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A Sharp Fannes-type Inequality for the von Neumann Entropy

Koenraad M. R. Audenaert

TL;DR

The paper addresses the problem of a sharp continuity bound for the von Neumann entropy in terms of the trace-norm distance between two $d$-dimensional quantum states. It reduces the quantum problem to a classical one via unitary invariance, then executes a constructive extremal optimization to show the entropy difference is bounded by $|S( ho)-S(\sigma)| \le T \log_2(d-1) + H((T,1-T))$, which is tight for all $T\in[0,1]$ and $d$. Equality is achieved by the commuting pair $\rho=\mathrm{Diag}(1-T, T/(d-1),\ldots,T/(d-1))$ and $\sigma=\mathrm{Diag}(1,0,\ldots,0)$, demonstrating sharpness. This bound improves on Fannes' original inequality by covering the full range of $T$ and providing an explicit equality case, with practical relevance in areas like entanglement theory where precise entropy continuity bounds are valuable.

Abstract

We derive an inequality relating the entropy difference between two quantum states to their trace norm distance, sharpening a well-known inequality due to M. Fannes. In our inequality, equality can be attained for every prescribed value of the trace norm distance.

A Sharp Fannes-type Inequality for the von Neumann Entropy

TL;DR

The paper addresses the problem of a sharp continuity bound for the von Neumann entropy in terms of the trace-norm distance between two -dimensional quantum states. It reduces the quantum problem to a classical one via unitary invariance, then executes a constructive extremal optimization to show the entropy difference is bounded by , which is tight for all and . Equality is achieved by the commuting pair and , demonstrating sharpness. This bound improves on Fannes' original inequality by covering the full range of and providing an explicit equality case, with practical relevance in areas like entanglement theory where precise entropy continuity bounds are valuable.

Abstract

We derive an inequality relating the entropy difference between two quantum states to their trace norm distance, sharpening a well-known inequality due to M. Fannes. In our inequality, equality can be attained for every prescribed value of the trace norm distance.

Paper Structure

This paper contains 9 sections, 1 theorem, 33 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof
  3. Reduction to classical case
  4. Proof Strategy
  5. The case $d=2$
  6. Optimal $\delta^+$
  7. Optimal $(1-p_1)r-Ts$
  8. Optimal $s_1$
  9. Optimal $p_1$

Key Result

Theorem 1

For all $d$-dimensional states $\rho$, $\sigma$ such that their trace norm distance is given by $T$,

Figures (3)

  • Figure 1: Scatter plot of 20000 randomly generated pairs $(\rho,\sigma)$ of qubit states ($d=2$); shown is the trace norm distance $T=||\rho-\sigma||_1/2$ versus the difference $\Delta=|S(\rho)-S(\sigma)|$ of the vN entropies. The upper curve in the interval $0\le T\le 1/(2e)$ represents the Fannes bound (\ref{['eq:fannes1']}). The lower curve represents our sharp bound (\ref{['eq:sharp']}) and is seen to follow the boundary of the set of scatter points tightly.
  • Figure 2: Same as Fig. 1, but for qutrits ($d=3$).
  • Figure 3: Same as Fig. 1, but for 4-dimensional quantum systems ($d=4$).

Theorems & Definitions (1)

  • Theorem 1