Invariance of the bit error rate in the ancilla-assisted homodyne detection

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.