of $D_j$ is given by another application of the Born Rule, which becomes $$\label{ltpanalogindices}
                Q(D_j)=\sum_{i=1}^{d^2}\left[\sum_{k=1}^{d^2}[\Phi]_{ik}P(H_k)\right]\! P(D_j|H_i)\;,$$ where $P(D_j|H_i)=\tr D_j\sigma_i$ is the probability for outcome $D_j$ conditioned on obtaining $H_i$ in the reference measurement. In more compact matrix notation, we can write $$\label{ltpanalog}
    Q(D)=P(D|H)\,\Phi\, P(H)\;,$$ where $P(D|H)$ is a matrix of conditional probabilities.

![The solid and dashed lines represent two hypothetical procedures an agent contemplates for a system assigned state $\rho$. The solid line represents making a direct measurement of a POVM $\{D_j\}$. The dotted line represents making the MIC measurement $\{H_i\}$ first, preparing a post-measurement state $\sigma_i$, and then finally making the $\{D_j\}$ measurement. For the solid path, the agent assigns one set of probabilities $Q(D_j)$. For the dotted path, she assigns two sets of probabilities: $P(H_i)$ and $P(D_j|H_i)$. Unadorned by physical assumptions, probability theory does not suggest a relation between these paths. The Born Rule in the form of Eq.  is such a relation.[]{data-label="figure1"}](groundsky.png){width="\linewidth"}

A SIC [@Zauner:1999; @Renes:2004; @Scott:2010; @Appleby:2014; @Appleby:2016; @Appleby:2017; @Scott:2017; @Grassl:2017; @Fuchs:2017a; @Stacey:2017] is a MIC for which all the $H_i$ are rank-1 and $$\tr H_i H_j = \frac{1}{d^2} \frac{d\delta_{ij} + 1}{d+1}\;.$$ SICs have yet to be proven to exist in all finite dimensions $d$, but 