Equivalence classes and canonical forms for two-qutrit entangled states of rank four having positive partial transpose
Bibliographic record
Abstract
Let \documentclass[12pt]{minimal}\begin{document}${\cal E}^{\prime }$\end{document}E′ denote the set of non-normalized two-qutrit entangled states of rank four having positive partial transpose (PPT). We show that the set of stochastic local operations and classical communications (SLOCC) equivalence classes of states in \documentclass[12pt]{minimal}\begin{document}${\cal E}^{\prime }$\end{document}E′, equipped with the quotient topology, is homeomorphic to the quotient R/A5 of the open rectangular box R⊂ R4 by an action of the alternating group A5. We construct an explicit map \documentclass[12pt]{minimal}\begin{document}$\omega :\Omega \rightarrow {\cal E}^{\prime }$\end{document}ω:Ω→E′, where Ω is the open positive orthant in R4, whose image ω(Ω) meets every SLOCC equivalence class \documentclass[12pt]{minimal}\begin{document}$E\subseteq {\cal E}^{\prime }$\end{document}E⊆E′. Although the intersection ω(Ω) ∩ E is not necessarily a singleton set, it is always a finite set of cardinality at most 60. By abuse of language, we say that any state in ω(Ω) ∩ E is a canonical form of any ρ ∈ E. In particular, we show that all checkerboard PPT entangled states can be parametrized up to SLOCC equivalence by only two real parameters. We also summarize the known results on two-qutrit extreme PPT states and edge states, and examine which other interesting properties they may have. Thus we find the first examples of extreme PPT states whose rank is different from the rank of its partial transpose.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".