  This paper gives a precise characterization of the fundamental limits of
adaptive sensing for diverse estimation and testing problems concerning sparse
signals. We consider in particular the setting introduced in (IEEE Trans.
Inform. Theory 57 (2011) 6222-6235) and show necessary conditions on the
minimum signal magnitude for both detection and estimation: if ${\mathbf
{x}}\in \mathbb{R}^n$ is a sparse vector with $s$ non-zero components then it
can be reliably detected in noise provided the magnitude of the non-zero
components exceeds $\sqrt{2/s}$. Furthermore, the signal support can be exactly
identified provided the minimum magnitude exceeds $\sqrt{2\log s}$. Notably
there is no dependence on $n$, the extrinsic signal dimension. These results
show that the adaptive sensing methodologies proposed previously in the
literature are essentially optimal, and cannot be substantially improved. In
addition, these results provide further insights on the limits of adaptive
compressive sensing.
