  We introduce a concept of distance for a space-time where the notion of point
is replaced by the notion of physical states e.g. probability distributions. We
apply ideas of information theory and compute the Fisher information matrix on
such a space-time. This matrix is the metric on that manifold. We apply these
ideas to a simple model and show that the Lorentzian metric can be obtained if
we assumed that the probability distributions describing space-time
fluctuations have complex values. Such complex probability distributions appear
in non-Hermitian quantum mechanics.
