Inherent probabilistic nature of measurement in QM seems to impose a distinction between past and future and so a definite direction of time, similar to that occurred in statistical mechanics. This arrow of time is not present in the deterministic equation of evolution of quantum states. How to reconcile booth issues in a fundamental level ?
Consider a p-p collision at the LHC. Getting two protons to collide is much easier than getting all the production particles to collide in order to reverse the process. The arrow of time simply reflects the statistical improbability of decreasing disorder/entropy (compared to increasing disorder/entropy) when nature is left to itself. Time itself, as an observer measures it, arises from the way things change relative to each other. Quantum states that an observer sees as having a unique space-time co-ordinate may appear in different forms. (E.g. two protons or their collision products.) An observer that sees different forms interprets the difference as arising from time-based evolution.
The time evolution of the wave-function in qm is as deterministic as the deterministic evolution of a trajectory in phase space in classical mechanics. The difficulties arise with the interpretation of the wave function, e.g., in the framework of the Born rule.
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