Discrete Weyl Channels with Markovian Memory

We discuss discrete Weyl channels (DWCs) with correlated noise in consecutive uses. Some special cases of DWCs have been shown to have the so-called transition behavior, where the optimal signal states, which maximize the Holevo information between the transmitter and the receiver, sharply transition from product states to maximally entangled states as the degree of memory between the consecutive channel uses increases from a threshold value. We first show the general existence of this transition behavior for a class of DWCs where the product states are optimal for the memoryless case. We also provide two methods to estimate the degree of memory exhibited by the correlated noise. These estimators rely on generating classical data by measuring the channel output which shows the same memory characteristics as the consecutive uses of the channel. We specifically consider the Markovian noise but one of our proposed estimators can be straightforwardly generalized to an arbitrary type of noise as it provides the direct access to the sequence in which noise affects the consecutive input states.