  The trade-offs among various output fidelities of asymmetric universal
cloning machines are investigated. First we find out all the attainable optimal
output fidelities for the 1 to 3 asymmetric universal cloning machine and it
turns out that there are two kinds of extremal asymmetric cloning machines
which have to cooperate in order to achieve some of the optimal output
fidelities. Second we construct a family of extremal cloning machines that
includes the universal symmetric cloning machine as well as an asymmetric 1 to
$1+N$ cloning machine for qudits with two different output fidelities such that
the optimal trade-off between the measurement disturbance and state estimation
is attained in the limit of infinite $N$.
