Consider two quantum states $\rho$ and $\sigma$ and the probability distributions induced by measuring both of them in the standard basis. Let’s call the probability distributions $p_{\rho}$ and $p_{\sigma}$. What is the relation between the trace distance of $\rho$ and $\sigma$ and the total variation distance between $p_{\rho}$ and $p_{\sigma}$?

Consider two quantum states $\rho$ and $\sigma$ and the probability distributions induced by measuring both of them in the standard basis. Let’s call the probability distributions $p_{\rho}$ and $p_{\sigma}$ respectively. What is the relation between the trace distance between $\rho$ and $\sigma$ and the total variation distance between $p_{\rho}$ and $p_{\sigma}$?