Cong Xu ,Zhaoqi Wu and Shao-Ming Fei,3,*

1 School of Mathematical Sciences,Capital Normal University,Beijing 100048,China

2 Department of Mathematics,Nanchang University,Nanchang 330031,China

3 Max-Planck-Institute for Mathematics in the Sciences,04103 Leipzig,Germany

Abstract We establish tighter uncertainty relations for arbitrary finite observables via (α,β,γ) weighted Wigner–Yanase–Dyson((α,β,γ)WWYD)skew information.The results are also applicable to the(α,γ)weighted Wigner–Yanase–Dyson((α,γ)WWYD)skew information and the weighted Wigner–Yanase–Dyson (WWYD) skew information.We also present tighter lower bounds for quantum channels and unitary channels via(α,β,γ)modified weighted Wigner–Yanase–Dyson((α,β,γ) MWWYD) skew information.Detailed examples are provided to illustrate the tightness of our uncertainty relations.

Keywords: quantum channels,(α,β,γ) WWYD skew information,(α,β,γ) MWWYD skew information,uncertainty relations

1.Introduction

As one of the cornerstones of quantum mechanics,the uncertainty principle reveals the insights that distinguish quantum theory from classical theory.Heisenberg [1] originally proposed the uncertainty principle in 1927,which indicates that the position and momentum of a particle cannot be precisely determined simultaneously.Robertson [2] generalized the variance-based uncertainty relation for position and momentum to arbitrary two observablesAandB.

where[A,B]=AB-BAandis the standard deviation of the observableMwith respect to a fixed state |ψ〉.

The uncertainty principle has attracted sustained attention.A host of methods have been presented to characterize the uncertainty [3–16].The skew information is one of the typical ways to describe the uncertainty principle.The skew information of an observableAwith respect to a quantum state ρ is given by [17].

which is called the Wigner–Yanase (WY) skew information.Later,Dyson proposed a one-parameter extension of the WY skew information,called Wigner–Yanase–Dyson (WYD)skew information.In [18] the WYD skew information is further generalized to the generalized Wigner–Yanase–Dyson(GWYD)skew information.The corresponding definitions of the above-mentioned skew information for arbitrary (not necessarily Hermitian) operators have also been introduced[9,10,19,20].The uncertainty relation related to WY skew information was initially introduced by Luo [21].Recently,uncertainty relations based on various generalized skew information have been explored intensely in [19,20,22–30].

By considering the arithmetic mean of ραand ρ1-α,Furuichietal[31] introduced another one-parameter skew information,

which is called the weighted Wigner–Yanase–Dyson(WWYD) skew information [20].A generalization of equation(3)was presented in[32]for an arbitrary operatorE,

which is termed as the modified weighted Wigner–Yanase–Dyson (MWWYD) skew information in [20].Note that equation (4)reduces to equation (14)in [9],and equation (3)reduces to equation (2) whenα=respectively.

Recently,the two-parameter extension of equation (3)was introduced in [33],

which is called the (α,γ) weighted Wigner–Yanase–Dyson((α,γ) WWYD) skew information in [34].

Motivated by the equation (5) above and equation (3) in[19],we introduced the (α,β,γ) weighted Wigner–Yanase–Dyson ((α,β,γ) WWYD) skew information [35],

and its non-Hermitian extension,the (α,β,γ) modified weighted Wigner–Yanase–Dyson((α,β,γ)MWWYD)skew information [35],

Here equation (7) reduces to equation (7) in [35] and equation (6) reduces to equation (5) when β=1-α,respectively.

As the broadest form of measurement [36,37],quantum channels play a crucial role in quantum theory.Chenetal[38] investigated the summation form of the uncertainty relations based on WY skew information for observables.Fuetal[39] explored the uncertainty relations for quantum channels in terms of WY skew information.Recently,the summation form of the uncertainty relations associated with skew information for arbitrary finite quantum observables and quantum channels has also been derived [40–45].

The paper is structured as follows.In section 2,by using operator norm inequalities,new uncertainty relations of observables are given in terms of the (α,β,γ)WWYD skew information.We present two distinct types of uncertainty relations for quantum channels with respect to the (α,β,γ)MWWYD skew information and establish an optimal lower bound in section 3.The tighter uncertainty relations of unitary channels are presented in section 4.We conclude with a summary in section 5.

2.Sum uncertainty relations for arbitrary finite N observables

For arbitrary finiteNobservablesA1,A2,…,AN,Xuetal[34]provided the following sum uncertainty relations,

where α,β ≥0,α+β ≤1,0 ≤γ ≤1,N＞2 for the inequality (8) andN≥2 for the inequalities (9–10).For convenience we denote byLb1,Lb2andLb3the right-hand sides of (8),(9) and (10),respectively.

The following relations are given in the appendix B in[45],

Theorem 1.ForarbitraryfiniteNobservablesA1,A2,…,AN(N≥2),wehavethefollowingsumuncertaintyrelationvia(α,β,γ)WWYDskewinformation,

α,β≥0,α+β≤1,0 ≤γ≤ 1,theparametersL,MintheexpressionsofLb1,Lb2andLb3satisfyM≥L＞ 0,L≥M＞ 0andL＞M＞ 0,respectively.

For convenience we denoteLb=max {Lb1,Lb2,Lb3}the right-hand side of (14).In [45] Lietalproved that (11),(12)and (13) are strictly tighter than those of norm inequalities related to (8),(9) and (10)for appropriateMandL.If we takeM=L,Lb1andLb2reduces to the cases ofLb3.For fxiedN(≥2),largerMand smallerLgive rise to larger right-hand sides of the inequalities (11) and (13).Conversely,smallerMand largerLresult in larger right-hand side of the inequality (12).

Corollary 1.ForfiniteNobservablesA1,A2,…,AN(N≥ 2),thesumuncertaintyrelationswithrespecttoWWYDskew informationaregivenby

and

where0 ≤α≤ 1,theparametersL,Min(18),(19)and(20)satisfyM≥L＞ 0,L≥M＞ 0andL＞M＞ 0,respectively.

Denoterhs18,rhs19 andrhs20 the right-hand sides of the inequalities (18),(19) and (20),respectively.Corollary 1 implies that≥ max {rhs1 9,rhs20,rhs21}.In particular,(33),(34)and(35)in[45]are just the special cases of the inequalities(18),(19)and(20)forα=respectively.Next we prove that our new lower boundis tighter than the existing ones by a detailed example.We consider the WWYD skew information as a special case,and takeM=2,L=1 forLb1,andM=1,L=2 forLb2andLb3.

3.Sum uncertainty relations for finite quantum channels

In this section,we give two different types of uncertainty relations associated with arbitrary finite number of quantum channels based on(α,β,γ)MWWYD skew information.We derive an optimal lower bound and show that our bounds are tighter than the existing ones by a detailed example.

Let Φ be a quantum channel with Kraus representation,In [34] the authors have presented the uncertainty quantification with respect to a channel Φ via (α,β,γ) MWWYD skew information,

where α,β ≥0,α+β ≤1,0 ≤γ ≤1.For arbitraryNquantum channels,Φ1,…,ΦNwith Kraus representationst=1,2,…,N(N＞2 for the inequality (22) andN≥2 for the inequalities (23–24)).Xuetal.[34] gave the following sum uncertainty quantifications associated with the channels,

where α,β ≥0,α+β ≤1,0 ≤γ ≤1,Snis then-element permutation group and πt,πs∈Snare arbitraryn-element permutations.We denote byLB1,LB2andLB3the right-hand sides of (22),(23) and (24),respectively.

α,β≥0,α+β≤1,0 ≤γ≤ 1,πt,πs∈Snarearbitraryn-elementpermutations,theparametersL,MinandsatisfyM≥L＞ 0,L≥M＞ 0andL＞M＞ 0,respectively.

Proof.According to the inequalities(11–13)and equation(7),we have

forL＞M＞ 0.Summing over the indexi,which implies theorem 2. □

Corollary 2.LetΦ1,…,ΦNbeNquantumchannelswithKrausrepresentationst=1,2,…,N(N≥2),wehavethesumuncertaintyrelationswithrespecttoWWYDskewinformation,

where0 ≤α≤ 1,theparametersL,Min(29),(30)and(31)satisfyM≥L＞ 0,L≥M＞ 0andL＞M＞ 0,respectively.

The uncertainty quantification of quantum channel Φ based on (α,β,γ) MWWYD skew information can also be written as [35],

α,β≥0,α+β≤1,0 ≤γ≤ 1,πt,πs∈Snarearbitraryn-elementpermutations,andtheparametersL,MinLB1,satisfyM≥L＞ 0,L≥M＞ 0andL＞M＞0,respectively.

According to appendix C in [45],it is not hard to prove that our lower bound max {LB1,LB2,LB3}is tighter than the lower boundLB=max {LB1,LB2,LB3}given in [35].

Motivated by the results given in appendix D of[45],we have an optimal lower bound,

To illustrate our results,we consider the MWWYD skew information as a special case,and takeM=2,L=1 forLB1,andM=1,L=2 forLB2 andLB3.

4.Sum uncertainty relations for finite unitary channels

In [35] we introduced the (α,β,γ) MWWYD skew information of a unitary operatorU,

Theorem 4.LetU1,…,UNbeNunitaryoperators,wehave

operators generated by Pauli matrices,

5.Conclusions

We have presented tighter uncertainty relations via (α,β,γ)WWYD skew information for multiple observables,quantum channels and unitary channels.By explicit examples,we have shown that our uncertainty inequalities are tighter than the existing results given in[34,35].Our results are also valid for the WY,WWYD and(α,γ)WWYD skew information as the special cases.Uncertainty relations give rise to fundamental limitations on quantum physical quantities,and our results may shed new light on the understanding of uncertainty relations and their applications in quantum information processing such as communication security.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant Nos.12161056,12075159,12171044);Jiangxi Provincial Natural Science Foundation(Grant No.20232ACB211003);and the Academician Innovation Platform of Hainan Province.

Conflict of interest

The authors declare that they have no conflict of interest.