Notes on super-operator norms induced by Schatten norms

Let be a super-operator, i. e. , a linear mapping of the form: L (F) (G) for finite dimensional Hilbert spaces F and G. This paper considers basic properties of the super-operator norms defined by \|\|ₐ ₏= \\| (X) \|ₚ/\|X\|q\,: \, X=0\, induced by Schatten norms for 1 p, q. These super-operator norms arise in various contexts in the study of quantum information. In this paper it is proved that if is completely positive, the value of the supremum in the definition of \|\|ₐ ₏ is achieved by a positive semidefinite operator X, answering a question recently posed by King and Ruskai~KingR04. However, for any choice of p 1, , there exists a super-operator that is the difference of two completely positive, trace-preserving super-operators such that all Hermitian X fail to achieve the supremum in the definition of \|\|₁ ₏. Also considered are the properties of the above norms for super-operators tensored with the identity super-operator. In particular, it is proved that for all p 2, q 2, and arbitrary, the norm \| \|ₐ ₏ is stable under tensoring with the identity super-operator, meaning that \| \|ₐ ₏ = \| I\|ₐ ₏. For 1 p < 2, the norm \|\|₁ ₏ may fail to be stable with respect to tensoring with the identity super-operator as just described, but \| I\|₁ ₏ is stable in this sense for I the identity super-operator on L (H) for dim (H) = dim (F). This generalizes and simplifies a proof due to Kitaev Kitaev97 that established this fact for the case p=1.