Quantum information inequalities via tracial positive linear maps

We present some generalizations of quantum information inequalities involving tracial positive linear maps between C⁎C⁎-algebras. Among several results, we establish a noncommutative Heisenberg uncertainty relation. More precisely, we show that if Φ:A→BΦ:A→B is a tracial positive linear map between C⁎C⁎-algebras, ρ∈Aρ∈A is a Φ-density element and A,BA,B are self-adjoint operators of AA such that sp(-iρ12[A,B]ρ12)⊆[m,M] for some scalers 0