Two-qubit copying machine for economical quantum eavesdropping

We study the mapping that occurs when a single qubit in an arbitrary state interacts with another qubit in a given, fixed state resulting in some unitary transformation on the two qubit system that, in effect, makes two copies of the first qubit. The general problem of the quality of the resulting copies is discussed using a special representation, a generalization of the usual Schmidt decomposition, of an arbitrary two-dimensional subspace of a tensor product of two two-dimensional Hilbert spaces. We exhibit quantum circuits that can reproduce the results of any two-qubit copying machine of this type. A simple stochastic generalization (using a classical random signal) of the copying machine is also considered. These copying machines provide simple embodiments of previously proposed optimal eavesdropping schemes for the BB84 and B92 quantum cryptography protocols.