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 ( -i ρ 1 2 [ A , B ] ρ 1 2 ) ⊆ [ m , M ] for some scalers 0 m M , then under some conditions (0.1) V ρ , Φ ( A ) ♯ V ρ , Φ ( B ) ≥ 1 2 K m , M ( ρ [ A , B ] ) | Φ ( ρ [ A , B ] ) | , where K m , M ( ρ [ A , B ] ) is the Kantorovich constant of the operator -i ρ 1 2 [ A , B ] ρ 1 2 and V ρ , Φ ( X ) is the generalized variance of X. In addition, we use some arguments differing from the scalar theory to present some inequalities related to the generalized correlation and the generalized Wigner–Yanase–Dyson skew information.