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Uncertainty relations for generalized metric adjusted skew information and generalized metric adjusted correlation measure
Journal of Uncertainty Analysis and Applications volume 1, Article number: 12 (2013)
Abstract
Abstract
In this paper, we give a Heisenberg-type or a Schrödinger-type uncertainty relation for generalized metric adjusted skew information or generalized metric adjusted correlation measure. These results generalize the previous result of Furuichi and Yanagi (J. Math. Anal. Appl. 388:1147-1156, 2012).
AMS
Primary: 15A45, 47A63; secondary: 94A17
Introduction
We start from the Heisenberg uncertainty relation [1]:
for a quantum state (density operator) ρ and two observables (self-adjoint operators) A and B. The further stronger result was given by Schrödinger in [2], [3]:
where the covariance is defined by Cov ρ (A,B)≡T r [ ρ(A−T r [ ρ A]I)(B−T r [ ρ B]I) ].
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 ρ (H) representing a quantum uncertainty excluding the classical mixture [4]:
with the Wigner-Yanase skew information [5]:
and then he successfully showed a new Heisenberg-type uncertainty relation on U ρ (H) in [4]:
As stated in [4], the physical meaning of the quantity U ρ (H) can be interpreted as follows. For a mixed state ρ, the variance V ρ (H) has both classical mixture and quantum uncertainty. Also, the Wigner-Yanase skew information I ρ (H) represents a kind of quantum uncertainty [6, 7]. Thus, the difference V ρ (H)−I ρ (H) has a classical mixture so that we can regard that the quantity U ρ (H) has a quantum uncertainty excluding a classical mixture. Therefore, it is meaningful and suitable to study an uncertainty relation for a mixed state by the use of the quantity U ρ (H).
Recently, a one-parameter extension of the inequality (1) was given in [8]:
where
with the Wigner-Yanase-Dyson skew information I ρ,α (H) defined by
It is notable that the convexity of I ρ,α (H) with respect to ρ was successfully proven by Lieb in [9]. The further generalization of the Heisenberg-type uncertainty relation on U ρ (H) has been given in [10] using the generalized Wigner-Yanase-Dyson skew information introduced in [11]. Recently, it is shown that these skew informations are connected to special choices of quantum Fisher information in [12]. The family of all quantum Fisher informations is parametrized by a certain class of operator monotone functions which were justified in [13]. The Wigner-Yanase skew information and Wigner-Yanase-Dyson skew information are given by the following operator monotone functions:
respectively. In particular, the operator monotonicity of the function f WYD was proved in [14] (see also [15]). On the other hand, the uncertainty relation related to the Wigner-Yanase skew information was given by Luo [4], and the uncertainty relation related to the Wigner-Yanase-Dyson skew information was given by Yanagi [8]. In this paper, we generalize these uncertainty relations to the uncertainty relations related to quantum Fisher informations by using (generalized) metric adjusted skew information or correlation measure.
Operator monotone functions
Let (respectively ) be the set of all n×n complex matrices (respectively all n×n self-adjoint matrices), endowed with the Hilbert-Schmidt scalar product 〈A,B〉=T r (A∗ B). Let be the set of strictly positive elements of and be the set of stricly positive density matrices, that is . If it is not otherwise specified, from now on, we shall treat the case of faithful states, that is ρ>0.
A function is said to be operator monotone if, for any and such that 0≤A≤B, the inequalities 0≤f(A)≤f(B) hold. An operator monotone function is said to be symmetric if f(x)=x f(x−1) and normalized if f(1)=1.
Definition 1
is the class of functions f:(0,+∞)→(0,+∞) such that
-
1.
f(1)=1,
-
2.
t f(t −1)=f(t),
-
3.
f is operator monotone.
Example 1
Examples of elements of are given by the following list:
Remark 1
Any satisfies
For , define f(0)=lim x→0 f(x). We introduce the sets of regular and non-regular functions
and notice that trivially .
Definition 2
For , we set
Theorem 1
([[12] , [16] , [17]]) The correspondence is a bijection between and .
Metric adjusted skew information and correlation measure
In the Kubo-Ando theory of matrix means, one associates a mean to each operator monotone function by the formula
where . Using the notion of matrix means, one may define the class of monotone metrics (also called quantum Fisher informations) by the following formula:
where L ρ (A)=ρ A,R ρ (A)=A ρ. In this case, one has to think of A,B as tangent vectors to the manifold at the point ρ (see [12, 13]).
Definition 3
For and , we define the following quantities:
The quantity is known as metric adjusted skew information [18], and the metric adjusted correlation measure was also previously defined in [18].
Then we have the following proposition.
Proposition 1
([16, 19]) For and , we have the following relations, where we put A 0=A−T r [ ρ A]I and B 0=B−T r [ ρ B]I:
-
1.
,
-
2.
,
-
3.
,
-
4.
,
-
5.
,
-
6.
.
Now we modify the uncertainty relation given by [20].
Theorem 2
For , it holds
where and .
Remark 2
Since Theorem 2 is easily given by using the Schwarz inequality, we omit the proof. In [20] we gave the uncertainty relation
But since 4 f(0)≤1 and , it is easily given by Theorem 2.
Theorem 3
then it holds
where and .
Remark 3
Though we cannot use the Schwarz inequality, we can get (4) in Theorem 3 by modifying the proof given by [20].
By putting
we obtain the following uncertainty relation.
Corollary 1
For and ,
where
Remark 4
Even if (2) does not necessarily hold, then
where and . Since f(0)<1, it is easy to show that (5) and (6) are weaker than (3) and (4), respectively.
Generalized metric adjusted skew information and correlation measure
We give some generalizations of Heisenberg or Schrd̈inger uncertainty relations which include Theorem 3 as corollary.
Definition 4
([22]) Let satisfy
for some k>0. We define
Definition 5
For and , we define the following quantities:
The quantity and are said to be generalized metric adjusted skew information and generalized metric adjusted correlation measure, respectively.
Then we have the following proposition.
Proposition 2
For and , we have the following relations, where we put A 0=A−T r [ ρ A]I and B 0=B−T r [ ρ B]I:
-
1.
,
-
2.
,
-
3.
,
-
4.
.
Theorem 4
For , it holds
where and .
Proof of Theorem 4. We define for
Since
it is easy to show that is an inner product in . Then we can get the result by using the Schwarz inequality.
Theorem 5
For , if
for some ℓ>0, then it holds
where and .
In order to prove Theorem 5, we need the following lemmas.
Lemma 1
If (7) and (8) are satisfied, then we have the following inequality:
Proof of Lemma 1
By (7) and (8), we have
Therefore, by (10) and (11),
We have the following expressions for the quantities , , , and by using Proposition 2 and a mean .
Lemma 2
Let {|ϕ 1〉,|ϕ 2〉,…,|ϕ n 〉} be a basis of eigenvectors of ρ, corresponding to the eigenvalues {λ 1,λ 2,…,λ n }. We put a jk =〈ϕ j |A 0|ϕ k 〉,b jk =〈ϕ j |B 0|ϕ k 〉, where A 0≡A−T r [ ρ A]I and B 0≡B−T r [ ρ B]I for and . Then we have
and
We are now in a position to prove Theorem 5.
Proof of Theorem 5. At first we prove (9). Since
Then by Lemma 1, we have
By a similar way, we also have
Hence, we have the desired inequality (9). □
We give some examples satisfying the condition (8).
Example 2
Let
Then
Proof of Example 2. In [10, 21] we give
for x>0 and 0≤α≤1. Then we have
□
Example 3
Let
Then
holds for 0<α<1.
Proof of Example 3. Since
we have
Since
we have
Then we have
□
Example 4
Let
Then .
In order to prove Example 4, we need the following lemma.
Lemma 3
For x>0, we set the function of y as
Then F(y) has following properties:
-
1.
F(y) is monotone increasing for .
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2.
F(y) is convex for y<0.
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3.
F(y) is concave for y≥1/2.
We give the proof of Lemma 3 in the Appendix.
Proof of Example 4. By Lemma 3,
It follows from the monotonicity that
for y∈ [ 3/4,1]. Then
for y∈[ 3/4,1]. Therefore, we have
Hence, we have
□
Appendix
Proof of Lemma 3.
-
(i)
Since F(y)>0 for x>0 and , it is sufficient to prove for the proof of F ′(y)>0. We have
Then we put
where we put xy≡r>0. From elementary calculations, we have G(r)≥G(1)=0 which implies .
-
(ii)
We firstly set f(y)≡ logF(y). Since F(y)>0, we have only to prove f ′′(y)>0 for the proof of F ′′(y)>0. We set again Then we have In addition, by , we have
By , we have
We prove f′′(y)>0 for y<0. We calculate
Thus, if we put
then we have only to prove h(y)<0 for y<0. Since we have h(0)=0, we have only to prove h′(y)>0 for y<0. Here we have
If we set again
where we put xy≡t>0, then we prove the following cases:
-
(a)
If x<1(i.e., t>1), then l(t)>0.
-
(b)
If x>1(i.e., 0<t<1), then l(t)<0.
For case (a), we calculate
and
Thus, we have l′(t)≥l′(1)=0, and then we have l(t)≥l(1)=0. For case (b), we easily find that
Thus, we have l′(t)≥l′(1)=0, and then we have l(t)≤l(1)=0.
-
(iii)
We calculate
where
We prove h(x,y)≤0 for x>0 and y≥1/2. Then we have
Here we note that . We also put
If we have g(x,y)≥0 for x>0 and y≥1/2, then we have for 0<x≤1 and for x≥1. Thus, we then obtain h(x,y)≤h(1,y)=0 for y≥1/2, due to . Therefore, we have only to prove g(x,y)≥0 for x>0 and y≥1/2.
-
(a)
For the case 0<x≤1, we have
Since g(1,y)=0, if we prove , then we can prove g(x,y)≥g(1,y)=0 for y≥1/2 and 0<x≤1. Since we have the relations
for 0<x≤1, we calculate
Thus, we have only to prove
for 0<x≤1 and y≥1/2. Since it is trivial k(y)≥0 for y≥2, we assume 1/2≤y<2 from here. To this end, we prove that k 1(y)≡3(y−2)xy/2 +(y−2)x3y/2 is monotone increasing for 1/2≤y<2 and k 2(y)≡3y +(y+4)xy is also monotone increasing for 1/2≤y<2. We easily find that
for 0<x≤1 and 1/2≤y<2.
We also have
Here we prove for 0<x≤1 and 1/2≤y<2. We put again
then we have
Thus, we have
Since for 0<x<α y and for α y <x≤1, we have
Since we have , the function k 4(y) is monotone increasing for y. Thus, we have
since e10/9≃3.03773. Therefore, k 2(y) is also a monotone increasing function of y for 0<x≤1 and 1/2≤y<2. Thus, k(y) is monotone increasing for y≥1/2, and then we have
-
(b)
For the case x ≥ 1, we firstly calculate
We put
Then we calculate
Then we put
We have
and then
Since we find
for x≥1, we have q(x,y)≥q(x,1/2)≥q(1,1/2)=0. Therefore, we have , which implies p(x,y)≥p(1,y)=0. Thus, we have , and then we have g(x,y)≥g(x,1/2), where
To prove g(x,1/2)≥0 for x≥1 and y≥1/2, we put x1/2≡z≥1 and
Since we have and
we have r′(1)=0 and then we have r′(z)≥0 for z≥1. Thus, we have r(z)≥0 for z≥1 by r(1)=0. Finally, we have g(x,y)≥g(x,1/2)≥0, for x≥1 and y≥1/2.
□
References
Heisenberg W: Uber den anschaulichen Inhalt der quantummechanischen Kinematik und Mechanik. Zeitschrift für Physik 1927, 43: 172–198. 10.1007/BF01397280
Robertson HP: The uncertainty principle. Phys. Rev 1929, 34: 163–164. 10.1103/PhysRev.34.163
Schrödinger E: About Heisenberg uncertainty relation. Proc. Prussian Acad. Sci. Phys. Math 1930, XIX: 293.
Luo S: Heisenberg uncertainty relation for mixed states. Phys. Rev. A 2005, 72: 042110.
Wigner EP, Yanase MM: Information content of distribution. Proc. Nat. Acad. Sci 1963, 49: 910–918. 10.1073/pnas.49.6.910
Luo S, Zhang Q: Informational distance on quantum-state space. Phys. Rev. A 2004, 69: 032106.
Luo S: Quantum versus classical uncertainty. Theor. Math. Phys 2005, 143: 681–688. 10.1007/s11232-005-0098-6
Yanagi K: Uncertainty relation on Wigner-Yanase-Dyson skew information. J. Math. Anal. Appl 2010, 365: 12–18. 10.1016/j.jmaa.2009.09.060
Lieb EH: Convex trace functions and the Wigner-Yanase-Dyson conjecture. Adv. Math 1973, 11: 267–288. 10.1016/0001-8708(73)90011-X
Yanagi K: Uncertainty relation on generalized Wigner-Yanase-Dyson skew information. Linear Algebra Appl 2010, 433: 1524–1532. 10.1016/j.laa.2010.05.024
Cai L, Luo S: On convexity of generalized Wigner-Yanase-Dyson information. Lett. Math. Phys 2008, 83: 253–264. 10.1007/s11005-008-0222-2
Gibilisco P, Hansen F, Isola T: On a correspondence between regular and non-regular operator monotone functions. Linear Algebra Appl 2009, 430: 2225–2232. 10.1016/j.laa.2008.11.022
Petz D: Monotone metrics on matrix spaces. Linear Algebra Appl 1996, 244: 81–96.
Petz D, Hasegawa H: On the Riemannian metric of α-entropies of density matrices. Lett. Math. Phys 1996, 38: 221–225. 10.1007/BF00398324
Furuta T: Elementary proof of Petz-Hasegawa theorem. Lett. Math. Phys 2012, 101: 355–359. 10.1007/s11005-012-0568-3
Gibilisco P, Imparato D, Isola T: Uncertainty principle and quantum Fisher information, II. J. Math. Phys 2007, 48: 072109. 10.1063/1.2748210
Kubo F, Ando T: Means of positive linear operators. Math. Ann 1980, 246: 205–224. 10.1007/BF01371042
Hansen F: Metric adjusted skew information. Proc. Nat. Acad. Sci 2008, 105: 9909–9916. 10.1073/pnas.0803323105
Gibilisco P, Isola T: On a refinement of Heisenberg uncertainty relation by means of quantum Fisher information. J. Math. Anal. Appl 2011, 375: 270–275. 10.1016/j.jmaa.2010.09.029
Furuichi S, Yanagi K: Schrödinger uncertainty relation, Wigner-Yanase-Dyson skew information and metric adjusted correlation measure. J. Math. Anal. Appl 2012, 388: 1147–1156. 10.1016/j.jmaa.2011.10.061
Yanagi K: Metric adjusted skew information and uncertainty relation. J. Math. Anal. Appl 2011, 380: 888–892. 10.1016/j.jmaa.2011.03.068
Gibilisco P, Hiai F, Petz D: Quantum covariance, quantum Fisher information, and the uncertainty relations. IEEE Trans. Inf. Theory 2009, 55: 439–443.
Acknowledgements
The first author (KY) was partially supported by JSPS KAKENHI Grant Number 23540208. The second author (SF) was partially supported by JSPS KAKENHI Grant Number 24540146.
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Yanagi, K., Furuichi, S. & Kuriyama, K. Uncertainty relations for generalized metric adjusted skew information and generalized metric adjusted correlation measure. J. Uncertain. Anal. Appl. 1, 12 (2013). https://doi.org/10.1186/2195-5468-1-12
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DOI: https://doi.org/10.1186/2195-5468-1-12