A geometrical description of the universal 1 → 2 asymmetric quantum cloning region

We consider the problem of determining the achievable region of the universal 1 → 2 asymmetric quantum cloning problem. This concerns the quantum cloning of any quantum state to two approximate clones of different qualities. Measuring the cloning performance with the probabilities of the mixed clone states as the figure of merit, we show that the physical region is a union of ellipses in the plane. The study of these regions has consequences for the eavesdropping on quantum cryptography, and a wide variety of tasks. Equivalently, we characterize the region of quantum-state compatibility of two possibly different isotropic states, considering, for the first time, negative figures of merit.