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Notes on super-operator norms induced by Schatten norms
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This paper considers basic properties of super-operator norms induced by Schatten p-norms. Such super-operator norms arise in various contexts in the study of quantum information. It is proved that for completely positive super-operators, the value of any such norm is achieved by a positive semidefinite input, answering a question recently posed by King and Ruskai. However, for any choice of p, there exists a super-operator that is the difference of two completely positive, trace-preserving super-operators such that the value of the super-operator norm is not even achieved on a Hermitian input operator. Also considered are the properties of the above norms for super-operators tensored with the identity super-operator.
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