There are several different criteria used for distortion measure (especially for equalization), including mean-squared error (MSE, to minimize the MSE of each symbol), Zero-Force (to minimize peak distortion), least-squares (LS, to minimize the sum of the squared errors of all past symbols), among others. More information in details can be found from some good textbooks.
With the Zero-Force equalization, if the peak distortion is really minimized to zero, then zero distortion is achieved. However, I am not sure whether this is achievable in reality.
It is possible to reduce distortions in a signal to ideally zero, but reducing noise to zero is not possible.
Distortion in a signal can be caused by various factors such as imperfect transmission or processing of the signal, nonlinearities in electronic components, or interference from other signals. By carefully designing and optimizing the signal processing system, it is possible to reduce the distortion to a negligible level.
Noise, on the other hand, is a random variation in the signal that is introduced by various sources such as thermal noise, shot noise, or quantization noise. While it is possible to reduce the noise level to some extent by using various noise reduction techniques such as filtering or averaging, it is not possible to completely eliminate the noise without losing some information from the signal.
In practical signal processing systems, a trade-off must be made between reducing the noise to an acceptable level while minimizing the distortion introduced by the processing system.