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
KeywordsTrace 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.
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.
t f(t −1)=f(t),
f is operator monotone.
and notice that trivially .
Metric adjusted skew information and correlation measure
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.
Generalized metric adjusted skew information and correlation measure
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.
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