Uncertainty relation associated with a monotone pair skew information

Abstract

We derive a trace inequality leading to an uncertainty relation based on the monotone pair skew information introduced by Furuichi. As the monotone pair skew information generalizes the Wigner-Yanase-Dyson skew information as well as some other skew information, our result also extends a few known results on the uncertainty relations. Particularly it reduces to that of Luo, Yanagi, and Furuichi et al. in the special cases.

Schrödinger Uncertainty Relation for the Monotone Triple Skew Information

“…The variance V ρ (A) for ρ and A is defined by V ρ (A) = Tr(ρA 2 ) − (Tr(ρA)) 2 = Tr(ρA 2 0 ). The following was introduced in [4]. Definition 2.1.…”

“…In order to get Heisenberg-type uncertainty relation associated to the monotone triple skew information, which improves the result of [4], Yanagi and Kajihara considered the following assumption [10]: Assumption 2.4. (f, g) and (f, h) are CLI monotone pairs satisfying…”

“…In [4], we have generalized the uncertainty relation on Wigner-Yanase-Dyson skew information as follows:…”

“…For the details we refer to [5]. For the monotone pair (f, g)-skew information I ρ,(f,g) (A), the present authors obtained the Heisenberg-type uncertainty relation in [4], which generalized the uncertainty relation associated with Wigner-Yanase-Dyson skew information shown by Yanagi in [9]. Recently this uncertainty relation was generalized by Yanagi and Kajihara via the monotone triple skew information [10] (see (2.12) and (2.13)).…”

Noncommutative versions of inequalities in quantum information theory

“…In quantum information theory, the classical expectation value of an observable (self-adjoint element) A in a quantum state (density element) ρ is defined by Tr(ρA), and the classical variance is expressed by Var ρ (A) := Tr(ρA 2 )−(Tr(ρA)) 2 . The Heisenberg uncertainty relation [5,9] where ρ is a quantum state and A and B are two observables. The Heisenberg uncertainty relation gives a fundamental limit for the measurements of incompatible observables.…”