Fidelity is defined as follows: $$F(\rho, \sigma) = \left( \operatorname{tr} \sqrt{\sqrt{\rho} \sigma \sqrt{\rho}} \right)^2$$
A Bregman divergence $D_B$ is defined as: $$D_B(\rho||\sigma) = f(\rho) - f(\sigma) - \langle \nabla f(\sigma), \rho-\sigma\rangle$$
I am attempting to find an expression that represents fidelity in terms of a Bregman divergence. I have explored various methods to express fidelity using the Bures-Wasserstein distance and Rényi divergence, but so far, I have not achieved my goal. I specifically need this expression for mixed states of a two-level quantum system (qubit), but a general case would also be highly appreciated.
Thank you.
Edit:
I want to find $f$ and $<\cdot,\cdot>$ so: $\left( \operatorname{tr} \sqrt{\sqrt{\rho} \sigma \sqrt{\rho}} \right)^2= f(\rho) - f(\sigma) - \langle \nabla f(\sigma), \rho-\sigma\rangle$