Notes on super-operator norms induced by schatten norms
Let (D be a super-operator, i.e., a linear mapping of the form Phi : L(Y) -> L(G) for finite dimensional Hilbert spaces F7 and G. This paper considers basic properties of the super-operator norms defined by parallel to Phi parallel to(q -> p) = sup{parallel to Phi(X)parallel to(p)/parallel to X parallel to(q) : X not equal 0}, induced by Schatten norms for 1   p) is achieved by a positive semidefinite operator X, answering a question recently posed by King and Ruskai [9]. However, for any choice of p G [1, 00], there exists a super-operator 4) 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 parallel to Phi parallel to(1 -> p). 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   p) is stable under tensoring Phi with the identity super-operator, meaning that parallel to Phi parallel to(q -> p) = parallel to Phi circle times I parallel to(q -> p). For 1   p) may fail to be stable with respect to tensoring 4 with the identity super-operator as just described, but parallel to Phi circle times I parallel to(1 -> p) 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 [10] that established this fact for the case p = 1.
super-operator norms schatten norms quantum information