  We identify a practical vector quantizer design problem where any
fixed-length quantizer (FLQ) yields non-zero distortion at any finite rate,
while there is a variable-length quantizer (VLQ) that can achieve zero
distortion with arbitrarily low rate. The problem arises in a $t \times 1$
multiple-antenna fading channel where we would like to minimize the channel
outage probability by employing beamforming via quantized channel state
information at the transmitter (CSIT). It is well-known that in such a
scenario, finite-rate FLQs cannot achieve the full-CSIT (zero distortion)
outage performance. We construct VLQs that can achieve the full-CSIT
performance with finite rate. In particular, with $P$ denoting the power
constraint of the transmitter, we show that the necessary and sufficient VLQ
rate that guarantees the full-CSIT performance is $\Theta(1/P)$. We also
discuss several extensions (e.g. to precoding) of this result.
