The coherent information on the manifold of positive definite density matrices
Bibliographic record
Abstract
This paper will explore the restriction of the coherent information to the positive definite density matrices in the special case where the quantum channels are strictly positive linear maps. The space of positive definite density matrices is equipped with an embedded submanifold structure of the real vector space of Hermitian matrices. These ensure that the n-shot coherent information is differentiable and allows for the computation of its gradient and Hessian. We show that any tensor products of critical points preserve being a critical point of the coherent information. Furthermore, we show that for any positive integer n, the maximally mixed state is always a critical point for the class of mixed unitary quantum channels with orthogonal, unitary Kraus operators. We determine when the maximally mixed state is a local maximum/minimum or saddle point, including its eigenvectors, for the class of Pauli-erasure channels when n is equal to 1. This class includes the dephrasure channel and Pauli channel and refines potential regions where super-additivity is thought to occur. These techniques can be used to study other optimization problems over density matrices and allow the use of manifold optimization algorithms and a better understanding of the quantum capacity problem by utilizing the first and second order geometry.
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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".