  Quantum cloning is a fundamental protocol of quantum information theory.
Perfect universal quantum cloning is prohibited by the laws of quantum
mechanics, only imperfect copies being reachable. Symmetric quantum cloning is
concerned with case when the quality of the clones is identical. In this work,
we study the general case of $1 \to N$ asymmetric cloning, where one asks for
arbitrary qualities of the clones. We characterize, for all Hilbert space
dimensions and number of clones, the set of all possible clone qualities. This
set is realized as the nonnegative part of the unit ball of a newly introduced
norm, which we call the $\mathcal{Q}$-norm. We also provide a closed form
expression for the quantum cloner achieving a given clone quality vector. Our
analysis relies on the Schur-Weyl duality and on the study of the spectral
properties of partially transposed permutation operators.
