We obtain the maximum average data rates achievable over block-fading channels when the receiver has perfect channel state information (CSI), and only an entropy-constrained quantized approximation of this CSI is available at the transmitter. We assume channel gains in consecutive blocks are independent and identically distributed and consider a short term power constraint. Our analysis is valid for a wide variety of channel fading statistics, including Rician and Nakagami-m fading. For this situation, the problem translates into designing an optimal entropy-constrained quantizer to convey approximated CSI to the transmitter and to define a rate-adaptation policy for the latter so as to maximize average downlink data rate. A numerical procedure is presented which yields the thresholds and reconstruction points of the optimal quantizer, together with the associated maximum average downlink rates, by finding the roots of a small set of scalar functions of two scalar arguments. Utilizing this procedure, it is found that achieving the maximum downlink average capacity C requires, in some cases, time sharing between two regimes. In addition, it is found that, for an uplink entropy constraint H̅2(L), a quantizer with more than L cells provides only a small capacity increase, especially at high SNRs.