I'm looking for an upper bound for the Lipschitz constant of entanglement entropy between two subsystems with respet to the standard distance measure of pure states in the Hilbert space of the full system.
The whole system is in a pure state $\psi \in \mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B$. The entanglement entropy of a state $\psi$ is given by $S_A(\psi) = -\text{Tr}\rho_A log \rho_A$ where $\rho_A$ is the reduced density matrix given by $\rho_A = \text{Tr}_B |\psi\rangle\langle\psi|$.
What I'm looking for is some $\eta > 0$ for which $S_A(\psi′)−S_A(\phi)| ≤ η || \psi - \phi||$ where $\psi$ and $\phi$ are two normalised vectors in $\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B$ and $||\psi|| = \sqrt{\langle \psi, \psi \rangle}$
I tought I could do this by using perturbation theory for the eigenvalues of the reduced density matrix. I take a normalised pure state $\psi$ and perturb it to $\psi' = \psi + \epsilon \phi$ with $\langle \psi, \phi \rangle = 0$, the reduced density matrix changes to $\rho_A'= \text{Tr}_B |\psi\rangle\langle\psi| + \epsilon \text{Tr}_B |\phi\rangle\langle\phi|$. The eigenvalues of the reduced density matrix change from $r_i$ with corresponding eigenvectors $e_i$ to $r_i' = r_i + \epsilon \langle e_i,\text{Tr}_B |\phi\rangle\langle\phi| \;e_i\rangle + \mathcal{O}(\epsilon)$.
Then I should be able to find a upper bound of the form $|S_A(\psi') - S_A(\psi)| \leq \eta\epsilon$ by using $S_A(\psi) = \sum_i r_i \log r_i$ and $S_A(\psi') = \sum_i r_i' \log r_i'$.
For some reason I'm not getting there. How should I proceed?
I found something in this article that uses the gradient directly, but I don't really see how it is done. They find upper bound $\eta < \sqrt{8} \log N_A$ for $N_a \geq 3$