A semiclassical reversibility paradox in simple chaotic systems

Using semiclassical methods, it is possible to construct very accurate approximations in the short-wavelength limit of quantum dynamics that rely exclusively on classical dynamical input. For systems whose classical realization is strongly chaotic, there is an exceedingly short logarithmic Ehrenfest time scale, beyond which the quantum and classical dynamics of a system necessarily diverge, and yet the semiclassical construction remains valid far beyond that time. This fact leads to a paradox if one ponders the reversibility and predictability properties of quantum and classical mechanics. They behave very differently relative to each other, with classical dynamics being essentially irreversible/unpredictable, whereas quantum dynamics is reversible/stable. This begs the question: 'How can an accurate approximation to a reversible/stable dynamics be constructed from an irreversible/unpredictable one?' The resolution of this incongruity depends on a couple of key ingredients: a well-known, inherent, one-way structural stability of chaotic systems; and an overlap integral not being amenable to the saddle point method