Asymmetric quantum cloning machines

Abstract

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 $(\\sqrt{p},\\sqrt{p'})$-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 $N\\to\\infty$, 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 $p_x$, $p_y$ and $p_z$. The capacity is proven to be vanishing if $(\\sqrt{p_x},\\sqrt{p_y},\\sqrt{p_z})$ 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.