Figure 6

The encoding process (an arbitrary-BS and the final gate) in Fig. 2. To encode the arbitrary quantum information onto four-photon decoherence-free states, (single logical qubit information), the arbitrary-BS, having a transmission (\(\tau_{1}\)) and reflection (\(\tau_{2}\)) of arbitrary probabilities, is applied to photon D. The final gate then interacts with photon D using QD5, by the reflection operator \({\hat{\text{R}}}_{2}^{{{\text{Id}}}} \left( \omega \right)\), with \(\omega - \omega_{c} = 0\), and can discriminate the path of photon D, according to the measurement outcome of QD5.