Tight entropy bound based on p-quasinorms

TL;DR

The paper addresses bounding the Shannon entropy $S(\mathbf{p})$ and the von Neumann entropy $H(\rho)$ via $p$-quasinorms in $\ell^p$ spaces, enabling analysis in both finite and infinite dimensions. It develops a logarithm inequality with tunable parameter $\sigma\in(0,1)$ and proves optimal constants $C_1=C_2=\frac{1}{\ln(2)(1-\sigma)}$, yielding explicit bounds $\frac{1}{\ln(2)(1-\sigma)}(1-||\mathbf{p}||_{2-\sigma}) \le S(\mathbf{p}) \le \frac{1}{\ln(2)(1-\sigma)} (||\mathbf{p}||_\sigma -1)$, extended to infinite distributions with $\mathbf{p}\in \ell^\sigma$ and to $H(\rho)$, and used to bound entropy differences. The bounds are tight as $\sigma\to1^-$ and provide a finite-entropy criterion aligned with Bacetti; numerical tests, e.g., with $\sigma=0.9$, validate practical accuracy for finite distributions. Overall, the work offers a simple, analyzable entropy estimator applicable to classical and quantum systems, with potential for uniform continuity bounds on entropy differences.

Abstract

In the present paper we prove a family of tight upper and lower bounds for the Shannon entropy and von Neumann entropy based on the p-norms. This allows us to have an entropy estimate, a criterion for the finiteness of it and a bound on the difference of entropy, additionally, we did some numerical tests that show the efficiency of our approximations.

Figures (4)

• Figure 1: Upper and Lower entropy bounds.
• Figure 2: Entropy bounds absolute and relative errors
• Figure 3: Entropy difference bound .
• Figure 4: Entropy difference bound absolute and relative errors.

Theorems & Definitions (8)

• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 3
• proof
• Corollary 4
• Corollary 5