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We study the mapping which occurs when a single qubit in an arbitrary stateinteracts with another qubit in a given, fixed state resulting in some unitarytransformation on the two qubit system which, in effect, makes two copies ofthe first qubit. The general problem of the quality of the resulting copies isdiscussed using a special representation, a generalization of the usual Schmidtdecomposition, of an arbitrary two-dimensional subspace of a tensor product oftwo 2-dimensional Hilbert spaces. We exhibit quantum circuits which canreproduce the results of any two qubit copying machine of this type. A simplestochastic generalization (using a ``classical'' random signal) of the copyingmachine is also considered. These copying machines provide simple embodimentsof previously proposed optimal eavesdropping schemes for the BB84 and B92quantum cryptography protocols.Comment: Minor changes. 26 pages RevTex including 7 PS figure

Two-qubit copying machine for economical quantum eavesdropping