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Sum Types of Uncertainty Relations for Generalized Quasi-metric Adjusted Skew Informations

研二郎柳

Published Dec 1, 2018• 0 citations

Abstract

It is well known that almost all uncertainty relations including Heisenberg uncertainty relation and Schrödinger uncertainty relation were given by product types of trace inequalities.This is why these results were proved by Schwarz's inequality.These product types of uncertainty relations were extended to the case of not necessarily hermitian quantum mechanical observables and positive operators representing quantum states.On the other hand sum types of uncertainty relations were given for arbitrary finite N not necessarily hermitian quantum mechanical observables.Some uncertainty relations are presented by generalized quasi-metric adjusted skew informations for two different generalized states.These uncertainty relations are nontrivial as long as the observables are mutually noncommutative.The relations among these new and existing uncertainty inequalities have been investigated.Finally, the reverse inequalities of the sum types of uncertainty relations are obtained.

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It is well known that almost all uncertainty relations including Heisenberg uncertainty relation and Schrödinger uncertainty relation were given by product types of trace inequalities.This is why these results were proved by Schwarz's inequality.These product types of uncertainty relations were extended to the case of not necessarily hermitian quantum mechanical observables and positive operators representing quantum states.On the other hand sum types of uncertainty relations were given for arbitrary finite N not necessarily hermitian quantum mechanical observables.Some uncertainty relations are presented by generalized quasi-metric adjusted skew informations for two different generalized states.These uncertainty relations are nontrivial as long as the observables are mutually noncommutative.The relations among these new and existing uncertainty inequalities have been investigated.Finally, the reverse inequalities of the sum types of uncertainty relations are obtained.

In 2015 we obtained non-hermitian extensions of Heisenberg type and Schrödinger type uncertainty relations for generalized metric adjusted skew information or generalized metric adjusted correlation measure and gave the results of DouDu as corollaries. In this paper we define generalized quasimetric adjusted q skew information for different two generalized states and obtain corresponding uncertainty relation. The result is applied to the inequalities related to fidelity and trace distance for different two generalized states which were given by Audenaert et al; and Powers-St

\phi

rmer. Finally we state q uncertainty relations for generalized metric adjusted q skew information or generalized metric adjusted q correlation measure.

Iin 2015 we obtained non-hermitian extensions of Heisenberg type and Schrödinger type uncertainty relations for generalized metric adjusted skew information or generalized metric adjusted correlation measure and gave the results of Dou-Du in 2013 and 2014 as corollaries. In this paper, we define generalized quasi-metric adjusted Q skew information for different two generalized states and obtain corresponding uncertainty relation. The result is applied to the inequalities related to fidelity and trace distance for different two generalized states which were given by Audenaert, et al. in 2009 and 2008; and Powers-Strmer in 1970. 2010 Mathematics Subject Classification: 15A45, 47A63, 94A17.

In quantum mechanics it is well known that the Heisenberg-Schrödinger uncertainty relations hold for two non-commutative observables and density operator. Recently Dou and Du (J. Math. Phys. 54:103508, 2013 ; Int. J. Theor. Phys. 53:952-958, 2014 ) obtained several uncertainty relations for two non-commutative non-hermitian observables and density operators. In this paper, we show that their results can be given as corollaries of our non-hermitian extensions of Heisenberg type or Schrödinger type uncertainty relations for the generalized metric adjusted skew information or generalized metric adjusted correlation measures which were obtained in Furuichi and Yanagi (J. Math. Anal. Appl. 388:1147-1156, 2012 ).

In quantum mechanics it is well known that observables are represented by hermitian matrices (or operators). Uncertainty relations are represented as some kinds of trace inequalities satisfied by two observables and one density matrix (or operator). By releasing the hermitian restriction on observables, we obtained non-hermite uncertainty relations. In this paper we give several non-hermitian extensions of Heisenberg or Schrödinger type q uncertainty relations for generalized skew information under some conditions.

Abstract In this paper, we first provide three general norm inequalities, which are used to give new uncertainty relations of any finite observables and quantum channels via metric-adjusted skew information. The results are applicable to its special cases as Wigner-Yanase-Dyson skew information. In quantifying the uncertainty of channels, we discuss two types of lower bounds and compare the tightness between them, meanwhile, a tight lower bound is given. The uncertainty relations obtained by us are stronger than the existing ones. To illustrate our results, we give several specific examples.