Fixed space of positive trace-preserving super-operators

Ansis Rosmanis

We examine the fixed space of positive trace-preserving super-operators. We describe a specific structure that this space must have and what the projection onto it must look like. We show how these results, in turn, lead to an alternative proof of the complete characterization of the fixed space of completely positive trace-preserving super-operators.

Shuhui Gao, Yuan Li, Nana Wei

G. Dattoli, A. Torre

John Watrous

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 $p=1$.

Michael R. Geller

Abstract Models of nonlinear quantum computation based on deterministic positive trace‐preserving (PTP) channels and evolution equations are investigated. The models are defined in any finite Hilbert space, but the main results are for dimension . For every normalizable linear or nonlinear positive map ϕ on bounded linear operators X , there is an associated normalized PTP channel . Normalized PTP channels include unitary mean field theories, such as the Gross–Pitaevskii equation for interacting bosons, as well as models of linear and nonlinear dissipation. They classify into four types, yielding three distinct forms of nonlinearity whose computational power are explored. In the qubit case, these channels support Bloch ball torsion and other distortions studied previously, where it has been shown that such nonlinearity can be used to increase the separation between a pair of close qubit states, suggesting an exponential speedup for state discrimination. Building on this idea, the authors argue that this operation can be made robust to noise by using dissipation to induce a bifurcation to a novel phase where a pair of attracting fixed points create an intrinsically fault‐tolerant nonlinear state discriminator.

Anatolij Dvurečenskij, PEKKA J. LAHTI, Kari Ylinen