Occam's Razor asserts that when two alternative models result in the same predictions, then, in the absence of any further knowledge, preference should be given to the `simpler' of the two. For example, while it is possible to postulate the falling motion of an apple by postulating that it is an intelligent entity that decides to move in exact accordance the theory of gravity, the theory of gravity itself is preferable , since it makes the same predictions, without the need to decribe the state of an external intelligence.
This motivates us to consider the question, does there exist situations such that the simplest model that makes a particular prediction is based on quantum, rather than classical information?
Of course, in many situations, the decision of which of two theories is `simpler' is not so black and white, and one needs carefully define what simplicity means in context. In his article, I will argue that when a particular view of simplicity is considered, quantum systems can indeed be model certain observations with greator simplicity than their best classical counterpart.
The specific situation we consider involves a some discrete dynamical system incased within a black box, such that it emits a number at discrete intervals of time. For the sake of this article, we limit ourselves to the case where the resulting sequence is a bitstring, though our argument can be easily generalised. In addition, we assume that the systems stationary - the behaviour of the process does not change with time. We now pose the question:
Given the probability distribution that characterizes the distribution of bitstrings such a system outputs, what is the simplest physical system that can replicate the same behaviour?
Before we can answer this question, we need a formal notion of `simplicity'. There exists many valid choices. In this article, we adopt idea of Process memory. Intuitively, this is the amount of information required to specify the state of a given discrete dynamical system, or alternatively, the average amount of information obtained when we gain knowledge about the exact state of the system.
For example, any static system that remains in the same state for all time would have process memory of 0, since no information is required to specify its current state. Meanwhile, the physical system that consists of a coin that alternates between head and tails has a process memory of 1, since it requires exact a single bit of information to determine the state of the coin.
Process Memory: A Measure of Complexity
In this article, we will adopt idea of Process memory. Intuitively, this is the amount of information that is required to describe the state of a physical system averaged over all time.
More generally, the process memory of a physical system is the average amount of information you would gain about the system, should you be told exactly what configuration it is in
Formally,
