As discussed e.g. in this post, given two states $\rho$ and $\sigma$, the measurement that allows to optimally discriminate between them (i.e. the measurement providing the highest average probability of figuring out which of the two states the observer was given) is the one projective measurement corresponding to the eigenstates of $\rho-\sigma$. Or more generally of $p\rho-(1-p)\sigma$ if there is a known prior probability $p$ of receiving $\rho$.
Geometrically, as discussed e.g. here, this result can be understood as saying that the optimal POVM corresponds to the direction (in the Bloch representation) $p\mathbf v_\rho-(1-p)\mathbf v_\sigma$, with $\mathbf v_\rho$ Bloch vector corresponding to $\rho$. For example, for $p=1/2$, fixing $\rho=|+\rangle\!\langle+|$, and computing the measurement optimally discriminating $\rho$ from $\sigma_p=p |0\rangle\!\langle0| + (1-p) |1\rangle\!\langle1|$ varying $p$, we get the following:
where we are looking at the plane of the Bloch sphere spanned by the $Z$ and $X$ directions, and the black arrows correspond to $\rho$ and $\sigma_p$, and the red arrows to the associated optimal measurement.
Consider now the case where we want to discriminate between more than two states. Given an ensemble of states $\rho_i$ with prior probabilities $p_i$, this amounts to the POVM $\mu$ that maximises the discrimination probability $$p_{\rm disc}(\mu) \equiv \sum_i p_i \langle\mu_i, \rho_i\rangle.$$ For ensembles with more than two states, this won't in general have a nice closed expression. Nevertheless, there might be a way to understand how the structure of the resulting optimal measurement relates to that of the given ensemble. Is there any result to this end?
