Notes on super-operator norms induced by schatten norms

Let Φ be a super-operator, i.e., a linear mapping of the form Φ : L(F) → L(G) for finite dimensional Hilbert spaces F and G. This paper considers basic properties of the super-operator norms defined by ‖Φ‖q→p = sup{‖Φ(X)‖p/‖X‖q : X 6= 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 Φ is completely positive, the value of the supremum in the definition of ‖Φ‖q→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 ∈ [1,∞], there exists a super-operator Φ 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 ‖Φ‖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 ≤ 2, and arbitrary Φ, the norm ‖Φ‖q→p is stable under tensoring Φ with the identity super-operator, meaning that ‖Φ‖q→p = ‖Φ⊗ I‖q→p. For 1 ≤ p < 2, the norm ‖Φ‖1→p may fail to be stable with respect to tensoring Φ with the identity superoperator as just described, but ‖Φ ⊗ I‖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.