  We consider the statistical problem of `compressive' estimation of low rank
states with random basis measurements, where the estimation error is expressed
terms of two metrics - the Frobenius norm and quantum infidelity. It is known
that unlike the case of general full state tomography, low rank states can be
identified from a reduced number of observables' expectations. Here we
investigate whether for a fixed sample size $N$, the estimation error
associated to a `compressive' measurement setup is `close' to that of the
setting where a large number of bases are measured. In terms of the Frobenius
norm, we demonstrate that for all states the error attains the optimal rate
$rd/N$ with only $O(r \log{d})$ random basis measurements. We provide an
illustrative example of a single qubit and demonstrate a concentration in the
Frobenius error about its optimal for all qubit states. In terms of the quantum
infidelity, we show that such a concentration does not exist uniformly over all
states. Specifically, we show that for states that are nearly pure and close to
the surface of the Bloch sphere, the mean infidelity scales as $1/\sqrt{N}$ but
the constant converges to zero as the number of settings is increased. This
demonstrates a lack of `compressive' recovery for nearly pure states in this
metric.
