Bibliographic record
Abstract
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 $p=1$.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".