  It is a fundamental property of quantum mechanics that information is lost as
a result of performing measurements. Indeed, with every quantum measurement one
can associate a number -- its POVM norm constant -- that quantifies how much
the distinguishability of quantum states degrades in the worst case as a result
of the measurement. This raises the obvious question which measurements
preserve the most information in these sense of having the largest norm
constant. While a number of near-optimal schemes have been found (e.g. the
uniform POVM, or complex projective 4-designs), they all seem to be difficult
to implement in practice. Here, we analyze the distinguishability of quantum
states under measurements that are orbits of the Clifford group. The Clifford
group plays an important role e.g. in quantum error correction, and its
elements are considered simple to implement. We find that the POVM norm
constants of Clifford orbits depend on the effective rank of the states that
should be distinguished, as well as on a quantitative measure of the "degree of
localization in phase space" of the vectors in the orbit. The most important
Clifford orbit is formed by the set of stabilizer states. Our main result
implies that stabilizer measurements are essentially optimal for distinguishing
pure quantum states. As an auxiliary result, we use the methods developed here
to prove new entropic uncertainty relations for stabilizer measurements. This
paper is based on a very recent analysis of the representation theory of tensor
powers of the Clifford group.
