Golden-Thompson via pinching inequality

For two hermitian matrices $A$ and $B$, Golden-Thompson inequality \cite{golden1965lower, thompson1965inequality} states that $$\mathrm{t}\mathrm{r}[\exp (A+B)]\le \mathrm{t}\mathrm{r}[\exp (A)\exp (B)]\mathrm{.}$$ We elaborate here the proof from Sutter et al. (2017).