This is related to a broader question about quantum information geometry - this question is more specific.
A fundamental concept in Amari's treatment of information geometry is that of a Bregman divergence. Given a convex function $\psi(\mathbf{x})$, a Bregman divergence is defined as $$ D_\psi(\mathbf{x}\|\mathbf{x}_0) = \psi(\mathbf{x}) - \psi(\mathbf{x}_0) - \nabla \psi(\mathbf{x}_0)(\mathbf{x}-\mathbf{x}_0). $$ Bregman divergences are non-negative and are zero when $\mathbf{x}=\mathbf{x}_0$, and they behave in some ways like the square of a distance between $\mathbf{x}_0$ and $\mathbf{x}$. (Though taking the square root does not form a metric.)
An example is the Kullback-Leibler divergence between two probability distributions, $D_{KL}(P\|P^0)=\sum_{i=1}^N p_i\log\frac{p_i}{p^0_i}$. In the case of finite support can be seen (for example) by parameterising each distribution by its first $N-1$ probabilities (so $p_N=1-\sum_{i=1}^{N-1}p_i$) and then taking $\psi$ to be the negative entropy, $\psi=\sum_{i=1}^{N}p_i\log p_i$.
In quantum theory, the natural analog of the Kullback-Leibler divergence is the quantum relative entropy, $$ D_Q(\rho\|\rho_0) = \mathrm{Tr}\,\rho(\log\rho - \log\rho_0), $$ where $\rho$ and $\rho_0$ are density matrices and $\log$ is the matrix logarithm.
My question is, in the case of a finite Hilbert space (or indeed in general), can the quantum relative entropy be expressed as a Bregman divergence? If so, what is the right way to parameterise $\rho$ so that this becomes clear?
