  We study the R\'enyi entanglement entropy and the Shannon mutual information
for a class of spin-1/2 quantum critical XXZ chains with random coupling
constants which are partially correlated. In the XX case, distinctly from the
usual uncorrelated random case where the system is governed by an
infinite-disorder fixed point, the correlated-disorder chain is governed by
finite-disorder fixed points. Surprisingly, we verify that, although the system
is not conformally invariant, the leading behavior of the R\'enyi entanglement
entropies are similar to those of the clean (no randomness) conformally
invariant system. In addition, we compute the Shannon mutual information among
subsystems of our correlated-disorder quantum chain and verify the same leading
behavior as the $n=2$ R\'enyi entanglement entropy. This result extends a
recent conjecture concerning the same universal behavior of these quantities
for conformally invariant quantum chains. For the generic spin-1/2 quantum
critical XXZ case, the true asymptotic regime is identical to that in the
uncorrelated disorder case. However, these finite-disorder fixed points govern
the low-energy physics up to a very long crossover length scale and the same
results as in the XX case apply. Our results are based on exact numerical
calculations and on a numerical strong-disorder renormalization group.
