Relating disturbance, information and orthogonality

In the general theory of quantum measurement, one associates a positive semidefinite operator on a $d$-dimensional Hilbert space to each of the $n$ possible outcomes of an arbitrary measurement. In the special case of a projective measurement, these operators are pairwise Hilbert--Schmidt orthogonal, but when $n>d$, orthogonality is restricted by positivity. This restriction has consequences which are not present from a classical perspective; in particular, we find it allows us to more precisely state the old quantum adage that it is impossible to extract information from a system without disturbance. Restricting attention throughout to the L\"uders state updating rule, we consider three properties of a measurement: its disturbance, a measure of how the expected post-measurement state deviates from the input state, its purity gain, a measure of the expected reduction of uncertainty resulting from measurement, and its orthogonality, a measure of the degree to which the measurement operators differ from an orthonormal set. We show that these quantities satisfy a simple algebraic relation amounting to an expression of an information-disturbance trade-off. Finally, we assess several classes of measurements on these grounds and identify symmetric informationally complete quantum measurements as the unique quantum analogs of a perfectly informative and nondisturbing classical ideal measurement.