Asymmetric quantum cloning in any dimension

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 non-identical Heisenberg channels. The trade-off between the quality of the copies imposed by quantum mechanics is shown to result from a complementarity akin to the Heisenberg uncertainty principle. More specifically, the probability distributions of the error operators affecting the two copies are the square modulus of two functions related by a Fourier transform. A no-cloning inequality is derived for the special case of isotropic cloners, quantifying the impossibility of perfect cloning: if pi(a) and pi(b) are the depolarizing fractions associated with the two copies, the domain in (pi(a)(1/2), pi(b)(1/2))-space located inside a particular ellipse representing close-to-perfect cloning is forbidden. More generally, an entropic no-cloning uncertainty relation is also discussed. Finally, the class of asymmetric cloning machines for quantum bits is investigated in detail, and a connection with the capacity of the Pauli channel is displayed.