Quantum information inequalities via tracial positive linear maps

Abstract. We present some generalizations of quantum information inequalities involving tracial positive linear maps between C∗-algebras. Among several results, we establish a noncommutative Heisenberg uncertainty relation. More precisely, we show that if Φ : A → B is a tracial positive linear map between C∗-algebras , ρ ∈ A is a Φ-density element and A,B are self-adjoint operators of A such that sp(-iρ 1 2 [A,B]ρ 1 2 ) ⊆ [m,M ] for some scalers 0 < m < M , then under some conditions