Derivative of trace distance

Question

If we have two time-dependent density matrices $\rho(t)$ and $\sigma(t)$. The trace distance is $$ D(\rho(t),\sigma(t))=\frac{1}{2} \operatorname{Tr} \vert \rho(t)-\sigma(t) \vert. $$ Does an explicit formula for $$\frac{d D(\rho(t),\sigma(t))}{dt}$$ exist? Only the domain where $\rho \neq \sigma$ is asked for.

Answer 1

Define the new matrix variables $$\eqalign{ A &= \rho(t)-\sigma(t) \cr B &= A^HA \cr }$$

Then the function, differential, and gradient can be written as $$\eqalign{ D &= \frac{1}{2}{\rm tr}(B^{1/2}) \cr dD &= \frac{1}{4}B^{-1/2}:dB \,\,\,=\,\,\, \frac{1}{4}B^{-1/2}:(A^H\dot{A}+\dot{A}^HA)\,dt \cr \frac{dD}{dt} &= \frac{1}{4}B^{-1/2}:(A^H\dot{A}+\dot{A}^HA) \cr }$$ where the colon is a product notation for the trace, i.e. $\,\,A:B={\rm tr}(A^TB)$