In this paper we address the relationship between zero temperature Glauber dynamics and the diffusion - annihilation problem in the free fermion case. We show that the well known duality transformation between the two problems can be formulated as a similarity transformation if one uses appropriate (toroidal) boundary conditions. This allows us to establish and clarify the precise nature of the relationship between the two models. In this way we obtain a one-to-one correspondence between observables and initial states in the two problems. A random initial state in Glauber dynamics is related to a short-range correlated state in the annihilation problem. In particular, the long-time behaviour of the density in this state is seen to depend on the initial conditions. Hence, we show that the presence of correlations in the initial state determine the dependence of the long-time behaviour of the density on the initial conditions, even if such correlations are short ranged. We also apply a field-theoretical method to the calculation of multi-time correlation functions in this initial state.