Doesn't the multi-worlds interpretation of quantum mechanics define an arrow of time?

Imagine we have a random number generator which randomly adds or subtracts 1 from a given number, each with equal probability. Say our initial number is zero. After the first iteration, the space of possible values consists of {-1,1}. After two iterations, it expands to {-2,-1,0,1,2}. It seems clear that progressively larger iterations monotonically increase the space of possible outcomes. Doesn't this defines an arrow of time in the forward direction?

Likewise, suppose our universe comes into existence with some initial state, S(0). As you suggest in your article, this state can by fully specified by phi(r), knowing the configuration over space of all particles and potentials.

From this point, the universe will evolve (again, as you have suggested in your article). As it does, the space of possible configurations for the universe increases; however, at any given time, some configurations are not yet accessible, as they constitute a greater change than one which could have happened (across all possible states) in the time elapsed since S(0).

Ok, one could object. But this assumes that the universe can only change in finitely sized steps (like the random adding machine mentioned earlier). But with quantum mechanics, there is no such limitation; ie, across any physical dimension (that is, value of a observable quantity), an arbitrarily large change is always possible, albeit with increasingly varnishing odds.

Ok. Fine. Consider then, the probability that a change of given size has come to pass. For a change of a fixed size, this probability increases as the system evolves. Again, an arrow of time is specified.

Couldn't we define time along these lines? And if we did, wouldn't that establish a difference between, say, two worlds with exactly the same physical configuration of particles, one which occurs at a later time, and one which occurs at an earlier time?

Question:

Doesn't the multi-worlds interpretation of quantum mechanics define an arrow of time?

Imagine we have a random number generator which randomly adds or subtracts 1 from a given number, each with equal probability. Say our initial number is zero. After the first iteration, the space of possible values consists of {-1,1}. After two iterations, it expands to {-2,-1,0,1,2}. It seems clear that progressively larger iterations monotonically increase the space of possible outcomes. Doesn't this defines an arrow of time in the forward direction?

Likewise, suppose our universe comes into existence with some initial state, S(0). As you suggest in your article, this state can by fully specified by phi(r), knowing the configuration over space of all particles and potentials.

From this point, the universe will evolve (again, as you have suggested in your article). As it does, the space of possible configurations for the universe increases; however, at any given time, some configurations are not yet accessible, as they constitute a greater change than one which could have happened (across all possible states) in the time elapsed since S(0).

Ok, one could object. But this assumes that the universe can only change in finitely sized steps (like the random adding machine mentioned earlier). But with quantum mechanics, there is no such limitation; ie, across any physical dimension (that is, value of a observable quantity), an arbitrarily large change is always possible, albeit with increasingly varnishing odds.

Ok. Fine. Consider then, the probability that a change of given size has come to pass. For a change of a fixed size, this probability increases as the system evolves. Again, an arrow of time is specified.

Couldn't we define time along these lines? And if we did, wouldn't that establish a difference between, say, two worlds with exactly the same physical configuration of particles, one which occurs at a later time, and one which occurs at an earlier time?