This > The laws of nature are the same if we reverse the direction of time. and this > To restate it in another way once again: The function mapping "state o

This > The laws of nature are the same if we reverse the direction of time. and this > To restate it in another way once again: The function mapping "state of the universe at time t" to "state of the universe at time t+dt" is a one-to-one map are not the same. Even for deterministic theories, time-reversal invariance implies one-to-one mappings but the reverse is not true. And for a stochastic theory it breaks down completely; you can have time-reversal symmetry but still lose information about the past (e.g., each time step the system changes into a completely random state form the state space). Whether quantum mechanics is considered fundamentally stochastic or deterministic depends on interpretation. (Roughly, Copenhagen is stochastic and Many Worlds is deterministic.)