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Journal of Mathematical & Computer Applications

Non-Hermitian Extension of Q Uncertainty Relation

Kenjiro Yanagi

Published May 14, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.47363/jmca/2024(3)150

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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.
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 ).
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.
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 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-Stϕ\phiϕrmer. Finally we state q uncertainty relations for generalized metric adjusted q skew information or generalized metric adjusted q correlation measure.