The Theorem

Two states $\rho$ and $\sigma$ of the same quantum system, if close together, have nearly the same entropy; the sections below define the entropy and the distance, and Fannes’ inequality makes the statement quantitative.

A Quantum System

A quantum system $Q$ is a physical object whose state space $\mathcal{H}$ is a finite dimensional complex Hilbert space.^✦ The dimension $d$ of $Q$ is the complex dimension of $\mathcal{H}$, the number of vectors in any basis of it.^✦ A state $\rho$ of $Q$ is a positive operator of unit trace on $\mathcal{H}$, pure or mixed.^✦

Entropy and Trace Distance

The von Neumann entropy of a state $\rho$ of $Q$ is $$S(\rho) = -\operatorname{tr}(\rho \log \rho),$$(1)^✦ the Shannon entropy of the eigenvalues of $\rho$. The logarithm is the natural one, here and below, so entropies are measured in nats.

The trace distance between two states $\rho$ and $\sigma$ of $Q$ is half the trace norm of their difference, $$D(\rho,\sigma) = \tfrac{1}{2}\operatorname{tr}|\rho - \sigma| = \tfrac{1}{2}\sum_i |\lambda_i|,$$(2)^✦ where the $\lambda_i$ are the eigenvalues of $\rho - \sigma$.

The Claim

What Is Checked

Both claims are stated over the states of a quantum system of dimension $d$, through their von Neumann entropy (1) and their trace distance (2).