# A note on Bianchi-Don\`a's proof to the variance formula of von Neumann entropy

## TL;DR

This paper clarifies that Bianchi and Dona's recent proof of the von Neumann entropy variance formula is fundamentally similar to the original proof by Alicki and Fannes, despite different initial approaches.

## Contribution

It demonstrates that the new proof by Bianchi and Dona aligns with the original proof's core calculations, providing insight into the proof's structure.

## Key findings

- The proof's core calculations are essentially the same as in the original work.
- Different starting points lead to similar results in the proof.
- Clarifies the relationship between recent and original proofs of the variance formula.

## Abstract

Bianchi and Don\`{a} [1] have recently reported a proof to the variance formula of von Neumann entropy, which was conjectured in [2] and firstly proved in [3]. The purpose of this short note is to show that, despite having a different starting point, the subsequent calculations (omitted in [1]) leading to the result are essentially the same as in [3].