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(-iρ12[A,B]ρ12)⊆[m,M] for some scalers 0<m<M, then under some conditions(0.1)Vρ,Φ(A)♯Vρ,Φ(B)≥12Km,M(ρ[A,B])|Φ(ρ[A,B])|, where Km,M(ρ[A,B]) is the Kantorovich constant of the operator -iρ12[A,B]ρ12 and Vρ,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.