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Exact decoherence-free state of two distant quantum systems in a non-Markovian environment

Decoherence-free state (DFS) encoding supplies a useful way to avoid the detrimental influence of the environment on quantum information processing. The DFS was previously well established in either the two subsystems locating at the same spatial position or the dynamics under the Born--Markovian approximation. Here, we investigate the exact DFS of two spatially separated quantum systems consisting of two-level systems or harmonic oscillators coupled to a common non-Markovian zero-temperature bosonic environment. The exact distance-dependent DFS and the explicit criterion for forming the DFS are obtained analytically, which reveals that the DFS can arise only in one-dimensional environment. It is remarkable to further find that the DFS is just the system-reduced state of the famous bound state in the continuum (BIC) of the total system predicted by Wigner and von Neumann. On the one hand our result gives insight into the physical nature of the DFS, and on the other hand it supplies an experimentally accessible scheme to realize the mathematically curious BIC in the standard quantum optical systems.