  It is now well-known that, with high probability, the additivity of minimum
output entropy does not hold for a pair of a random quantum channel and its
complex conjugate. We investigate asymptotic behavior of output states of
$r$-tensor powers of such pairs, as the dimension of inputs grows. We compute
the limit output states for any sequence of well-behaved inputs, which consist
of a large class of input states having a nice set of parameters. Then, we show
that among these input states tensor products of Bell states give
asymptotically the least output entropy, giving positive mathematical evidence
towards additivity of above pairs of channels.
