)
then we have
U (A)U (B) 4α(1 α)Corr (A,B)2 0.28332.
ρ,α ρ,α ρ,α
− − | | ≃ −
Therefore the inequality (8) does not hold for α [0,1/2) in general.
∈
Corollary 2.5 Underthe same assumptions withTheorem 2.1, we have the following inequality:
U (A)U (B) 4α(1 α) Re Corr (A,B) 2 Im Tr[ραAρ1−αB] 2
ρ,α ρ,α ρ,α
− − | { }| −| |
α(1 α)Tr[ρ[A,B]]2. (11)
(cid:0) (cid:8) (cid:9) (cid:1)
≥ − | |
Proof: From
1
Im Corr (A,B) = Tr[ρ[A,B]] Im Tr[ραAρ1−αB] ,
ρ,α
{ } 2i −
(cid:8) (cid:9)
we have
1
Tr[ρ[A,B]] 2 Im Corr (A,B) 2+ Im Tr[ραAρ1−αB] 2.
ρ,α
4| | ≤ | { }| | |
Thus we have (cid:8) (cid:9)
Corr (A,B)2 = Re Corr (A,B) 2+ Im Corr (A,B) 2
ρ,α ρ,α ρ,α
| | | { }| | { }|
1
Re Corr (A,B) 2+ Tr[ρ[A,B]] 2 Im Tr[ραAρ1−αB] 2,
ρ,α
≥ | { }| 4| | −| |
(cid:8) (cid:9)
which proves the corollary.
Remark 2.6 The following inequality does not hold in general for α [1,1]:
∈ 2
Re Corr (A,B) 2 Im Tr[ραAρ1−αB] 2. (12)
ρ,α
| { }| ≥| |
Because we have a counter-example as follows. We t(cid:8)ake α = 2 and (cid:9)
3
1 2 3 2 2 i 2 i
ρ= ,A = − ,B = ,
7 3 5 2+i 1 i 1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) − (cid:19)
then we have
Re Corr (A,B) 2 Im Tr[ραAρ1−αB] 2 0.0548142.
ρ,α
| { }| −| | ≃ −
This shows Theorem 2.1 does not refine the i(cid:8)nequality (5) in(cid:9)general.
3 Two-parameter extensions
Inthissection,weintroducetheparametricextendedcorrelationmeasureCorr (X,Y)bythe
ρ,α,γ
convex combination between Corr (X,Y) and Corr (X,Y). Then we establish the para-
ρ,α ρ,1−α
metric extended Schro¨dinger-type uncertainty relation applying the parametric extended corre-
lation measure Corr (X,Y). In addition, introducing the symmetric extended correlation
ρ,α,γ
(sym)
measureCorr (X,Y)bytheconvexcombination betweenCorr (X,Y)andCorr (Y,X),
ρ,α,γ ρ,α ρ,α
we show its Schro¨dinger-type uncertainty relation.
5
Definition 3.1 We define the parametric extended correlation measure Corr (X,Y) for two
ρ,α,γ
parameters α,γ [0,1] by
∈
Corr (X,Y) γCorr (X,Y)+(1 γ)Corr (X,Y) (13)
ρ,α,γ ρ,α ρ,1−α
≡ −
for any operators X and Y.
Note that we have Corr (H,H) = I (H) for any observable H. Then we can prove the
ρ,α,γ ρ,α
following inequality.
Theorem 3.2 If 0 α,γ 1 or 1 α,γ 1, then we have
≤ ≤ 2 2 ≤ ≤
U (A)U (B) 4α(1 α)Corr (A,B)2
ρ,α ρ,α ρ,α,γ
≥ − | |
for two observables A, B and a quantum state ρ.
Proof: By the similar way of the proof of Theorem 2.1, we have Eq.(9) and we also have
Corr (A,B) = Tr[ρAB] Tr[ρ1−αAραB]
ρ,1−α
−
= (λ λ1−αλα)a b +(λ λ1−αλα)a b . (14)
i − i j ij ji j − j i ji ij
Xi<j n o
Thus we have
Corr (A,B) = γCorr (A,B)+(1 γ)Corr (A,B)
ρ,α,γ ρ,α ρ,α
−
= γλα(λ1−α λ1−α)+(1 γ)λ1−α(λα λα) a b
i i − j − i i − j ij ji
Xi<j n o
+ γλα(λ1−α λ1−α)+(1 γ)λ1−α(λα λα) a b .
j j − i − j j − i ji ij
Xi<j n o
Since we have a = a and b = b , we then have
ij ji ij ji
| | | | | | | |
Corr (A,B) γ(λα+λα)λ1−α λ1−α +(1 γ)(λ1−α +λ1−α)λα λα a b
| ρ,α,γ | ≤ i j | i − j | − i j | i − j| | ij|| ji|
Xi<j n o
λ λ a b ,
i j ij ji
≤ | − || || |
i<j
X
thanks to the inequality
γ(xα+yα)x1−α y1−α +(1 γ)(x1−α +y1−α)xα yα x y (15)
| − | − | − |≤ | − |
for 0 α,γ 1 or 1 α,γ 1, and x,y 0. The rest of the proof goes similar way to that of
≤ ≤ 2 2 ≤ ≤ ≥
Theorem 2.1.
Corollary 3.3 For any α [0,1], two observables A, B and a quantum state ρ, we have
∈
Uρ,α(A)Uρ,α(B) ≥ 4α(1−α)|Corrρ,α,21(A,B)|2.
Proof: If γ = 1, then the equality of the inequality (15) holds for any α [0,1] and x,y 0.
2 ∈ ≥
Therefore we have the present corollary from Theorem 3.2.
We may define the following correlation measure instead of Definition 3.1.
6
(sym)
Definition 3.4 We define a symmetric extended correlation measure Corr (X,Y) for two
ρ,α,γ
parameters α,γ [0,1] by
∈
Corr(sym)(X,Y) γCorr (X,Y)+(1 γ)Corr (Y,X) (16)
ρ,α,γ ≡ ρ,α − ρ,α
for any operators X and Y.
(sym) (sym)
Note that we have Corr (A,B) = Corr (B,A) for self-adjoint operators A and B.
ρ,α,γ ρ,α,γ
Then we have the following therem by the similar proof of the above using the inequality
(xα+yα)x1−α y1−α x y
| − |≤ | − |
for x,y 0 and α 1.
≥ ≥ 2
Theorem 3.5 For α [1,1] and γ [0,1], we have
∈ 2 ∈
U (A)U (B) 4α(1 α)Corr(sym)(A,B)2
ρ,α ρ,α ≥ − | ρ,α,γ |
for two observables A, B and a quantum state ρ.
4 A further generalization by metric adjusted correlation mea-
sure
Inspired by the recent results in [10] and the concept of metric adjusted skew information intro-
duced by Hansen in [12], we here give a further generalization for Schro¨dinger-type uncertainty
relation applying metric adjusted correlation measure introduced in [12]. We firstly give some
notations according to those in [10]. Let M (C) and M (C) be the set of all n n complex
n n,sa
×
matrices and all n n self-adjoint matrices, equipped with the Hilbert-Schmidt scalar product
×
A,B = Tr[A∗B], respectively. Let M (C) be the set of all positive definite matrices of
n,+
h i
M (C) and M (C) be the set of all density matrices, that is
n,sa n,+,1
M (C) ρ M (C)Trρ= 1,ρ > 0 M (C).
n,+,1 n,sa n,+
≡ { ∈ | } ⊂
Here X M (C) means we have φX φ 0 for any vector φ Cn. In the study of
n,+
∈ h | | i ≥ | i ∈
quantum physics, we usually use a positive semidefinite matrix with a unit trace as a density
operator ρ. In this section, we assume the invertibility of ρ.
A function f : (0,+ ) R is said operator monotone if the inequalities 0 f(A)
∞ → ≤ ≤
f(B) hold for any A,B M (C) such that 0 A B. An operator monotone function
n,sa
∈ ≤ ≤
f : (0,+ ) (0,+ ) is said symmetric if f(x) = xf(x−1) and normalized if f(1) = 1. We
∞ → ∞
represents the set of all symmetric normalized operator monotone functions by . We have
op
F
the following examples as elements of :
op
F
Example 4.1 ([12, 10, 6, 25])
2x x+1 x 1
f (x) = , f (x) = , f (x) = − ,
RLD SLD BKM
x+1 2 logx
√x+1 2 (x 1)2
f (x) = , f (x) = α(1 α) − , α (0,1).
WY 2 WYD − (xα 1)(x1−α 1) ∈
(cid:18) (cid:19) − −
7
The functions f (x) and f (x) are normalized in the sense that lim f (x) = 1
BKM WYD x→1 BKM
and lim f (x) = 1. Note that a simple proof of the operator monotonicity of f (x)
x→1 WYD WYD
was given in [6]. See also [30] for the proof of the operator monotonicity of f (x) by use of
WYD
majorization.
Remark 4.2 ([10, 15, 24, 25]) For any f , we have the following inequalities:
op
∈F
2x x+1
f(x) , x > 0.
x+1 ≤ ≤ 2
That is, all f lies in between the harmonic mean and the arithmetic mean.
op
∈ F
Forf wedefinef(0) = lim f(x). Wealsodenotethesetsofregularandnon-regular
op x→0
∈ F
functions by
r = f f(0) = 0 and n = f f(0) = 0 .
Fop { ∈ Fop| 6 } Fop { ∈ Fop| }
Definition 4.3 ([8, 10]) For f r , we define the function f˜by
∈ Fop
1 f(0)
f˜(x) = (x+1) (x 1)2 , (x >0).
2 − − f(x)
(cid:26) (cid:27)
Then we have the following theorem.
Theorem 4.4 ([8, 6, 26]) The correspondence f f˜is a bijection between r and n.
→ Fop Fop
We can use matrix mean theory introduced by Kubo-Ando in [15]. Then a mean m corre-
f
sponds to each operator monotone function f by the following formula
op
∈F
m (A,B) = A1/2f(A−1/2BA−1/2)A1/2,
f
for A,B M (C). By the notion of matrix mean, we may define the set of the monotone
n,+
∈
metrics [23] by the following formula
A,B = Tr[Am (L ,R )−1(B)],
ρ,f f ρ ρ
h i
where L (A) = ρA and R (A) = Aρ.
ρ ρ
Definition 4.5 ([12, 8]) For A,B M (C), ρ M (C) and f r , we define the
∈ n,sa ∈ n,+,1 ∈ Fop
following quantities:
f(0)
Corrf(A,B) i[ρ,A],i[ρ,B] , If(A) Corrf(A,A),
ρ ≡ 2 h iρ,f ρ ≡ ρ
Cf(A,B) Tr[m (L ,R )(A)B], Cf(A) Cf(A,A),
ρ ≡ f ρ ρ ρ ≡ ρ
Uf(A) V (A)2 (V (A) If(A))2.
ρ ≡ ρ − ρ − ρ
q
f
The quantity I (A) is known as metric adjusted skew information [12]. It is notable that the
ρ
metric adjusted correlation measure Corrc(A,B) was firstly introduced in [12] for a regular
ρ
Morozova-Chentsov function c. Recently the notation Ic(A,B) in [1] and the notation If(A,B)
ρ ρ
in [11] were used. In addition, it is useful for the readers to be noted that the correlation
f
I (A,B) can be expressed as a difference of covariances [11]. Throughout the present paper, we
ρ
f
usethe notation Corr (A,B) as the metric adjusted correlation measure, to avoid the confusion
ρ
of the readers. (In the previous sections, we have already used Corr (A,B), Corr (A,B) and
ρ ρ,α
Corr (A,B) as correlation measures and done I (H) and I (H) as skew informations.)
ρ,α,γ ρ ρ,α
Then we have the following proposition.
8
Proposition 4.6 ([8, 10]) For A,B M (C), ρ M (C) and f r , we have the
∈ n,sa ∈ n,+,1 ∈ Fop
following relations, where we put A A Tr[ρA]I and B B Tr[ρB]I.
0 0
≡ − ≡ −
(1) If(A) = Tr[ρA2] Tr[m (L ,R )(A )A ]= V (A) Cf˜(A ).
ρ 0 − f˜ ρ ρ 0 0 ρ − ρ 0
(2) Jf(A) = Tr[ρA2]+Tr[m (L ,R )(A )A ]= V (A)+Cf˜(A ).
ρ 0 f˜ ρ ρ 0 0 ρ ρ 0
f f
(3) 0 I (A) U (A) V (A).
ρ ρ ρ
≤ ≤ ≤
f f f
(4) U (A) = I (A)J (A).
ρ ρ ρ
q
(5) Corrf(A,B) = 1Tr[ρA B ] + 1Tr[ρB A ] Tr[m (L ,R )(A )B ] = 1Tr[ρA B ] +
ρ 2 0 0 2 0 0 − f˜ ρ ρ 0 0 2 0 0
1Tr[ρB A ] Cf˜(A ,B ).
2 0 0 − ρ 0 0
Thefollowinginequality isthefurthergeneralization ofCorollary 3.3bytheuseofthemetric
adjusted correlation measure.
Theorem 4.7 For f r , if we have
∈ Fop
x+1
+f˜(x) 2f(x), (17)
2 ≥
then we have
Uf(A)Uf(B) 4f(0)Corrf(A,B)2, (18)
ρ ρ ≥ | ρ |
for A,B M (C) and ρ M (C).
n,sa n,+,1
∈ ∈
In order to prove Theorem 4.7, we use the following two lemmas.
Lemma 4.8 ([35]) If Eq.(17) is satisfied, then we have the following inequality:
2
x+y
m (x,y)2 f(0)(x y)2.
2 − f˜ ≥ −
(cid:18) (cid:19)
Proof: By Eq.(17), we have
x+y
+m (x,y) 2m (x,y).
2 f˜ ≥ f
We also have
x
m (x,y) = yf˜
f˜ y
(cid:18) (cid:19)
2
y x x f(0)
= +1 1
2 y − y − f(x/y)
( )
(cid:18) (cid:19)
x+y f(0)(x y)2
= − .
2 − 2m (x,y)
f
Therefore
2
x+y x+y x+y
m (x,y)2 = m (x,y) +m (x,y)
2 − f˜ 2 − f˜ 2 f˜
(cid:18) (cid:19) (cid:26) (cid:27)(cid:26) (cid:27)
f(0)(x y)2
− 2m (x,y)
f
≥ 2m (x,y)
f
= f(0)(x y)2.
−
9
f f f f
We have the following expressions for the quantities I (A), J (A), U (A) and Corr (A,B)
ρ ρ ρ ρ
by using Proposition 4.6 and a mean m .
f˜
Lemma 4.9 ([10]) Let φ , φ , , φ be a basis of eigenvectors of ρ, corresponding to
1 2 n
{| i | i ··· | i}
the eigenvalues λ ,λ , ,λ . We put a = φ A φ ,b = φ B φ , where A A
1 2 n jk j 0 k jk j 0 k 0
{ ··· } h | | i h | | i ≡ −
Tr[ρA]I and B B Tr[ρB]I for A,B M (C) and ρ M (C). Then we have
0 n,sa n,+,1
≡ − ∈ ∈
1
If(A) = (λ +λ )a a m (λ ,λ )a a
ρ 2 j k jk kj − f˜ j k jk kj
j,k j,k
X X
λ +λ
= 2 j k m (λ ,λ ) a 2,
2 − f˜ j k | jk|
j<k(cid:26) (cid:27)
X
1
Jf(A) = (λ +λ )a a + m (λ ,λ )a a
ρ 2 j k jk kj f˜ j k jk kj
j,k j,k
X X
λ +λ
2 j k +m (λ ,λ ) a 2,
≥ 2 f˜ j k | jk|
j<k(cid:26) (cid:27)
X
2 2
1
Uf(A)2 = (λ +λ )a 2 m (λ ,λ )a 2
ρ 4 j k | jk|  − f˜ j k | jk| 
j,k j,k
X X
   
and
1 1
Corrf(A,B) = λ a b + λ a b m (λ ,λ )a b
ρ 2 j jk kj 2 k jk kj − f˜ j k jk kj
j,k j,k j,k
X X X
λ +λ λ +λ
j k k j
= m (λ ,λ ) a b + m (λ ,λ ) a b .
2 − f˜ j k jk kj 2 − f˜ k j kj jk
j<k(cid:18) (cid:19) j<k(cid:18) (cid:19)
X X
(19)
We are now in a position to prove Theorem 4.7.
Proof of Theorem 4.7: From Eq.(19), we have
λ +λ λ +λ
Corrf(A,B) j k m (λ ,λ ) a b + j k m (λ ,λ ) a b
| ρ | ≤ 2 − f˜ j k jk kj 2 − f˜ k j kj jk
j<k(cid:12)(cid:18) (cid:19) (cid:12) j<k(cid:12)(cid:18) (cid:19) (cid:12)
X(cid:12) (cid:12) X(cid:12) (cid:12)
(cid:12)λ +λ (cid:12) (cid:12)λ +λ (cid:12)
(cid:12) j k m (λ ,λ ) a b (cid:12)+ (cid:12) j k m (λ ,λ ) a b (cid:12)
≤ 2 − f˜ j k | jk|| kj| 2 − f˜ k j | kj|| jk|
j<k(cid:12) (cid:12) j<k(cid:12) (cid:12)
X(cid:12) (cid:12) X(cid:12) (cid:12)
(cid:12) λ +λ (cid:12) (cid:12) (cid:12)
= 2 (cid:12) j k m (λ ,λ(cid:12)) a b (cid:12) (cid:12)
2 − f˜ j k | jk|| kj|
j<k(cid:12) (cid:12)
X(cid:12) (cid:12)
(cid:12) (cid:12)
λ(cid:12)j λk ajk bkj . (cid:12)
≤ | − || || |
j<k
X
Then we have
2
f(0)Corrf(A,B)2 f(0)1/2 λ λ a b
| ρ | ≤  | j − k|| jk|| kj|
j<k
X
 
10