Disguising quantum channels by mixing and channel distance trade-off

We consider the reverse problem to the distinguishability of two quantum channels, which we call the disguising problem. Given two quantum channels, the goal here is to make the two channels identical by mixing with some other channels with minimal mixing probabilities. This quantifies how much one channel can disguise as the other. In addition, the possibility to trade off between the two mixing probabilities allows one channel to be more preserved (less mixed) at the expense of the other. We derive lower- and upper-bounds of the trade-off curve and apply them to a few example channels. Optimal trade-off is obtained in one example. We relate the disguising problem and the distinguishability problem by showing the the former can lower and upper bound the diamond norm. We also show that the disguising problem gives an upper bound on the key generation rate in quantum cryptography.