General theory of decoy-state quantum cryptography with source errors

The existing theory of decoy-state quantum cryptography assumes the exact control of each states from Alice's source. Such exact control is impossible in practice. We develop the theory of decoy-state method so that it is unconditionally secure even there are state errors of sources, if the range of a few parameters in the states are known. This theory simplifies the practical implementation of the decoy-state quantum key distribution because the unconditional security can be achieved with a slightly shortened final key, even though the small errors of pulses are not corrected.