inefficient measurements will also have this property, but only if the information which is thrown away is also uniform with respect to the state-space, and we do not wish to burden the treatment with this additional complication.

To proceed we must make precise the notion that the sensitivity of a measurement is uniform over state-space. This is captured by stating that such a measurement should be invariant under reversible transformations of the state-space. For classical measurements (which are simply quantum measurements in which all operators and density matrices commute [@pool]) this means that the set of measurement operators is invariant under all permutations of the classical states: we will refer to these as [*completely symmetric*]{} measurements. Note that in this classical case, this is equivalent to saying that the measurement distinguishes all states from all other states equally well. The quantum generalization of this is invariance under all unitary transformations. Such measurements are referred to as being unitarily covariant. [@Barnum; @Ucov]

We must also quantify what we mean by the observer’s lack of knowledge, or uncertainty, before the measurement is made. This is captured by the simple and elegant concept of [*majorization*]{}. [@MO; @Bhatia] If two sets of probabilities $p 