While reading some online lecture notes (Link to pdf) about state discrimination, I stumbled upon the following bound for the success probability of discriminating between a set of states $\rho_i$, with prior probabilities $p_i$, using the corresponding pretty good measurement (Eq. (8), page 7 in the above pdf): $$P_e^{\rm PGM} \le \sum_{i\neq j}\sqrt{p_i p_j} F(\rho_i,\rho_j),$$ where $F(\rho,\sigma)\equiv\|\sqrt\rho\sqrt\sigma\|_1$, and $P_e^{\rm PGM}$ is the error probability. For reference, the pretty good measurement is defined as the POVM with elements $\mu^{\rm PGM}_i \equiv \eta^{-1/2} p_i \rho_i \eta^{-1/2}$, where $\eta_i\equiv p_i \rho_i$, and $\eta^{-1}$ is understood as the pseudoinverse when $\eta$ is not invertible.
The text doesn't give a direct reference for this result. I usually see bounds for the success probability for pretty good measurements in the form $P_{\rm succ}^{\rm PGM}\ge \operatorname{opt}(\eta)^2$, with $\operatorname{opt}(\eta)\equiv\max_\mu \sum_i \langle\mu_i,\eta_i\rangle$ the optimal success probability to discriminate between the elements of the ensemble $\{\eta_i\}_i$. This bound looks instead slightly different, amounting to $$P_{\rm succ}^{\rm PGM} \ge 1- \sum_{i\neq j}\sqrt{p_i p_j} F(\rho_i,\rho_j).$$ Are these two different forms of the same bound? If not, how does one show this one?
