Notes on
super-operator norms induced by Schatten norms
(pp058-068)
John Watrous
doi:
https://doi.org/10.26421/QIC5.1-6
Abstracts:
Let $\Phi$ be a super-operator, i.e., a linear mapping of
the form $\Phi:\mathrm{L}(\mathcal{F})\rightarrow\mathrm{L}(\mathcal{G})$
for finite dimensional Hilbert spaces $\mathcal{F}$ and $\mathcal{G}$.
This paper considers basic properties of the super-operator norms
defined by $\|\Phi\|_{q\rightarrow p}= \sup\{\|\Phi(X)\|_p/\|X\|_q\,:\,X\not=0\}$,
induced by Schatten norms for $1\leq p,q\leq\infty$. 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 $\|\Phi\|_{q\rightarrow
p}$ is achieved by a positive semidefinite operator $X$, answering a
question recently posed by King and Ruskai~\cite{KingR04}. However, for
any choice of $p\in [1,\infty]$, 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 $\|\Phi\|_{1\rightarrow p}$.
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 $p\geq 2$, $q\leq 2$, and
arbitrary $\Phi$, the norm $\|\Phi \|_{q\rightarrow p}$ is stable under
tensoring $\Phi$ with the identity super-operator, meaning that $\|\Phi
\|_{q\rightarrow p} = \|\Phi \otimes I\|_{q\rightarrow p}$. For $1\leq p
< 2$, the norm $\|\Phi\|_{1\rightarrow p}$ may fail to be stable with
respect to tensoring $\Phi$ with the identity super-operator as just
described, but $\|\Phi\otimes I\|_{1\rightarrow p}$ 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 $p=1$.
Key words:
Schatten norms, super-operator norms,
quantum information |