Notes on super-operator norms induced by schatten norms

• Authors: John Watrous
• Venue: Quantum Information & Computation
• Year: 2005

Let Φ be a super-operator, i.e., a linear mapping of the form Φ : L(F,) → L(G) forfinite dimensionM Hilbert spaces F and G. This paper considers basic properties of thesuper-operator norms defined by ||Φ||q→p = sup{||Φ(X)||p/||X||q: X ≠0}, induced bySchatten norms for 1 ≤ p, q ≤ ∞. These super-operator norms arise in various contextsin the study of quantum information. In this paper it is proved that if Φ is completely positive, the value of the supremumin the definition of ||Φ||q→p is achieved by a positive semidefinite operator X, answeringa 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 inthe definition of ||Φ||1→p. Also considered are the properties of the above norms for super-operators tensoredwith 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 identiV super-operator, meaning that ||Φ||q→p = ||Φ⊗I||q→p. For 1≤p 1→p mayfail to be stable with respect to tensoring Φ with the identity super-operator as justdescribed, but ||Φ⊗I||1→p is stable in this sense for I the identiV super-operator onL(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.