While the no-cloning theorem, which forbids the perfect copying of quantum
states, is well-known as one of the defining features of quantum mechanics, the
question of how well the theory allows a state to be cloned is yet to be
completely solved. In this paper, rigorous solutions to the problem of M to N
asymmetric cloning of qudits are obtained in a number of interesting cases. The
central result is the solution to the 1 to N universal asymmetric qudit cloning
problem for which the exact trade-off in the fidelities of the clones for every
N and d is derived. Analogous results are proven for qubits when M=N-1. We also
consider state-dependent 1 to N qubit cloning, providing a general
parametrization in terms of a Heisenberg star Hamiltonian. In all instances, we
determine the feasibility of implementing the cloning economically, i.e.,
without an ancilla, and determine the dimension of the ancilla when an economic
implementation is not possible.Comment: 12 page
