  Entanglement and entanglement-assisted are useful resources to enhance the
mutual information of the Pauli channels, when the noise on consecutive uses of
the channel has some partial correlations. In this Paper, we study
quantum-communication channels in $d$-dimensional systems and derive the mutual
information of the quantum channels for maximally entangled states and product
states coding with correlated noise. Then, we compare fidelity between these
states. Our results show that there exists a certain fidelity memory threshold
which depends on the dimension of the Hilbert space $(d)$ and the properties of
noisy channels. We calculate the classical capacity of a particular correlated
noisy channel and show that in order to achieve Holevo limit, we must use $d$
particles with $d$ degrees of freedom. Our results show that entanglement is a
useful means to enhance the mutual information. We choose a special
non-maximally entangled state and show that in the quasi-classical depolarizing
and quantum depolarizing channels, maximum classical capacity in the higher
memory channels is given by the maximally entangled state. Hence, our results
show that for high error channels in every degree of memory, maximally
entangled states have better mutual information.
