  We investigated the minimum achievable bit error rate of discriminating
binary coherent states by homodyne measurements with the help of arbitrary
ancilla states coupled via beam splitters. We took into account arbitrary pure
ancillary states those were updated in real-time by those partial measurements
and classical feedforward operations where the partial measurements were made
by homodyne detections with a common fixed phase. It was shown that the minimum
bit error rate of the system was invariant under these operations. We also
discussed how generalize the homodyne detection beyond the scheme which made
the bit error rate invariant.
