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Notes on super-operator norms induced by Schatten norms

By John Watrous

Abstract

Let \Phi be a super-operator, i.e., a linear mapping of the form \Phi : L(F) ! L(G) for finite dimensionalHilbert spaces F and G. This paper considers basic properties of the super-operator norms defined by k\Phi kq!p = supfk\Phi (X)kp=kXkq: X 6 = 0g, induced by Schatten norms for 1 ^ p; q ^ 1. Thesesuper-operator norms arise in various contexts in the study of quantum information. In this paper it is proved that if \Phi is completely positive, the value of the supremum in the definitionof k\Phi kq!p is achieved by a positive semidefinite operator X, answering a question recently posed byKing and Ruskai [9]. However, for any choice of p 2 [1; 1], there exists a super-operator \Phi that is thedifference of two completely positive, trace-preserving super-operators such that all Hermitian X fail toachieve the supremum in the definition of k\Phi k1!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 \Phi, the norm k\Phi kq!pis stable under tensoring \Phi with the identity super-operator, meaning that k\Phi kq!p = k\Phi \Omega Ikq!p. For 1 ^ p! 2, the norm k\Phi k1!p may fail to be stable with respect to tensoring \Phi with the identity super-operator as just described, but k\Phi \Omega Ik1!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 establishedthis fact for the case p = 1

Year: 2004
OAI identifier: oai:CiteSeerX.psu:10.1.1.135.4050
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