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It is now well-known that, with high probability, the additivity of minimumoutput entropy does not hold for a pair of a random quantum channel and itscomplex conjugate. We investigate asymptotic behavior of output states of$r$-tensor powers of such pairs, as the dimension of inputs grows. We computethe limit output states for any sequence of well-behaved inputs, which consistof a large class of input states having a nice set of parameters. Then, we showthat among these input states tensor products of Bell states giveasymptotically the least output entropy, giving positive mathematical evidencetowards additivity of above pairs of channels.

Asymptotically well-behaved input states do not violate additivity for conjugate pairs of random quantum channels