The Entropy Power Inequality for Quantum Systems
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Bibliographic record
Abstract
When two independent analog signals, X and Y are added together giving Z=X+Y, the entropy of Z, H(Z), is not a simple function of the entropies H(X) and H(Y), but rather depends on the details of X and Y's distributions. Nevertheless, the entropy power inequality (EPI), which states that e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2H(Z)</sup> ≥ e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2H(X)</sup> +e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2H(Y)</sup> , gives a very tight restriction on the entropy of Z. This inequality has found many applications in information theory and statistics. The quantum analogue of adding two random variables is the combination of two independent bosonic modes at a beam splitter. The purpose of this paper is to give a detailed outline of the proof of two separate generalizations of the EPI to the quantum regime. Our proofs are similar in spirit to the standard classical proofs of the EPI, but some new quantities and ideas are needed in the quantum setting. In particular, we find a new quantum de Bruijin identity relating entropy production under diffusion to a divergence-based quantum Fisher information. Furthermore, this Fisher information exhibits certain convexity properties in the context of beam splitters.
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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 | 0.001 |
| 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