  In this paper, we obtain asymptotic expressions for the ergodic capacity of
the keyhole multiple-input multiple-output (MIMO) channel at low
signal-to-noise ratio (SNR) in independent and identically distributed
Nakagami-$m$ fading conditions with perfect channel state information at the
transmitter and receiver. We show that the low-SNR capacity of this keyhole
MIMO channel scales proportionally as $\frac{\textrm{SNR}}{4} \log^2
\left(1/{\textrm{SNR}}\right)$. Our main contribution is to identify a
surprising result that the low-SNR capacity of the MIMO fading channel
increases in the presence of keyhole degenerate condition, which is in direct
contrast to the well-known MIMO capacity degradation at high SNR under keyhole
conditions. To explain why rank-deficient keyhole fading channel outperforms
the full-rank MIMO fading channel at sufficiently low-SNR, we remark that the
rank of the MIMO channel matrix has no impact in the low-SNR regime and that
the double-faded (or double-scattering) nature of the keyhole MIMO channel
creates more opportunistic communications at low-SNR when compared with pure
MIMO fading channel which leads to increased capacity. Finally, we also show
that a simple one-bit channel information based on-off power control achieves
this low-SNR capacity; surprisingly, this power adaptation is robust against
both moderate and severe fading for a wide range of low SNR values. These
results also hold for the keyhole MIMO Rayleigh channel as a special case.
