characterized by the fact that $\Phi$ cannot equal the identity. Figure \[figure2\] illustrates the situation.

The weak log majorization result of Lemma \[maj\] has at least one more important implication for quantifying the quantum deviation from classicality. Instead of looking at the functional form of Eq.  and considering how much of a deviation from the LTP it represents, one may approach the problem from a geometric perspective.

Classically one can always imagine assigning probability 1 to an outcome of a putative “maximally informative measurement”—for instance when one knows the system’s exact phase space point. However, in an interpretation of quantum theory without hidden variables, whatever one might mean by “maximally informative,” one cannot mean that the reference measurement’s full probability simplex is available. Indeed, quantum mechanics does not allow probability 1 for the outcome of any MIC measurement [@Fuchs:2002]. Thus deviation from classicality can also be captured by the fact that the region of probabilities compatible with quantum states is strictly smaller than the full ($d^2-1$)-simplex. In this setting, the irreducible deviation from classicality is defined by the largest possible region for a reference measurement’s probability simplex. The following theorem establishes that a SIC measurement uniquely maximizes the Euclidean volume 