We present a unified universal quantum cloning machine, which combines several different existing universal cloning machines together, including the asymmetric case. In this unified framework, the identical pure states are projected equally into each copy initially constituted by input and one half of the maximally entangled states. We show explicitly that the output states of those universal cloning machines are the same. One importance of this unified cloning machine is that the cloning procession is always the symmetric projection, which reduces dramatically the difficulties for implementation. Also, it is found that this unified cloning machine can be directly modified to the general asymmetric case. Besides the global fidelity and the single-copy fidelity, we also present all possible arbitrary-copy fidelities.
A family of asymmetric cloning machines for quantum bits and
N-dimensional quantum states is introduced. These machines produce two approximate copies of a single quantum state that emerge from two distinct channels. In particular, an asymmetric Pauli cloning machine is defined that makes two imperfect copies of a quantum bit, while the overall input-to-output operation for each copy is a Pauli channel. A no-cloning inequality is derived, characterizing the impossibility of copying imposed by quantum mechanics. If
p and
p′ are the probabilities of the depolarizing channels associated with the two outputs, the domain in
(sqrtp,sqrtp′)-space located inside a particular ellipse representing close-to-perfect cloning is forbidden. This ellipse tends to a circle when copying an
N-dimensional state with
Ntoinfty, which has a simple semi-classical interpretation. The symmetric Pauli cloning machines are then used to provide an upper bound on the quantum capacity of the Pauli channel of probabilities
px,
py and
pz. The capacity is proven to be vanishing if
(sqrtpx,sqrtpy,sqrtpz) lies outside an ellipsoid whose pole coincides with the depolarizing channel that underlies the universal cloning machine. Finally, the tradeoff between the quality of the two copies is shown to result from a complementarity akin to Heisenberg uncertainty principle.