> 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.)
Until you start talking about entropy and chaos and statistics (which is a fascinating can of worms), the laws that you use to predict the next state of a system are the same that you would use to calculate the previous state of a system.
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, hence two possible pasts could not have created the same present.