  A family of asymmetric cloning machines for $N$-dimensional quantum states is
introduced. These machines produce two imperfect copies of a single state that
emerge from two distinct Heisenberg channels. The tradeoff between the quality
of these copies is shown to result from a complementarity akin to Heisenberg
uncertainty principle. A no-cloning inequality is derived for isotropic
cloners: if $\pi_a$ and $\pi_b$ are the depolarizing fractions associated with
the two copies, the domain in $(\sqrt{\pi_a},\sqrt{\pi_b})$-space located
inside a particular ellipse representing close-to-perfect cloning is forbidden.
More generally, a no-cloning uncertainty relation is discussed, quantifying the
impossibility of copying imposed by quantum mechanics. Finally, an asymmetric
Pauli cloning machine is defined that makes two approximate copies of a quantum
bit, while the input-to-output operation underlying each copy is a (distinct)
Pauli channel. The class of symmetric Pauli cloning machines is shown to
provide an upper bound on the quantum capacity of the Pauli channel of
probabilities $p_x$, $p_y$ and $p_z$.
