  In this work, we consider optimal state discrimination for a quantum system
that interacts with an environment, i.e., states evolve under a quantum
channel. We show the conditions on a quantum channel and an ensemble of states
such that a measurement for optimal state discrimination is preserved. In
particular, we show that when an ensemble of states with equal {\it a priori}
probabilities is given, an optimal measurement can be preserved over any
quantum channel by applying local operations and classical communication, that
is, by manipulating the quantum states before and after the channel
application. Examples are provided for illustration. Our results can be readily
applied to quantum communication protocols over various types of noise.
