Schur–Horn Shannon Entropy Monotonicity

Abstract

A unitary spectral readout cannot decrease finite Shannon entropy.

Theorem 1.1 (Unitary spectral readout cannot decrease Shannon entropy).

$$\mathrm{shannonEntropy}(x)\le \mathrm{shannonEntropy}\left(\mathrm{diagonalPart}\left(U\mathrm{diagonal}(x)U^{\ast }\right)\right)$$

Formalization. D5/S3/Quantum/Divergence/SchurHornEntropyMonotonicity.spectral_readout_entropy_le (validation pending).

Source. Repository-derived.

Commentary.

For a finite index type n, a unitary matrix U, and a nonnegative real spectrum x, form U diagonal(x) U* and read the real parts of its diagonal entries. The readout vector is the squared-entry-norm matrix of U applied to x.

The squared-entry-norm matrix is doubly stochastic. Row-wise Jensen for the convex function t log t, followed by the column-sum identity, shows that the readout’s x log x sum is no larger than the spectral sum. Negating gives the Shannon entropy inequality.

This is the finite Schur–Horn bridge used by the later pinching and Klein targets. No normalization, strict positivity, or distinct eigenvalue assumption is required.

References

• Truth anchor: D5/S3/Quantum/Divergence/SchurHornEntropyMonotonicity.spectral_readout_entropy_le
• Dependency: D5/S3/Entropy/MaxEntropy
• Dependency: D5/S3/Weil/ZetaLinear/VonNeumann