Vol.5 , No. 4, Publication Date: Dec. 24, 2018, Page: 85-94

[1] | Kenjiro Yanagi, Department of Mathematics, Josai University, Sakado, Japan. |

It is well known that almost all uncertainty relations including Heisenberg uncertainty relation and Schr¨odinger 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.

Keywords

Trace Inequality, Metric Adjusted Skew Information, Generalized Quasi-metric Adjusted Skew Information

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