The spurious interaction of quantum systems with their environment known as decoherence leads, as a function of time, to a decay of coherence of superposition states. Since the interactions between system and environment are local, they can also cause a loss of spatial coherence: correlations between spatially distant parts of the system are lost and the equilibrium states can become localized. This effect limits the distance over which quantum information can be transmitted, e.g., along a spin chain. We investigate this issue in a nuclear magnetic resonance quantum simulator, where it is possible to monitor the spreading of quantum information in a three-dimensional network: states that are initially localized on individual spins (qubits) spread under the influence of a suitable Hamiltonian apparently without limits. If we add a perturbation to this Hamiltonian, the spreading stops and the system reaches a limiting size, which becomes smaller as the strength of the perturbation increases. This limiting size appears to represent a dynamical equilibrium. We present a phenomenological model to describe these results.Comment: 12 pages, 13 figures. Revised versio
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