  In quantum mechanics without application of any superselection rule to the
set of the observables, a closed quantum system temporally evolves unitarily,
and this Lorentzian regime is characterized by von Neumann entropy of exactly
zero. In the holographic theory in the classicalized ground state, we argue
that the unitary real-time evolution of a non-relativistic free particle with
complex-valued quantum probability amplitude in this Lorentzian regime can be
temporally analytically continued to an imaginary-time classical stochastic
process with real-valued conditional probability density in the Euclidean
regime, where the von Neumann entropy of the classicalized hologram and the
information of a particle trajectory acquired by the classicalized hologram are
positive valued. This argument could shed light on the Euclidean regime of the
holographic Universe.
