The inequalities of quantum information theory
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Bibliographic record
Abstract
Let /spl rho/ denote the density matrix of a quantum state having n parts 1, ..., n. For I/spl sube/N={1, ..., n}, let /spl rho//sub I/=Tr/sub N/spl bsol/I/(/spl rho/) denote the density matrix of the state comprising those parts i such that i/spl isin/I, and let S(/spl rho//sub I/) denote the von Neumann (1927) entropy of the state /spl rho//sub I/. The collection of /spl nu/=2/sup n/ numbers {S(/spl rho//sub I/)}/sub I/spl sube/N/ may be regarded as a point, called the allocation of entropy for /spl rho/, in the vector space R/sup /spl nu//. Let A/sub n/ denote the set of points in R/sup /spl nu// that are allocations of entropy for n-part quantum states. We show that A~/sub n/~ (the topological closure of A/sub n/) is a closed convex cone in R/sup /spl nu//. This implies that the approximate achievability of a point as an allocation of entropy is determined by the linear inequalities that it satisfies. Lieb and Ruskai (1973) have established a number of inequalities for multipartite quantum states (strong subadditivity and weak monotonicity). We give a finite set of instances of these inequalities that is complete (in the sense that any valid linear inequality for allocations of entropy can be deduced from them by taking positive linear combinations) and independent (in the sense that none of them can be deduced from the others by taking positive linear combinations). Let B/sub n/ denote the polyhedral cone in R/sup /spl nu// determined by these inequalities. We show that A~/sub n/~=B/sub n/ for n/spl les/3. The status of this equality is open for n/spl ges/4. We also consider a symmetric version of this situation, in which S(/spl rho//sub I/) depends on I only through the number i=/spl ne/I of indexes in I and can thus be denoted S(/spl rho//sub i/). In this case, we give for each n a finite complete and independent set of inequalities governing the symmetric allocations of entropy {S(/spl rho//sub i/)}/sub 0/spl les/i/spl les/n/ in R/sup n+1/.
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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 | 0.002 | 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 | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.002 |
| Open science | 0.001 | 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 it