Conditional Entropy represents the amount of uncertainty remaining about the #channel input after the channel output has been observed (which is different from Entropy which measure the information in the source). It is defined mathematically by:
$$ H(A_X|A_Y) = \sum^{K-1}_{k=0} \sum^{J-1}_{j=0} p(x_j|y_k) p(y_k) \lg \frac{1}{p(x_j|y_k)} $$
Note: Notice the relationship of Join Probability Distribution#.
Since #Entropy represents the uncertainty about the channel input before observing the channel output, and #Conditional Entropy represents the uncertainty about the channel input after observing the channel output, we could remove the noise uncertainty by:
Unfortunately, in practice, there will always be noise and other transmission impairment that influence the source symbol in the channel, which retain some uncertainties even after observing the output. We described them with Conditional Entropy# and Mutual Information#.