Several experimental and theoretical studies report instances of concerted or correlated multiple proton tunnelling in solid phases of water. Here, we construct a pseudo-spin model for the quantum motion of protons in a hexameric H2O ring and extend it to open system dynamics that takes environmental effects into account in the form of O-H stretch vibrations. We approach the problem of correlations in tunnelling using quantum information theory in a departure from previous studies. Our formalism enables us to quantify the coherent proton mobility around the hexagonal ring by one of the principal measures of coherence, the l(1) norm of coherence. The nature of the pairwise pseudo-spin correlations underlying the overall mobility is further investigated within this formalism. We show that the classical correlations of the individual quantum tunnelling events in long-time limit is sufficient to capture the behaviour of coherent proton mobility observed in low-temperature experiments. We conclude that long-range intra-ring interactions do not appear to be a necessary condition for correlated proton tunnelling in water ice.