of this region, thereby answering a question raised by one of us in 2002 [@Fuchs:2014 pp. 475, 571].

\[vol\] For any MIC in dimension $d$, let $\mathcal{P}$ denote the image of $\mathcal{Q}_d$ under the Born Rule and let $\textnormal{vol}_\textnormal{E}(\mathcal{P})$ denote its Euclidean volume. Then $$\textnormal{vol}_\textnormal{E}(\mathcal{P})\leq\textnormal{vol}_\textnormal{E}(\mathcal{P_\textnormal{SIC}})\;,$$ with equality iff the MIC is a SIC. Furthermore, $$\label{volPsic}
    \textnormal{vol}_\textnormal{E}(\mathcal{P_\textnormal{SIC}})=\sqrt{\frac{(2\pi)^{d(d-1)}}{d^{d^2-2}(d+1)^{d^2-1}}}\frac{\Gamma(1)\cdots\Gamma(d)}{\Gamma(d^2)}\;.$$

The proof of Theorem \[vol\] involves methods of differential geometry which would be distracting here. We direct the interested reader to Appendix B for details.

The $(d^2-1)$-simplex $\Delta$ has Euclidean volume [@Caves:2001] $$\label{volsimplex}
      \text{vol}_\text{E}(\Delta)=\frac{d}{\Gamma(d^2)}\;,$$ so we can calculate the ratio of the Euclidean volumes of $\mathcal{P}_\text{SIC}$ and the simplex it lies within, $$\frac{\textnormal{vol}_E(\mathcal{P}_\text{SIC})}{\text{vol}_\text{E}(\Delta)}=\sqrt{\frac{(2\pi)^{d(d-1)}}{d^{d^2}(d+1)^{d^2-1}}}\Gamma(1)\cdots\Gamma(d)\;.$$

When $d=2$, quantum state space is the Bloch ball and $\mathcal{P}_\text{SIC}$ is the largest ball which can be inscribed in the regular tetrahedron $\Delta_3$, $$\frac{\text{vol}_\text{E}(\mathcal{P}_\text{SIC})}{\text{vol}_\text{E}(\Delta_3)}=\frac{\pi}{6\sqrt{3}}\approx 0.3023\;.$$ When $d=3$, $$\frac{\text{vol}_\text{E}(\mathcal{P}_\text{SIC})}{\text{vol}_\text{E}(\Delta_8)}=\frac{\pi^3}{1296\sqrt{3}}\approx 0.0138\;.$$ In general, the ratio is very rapidly decreasing, signifying a greater and greater deviation from classicality with each Hilbert space dimension.

Theorems \[unitarilyinvariant\] and \[vol\] show that the SICs provide a way of casting the Born Rule in wholly probabilistic terms, which by two different standards make the difference between classical and quantum as small as possible. Of all the representations 