  This paper studies the time-symmetry problem in quantum gravity. The issue
depends critically on the choice of the quantum state and has been considered
in this paper by restricting to the case of quantum wormholes. It is seen that
pure states represented by a wave functional are time symmetric. However, a
maximal analytic extension of the wormhole manifold is found that corresponds
to a mixed state describable by a nondegenerate density-matrix functional that
involves an extra quantum uncertainty for the three-metric, and is free from
the divergences encountered so far in statistical states formulated in quantum
gravity. It is then argued that, relative to one asymptotic region, the
statistical quantum state of single Euclidean wormholes in semiclassical
approximation is time-asymmetric and gives rise to a {\it topological} arrow of
time which will reflect in the set of all quantum fields at low energies of the
asymptotic flat region (To appear in Int. J. Mod. Phys. D)
