- Open Access
Uncertainty relations for generalized metric adjusted skew information and generalized metric adjusted correlation measure
© Yanagi et al.; licensee Springer. 2013
- Received: 10 September 2013
- Accepted: 24 October 2013
- Published: 11 November 2013
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).
Primary: 15A45, 47A63; secondary: 94A17
- Trace inequality
- Metric adjusted skew information
- Metric adjusted correlation measure
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) ].
As stated in , 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).
respectively. In particular, the operator monotonicity of the function f WYD was proved in  (see also ). On the other hand, the uncertainty relation related to the Wigner-Yanase skew information was given by Luo , and the uncertainty relation related to the Wigner-Yanase-Dyson skew information was given by Yanagi . 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.
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.
t f(t −1)=f(t),
f is operator monotone.
and notice that trivially .
Then we have the following proposition.
Now we modify the uncertainty relation given by .
where and .
But since 4 f(0)≤1 and , it is easily given by Theorem 2.
where and .
Though we cannot use the Schwarz inequality, we can get (4) in Theorem 3 by modifying the proof given by .
we obtain the following uncertainty relation.
where and . Since f(0)<1, it is easy to show that (5) and (6) are weaker than (3) and (4), respectively.
We give some generalizations of Heisenberg or Schrd̈inger uncertainty relations which include Theorem 3 as corollary.
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.
where and .
it is easy to show that is an inner product in . Then we can get the result by using the Schwarz inequality.
where and .
In order to prove Theorem 5, we need the following lemmas.
Proof of Lemma 1
We have the following expressions for the quantities , , , and by using Proposition 2 and a mean .
We are now in a position to prove Theorem 5.
Hence, we have the desired inequality (9). □
We give some examples satisfying the condition (8).
holds for 0<α<1.
In order to prove Example 4, we need the following lemma.
F(y) is monotone increasing for .
F(y) is convex for y<0.
F(y) is concave for y≥1/2.
We give the proof of Lemma 3 in the Appendix.
- (i)Since F(y)>0 for x>0 and , it is sufficient to prove for the proof of F ′(y)>0. We have
- (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
If x<1(i.e., t>1), then l(t)>0.
If x>1(i.e., 0<t<1), then l(t)<0.
- (iii)We calculate
- (a)For the case 0<x≤1, we have
for 0<x≤1 and 1/2≤y<2.
- (b)For the case x ≥ 1, we firstly calculate
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.
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.
- Heisenberg W: Uber den anschaulichen Inhalt der quantummechanischen Kinematik und Mechanik. Zeitschrift für Physik 1927, 43: 172–198. 10.1007/BF01397280View ArticleGoogle Scholar
- Robertson HP: The uncertainty principle. Phys. Rev 1929, 34: 163–164. 10.1103/PhysRev.34.163View ArticleGoogle Scholar
- Schrödinger E: About Heisenberg uncertainty relation. Proc. Prussian Acad. Sci. Phys. Math 1930, XIX: 293.Google Scholar
- Luo S: Heisenberg uncertainty relation for mixed states. Phys. Rev. A 2005, 72: 042110.View ArticleGoogle Scholar
- Wigner EP, Yanase MM: Information content of distribution. Proc. Nat. Acad. Sci 1963, 49: 910–918. 10.1073/pnas.49.6.910MathSciNetView ArticleGoogle Scholar
- Luo S, Zhang Q: Informational distance on quantum-state space. Phys. Rev. A 2004, 69: 032106.MathSciNetView ArticleGoogle Scholar
- Luo S: Quantum versus classical uncertainty. Theor. Math. Phys 2005, 143: 681–688. 10.1007/s11232-005-0098-6View ArticleGoogle Scholar
- Yanagi K: Uncertainty relation on Wigner-Yanase-Dyson skew information. J. Math. Anal. Appl 2010, 365: 12–18. 10.1016/j.jmaa.2009.09.060MathSciNetView ArticleGoogle Scholar
- Lieb EH: Convex trace functions and the Wigner-Yanase-Dyson conjecture. Adv. Math 1973, 11: 267–288. 10.1016/0001-8708(73)90011-XMathSciNetView ArticleGoogle Scholar
- Yanagi K: Uncertainty relation on generalized Wigner-Yanase-Dyson skew information. Linear Algebra Appl 2010, 433: 1524–1532. 10.1016/j.laa.2010.05.024MathSciNetView ArticleGoogle Scholar
- Cai L, Luo S: On convexity of generalized Wigner-Yanase-Dyson information. Lett. Math. Phys 2008, 83: 253–264. 10.1007/s11005-008-0222-2MathSciNetView ArticleGoogle Scholar
- 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.2008.11.022MathSciNetView ArticleGoogle Scholar
- Petz D: Monotone metrics on matrix spaces. Linear Algebra Appl 1996, 244: 81–96.MathSciNetView ArticleGoogle Scholar
- Petz D, Hasegawa H: On the Riemannian metric of α-entropies of density matrices. Lett. Math. Phys 1996, 38: 221–225. 10.1007/BF00398324MathSciNetView ArticleGoogle Scholar
- Furuta T: Elementary proof of Petz-Hasegawa theorem. Lett. Math. Phys 2012, 101: 355–359. 10.1007/s11005-012-0568-3MathSciNetView ArticleGoogle Scholar
- Gibilisco P, Imparato D, Isola T: Uncertainty principle and quantum Fisher information, II. J. Math. Phys 2007, 48: 072109. 10.1063/1.2748210MathSciNetView ArticleGoogle Scholar
- Kubo F, Ando T: Means of positive linear operators. Math. Ann 1980, 246: 205–224. 10.1007/BF01371042MathSciNetView ArticleGoogle Scholar
- Hansen F: Metric adjusted skew information. Proc. Nat. Acad. Sci 2008, 105: 9909–9916. 10.1073/pnas.0803323105View ArticleGoogle Scholar
- 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.2010.09.029MathSciNetView ArticleGoogle Scholar
- 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.2011.10.061MathSciNetView ArticleGoogle Scholar
- Yanagi K: Metric adjusted skew information and uncertainty relation. J. Math. Anal. Appl 2011, 380: 888–892. 10.1016/j.jmaa.2011.03.068MathSciNetView ArticleGoogle Scholar
- Gibilisco P, Hiai F, Petz D: Quantum covariance, quantum Fisher information, and the uncertainty relations. IEEE Trans. Inf. Theory 2009, 55: 439–443.MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.