  We consider sets of states in conformal quantum mechanics associated to
generators of time evolution whose orbits cover different regions of the time
domain. States labelled by a continuous global time variable define the
two-point correlation functions of the theory seen as a one-dimensional
conformal field theory. Such states exhibit the structure of a thermofield
double built on bipartite eigenstates of generators of non-global time
evolution. In terms of the correspondence between radial conformal symmetries
in Minkowski spacetime and time evolution in conformal quantum mechanics
proposed in arXiv:2002.01836, arXiv:2103.07228, such generators coincide with
conformal Killing vectors tangent to worldlines of Milne and diamond observers
at constant radius. The temperature of the thermofield double states in
conformal quantum mechanics reproduces the temperatures perceived by such
diamond and Milne observers. We calculate the entanglement entropy associated
to the thermofield double states and obtain a UV divergent logarithmic
behaviour analogous to known results in two-dimensional conformal field theory
in which the entangling boundary is point-like.
