Multiplicativity of Completely Bounded p-Norms Implies a New Additivity Result

AbstractWe prove additivity of the minimal conditional entropy associated with a quantum channel Φ, represented by a completely positive (CP), trace-preserving map, when the infimum of S(γ12) − S(γ1) is restricted to states of the form $$(\mathcal{I} \otimes \Phi)\left( | \psi \rangle \langle \psi | \right)$$. We show that this follows from multiplicativity of the completely bounded norm of Φ considered as a map from L1 → Lp for Lp spaces defined by the Schatten p-norm on matrices, and give another proof based on entropy inequalities. Several related multiplicativity results are discussed and proved. In particular, we show that both the usual L1 → Lp norm of a CP map and the corresponding completely bounded norm are achieved for positive semi-definite matrices. Physical interpretations are considered, and a new proof of strong subadditivity is presented.