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- W1491569431 abstract "Let $Phi$ be a super-operator, i.e., a linear mapping of the form $Phi:mathrm{L}(mathcal{F})rightarrowmathrm{L}(mathcal{G})$ for finite dimensional Hilbert spaces $mathcal{F}$ and $mathcal{G}$. This paper considers basic properties of the super-operator norms defined by $|Phi|_{qrightarrow p}= sup{|Phi(X)|_p/|X|_q,:,Xnot=0}$, induced by Schatten norms for $1leq p,qleqinfty$. These super-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 definition of $|Phi|_{qrightarrow p}$ is achieved by a positive semidefinite operator $X$, answering a question recently posed by King and Ruskai~cite{KingR04}. However, for any choice of $pin [1,infty]$, there exists a super-operator $Phi$ that is the {em difference} of two completely positive, trace-preserving super-operators such that all Hermitian $X$ fail to achieve the supremum in the definition of $|Phi|_{1rightarrow p}$. 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 $pgeq 2$, $qleq 2$, and arbitrary $Phi$, the norm $|Phi |_{qrightarrow p}$ is stable under tensoring $Phi$ with the identity super-operator, meaning that $|Phi |_{qrightarrow p} = |Phi otimes I|_{qrightarrow p}$. For $1leq p < 2$, the norm $|Phi|_{1rightarrow p}$ may fail to be stable with respect to tensoring $Phi$ with the identity super-operator as just described, but $|Phiotimes I|_{1rightarrow p}$ is stable in this sense for $I$ the identity super-operator on $mathrm{L}(mathcal{H})$ for $op{dim}(mathcal{H}) = op{dim}(mathcal{F})$. This generalizes and simplifies a proof due to Kitaev cite{Kitaev97} that established this fact for the case $p=1$." @default.
- W1491569431 created "2016-06-24" @default.
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- W1491569431 date "2005-01-01" @default.
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- W1491569431 title "Notes on super-operator norms induced by Schatten norms" @default.
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- W1491569431 doi "https://doi.org/10.26421/qic5.1-6" @default.
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