A simple analysis of the mixed-state information metric in AdS$_3$/CFT$_2$

We compute the quantum information metrics of a thermal CFT on $\mathbb R^{1,1}$ perturbed by the scalar primary operators of conformal dimension $\Delta=3,4,5,6$. In particular, we assume that the Hamiltonian of the mixed state commutes with each other and the temperature is fixed. Under these conditions, the evaluation is analogous to the pure state case. We also apply the method of [arXiv:1607.06519] to calculate the mixed state information metric for the scalar primary operator with conformal dimension $\Delta=4$ holographically. We find an exact agreement between the two results in our approach.