Is nature - deep down – actually discrete?
The question asked in this article is whether the state of a quantum system is really continuous, or whether it is actually - deep down - discrete. Such a discretization could be so fine that we have so far just not observed this in any experiment. Researchers have proposed such discretized models in the past, for example in the context of quantum gravity.
To see what such a discretization would mean take a look at a single quantum bit (for example a spin-1/2 particle). In standard quantum theory, the state space is a sphere. Discretization would mean that we discretize the sphere to obtain something that looks patchy like this
The goal of this article is to find a simple postulate, motivated by how we expect information to behave, that tells us whether such a discretization is possible. This can indeed be done and the result of this article is an information-theoretic postulate that rules out all such discretized models except classical theory. You can think of classical theory as an extreme form of discretization where the sphere is cut down to a single line, or in higher dimensions, to a probability simplex.
Let us now explain this postulate. In pretty much any physical theory, such as quantum mechanics, researchers assume that there would at least be some notion of making a measurement. You can think of this measurement simply as a device that takes an input (for example, the quantum state we wish to measure), and produces two kinds of output: the first output of this device is the measurement outcome. For example if you want to measure position within a room, then the outcome is simply a spatial coordinate. The second output of the device is the remainder of the quantum state. That is, whatever is left of itafter a measurement is performed. Let's call this the output state of the device.
We know from quantum theory that if we gain information by making the measurement, then the output state cannot equal the input state. Gaining information thereby means that we are not sure which measurement outcome our device will produce ahead of time, or more precisely, that more than one measurement outcome is possible. Possible means that there is some non-zero probability that the outcome occurs. This feature of quantum theory is also phrased as "Information gain causes disturbance" or more informally that measurements cause a collapse of the wavefunction.
Here, we postulate that the "opposite" is also true: namely that if we do not gain any information by making a measurement, then there is also no disturbance. In terms of our measurement device this means that if the input state is such that we could already know the measurement outcome, ie, only one measurement outcome is possible, then the output state is in fact equal to the input state.
You could think of this intuitively in that we can verify "for free" information that we already know. If you want to use the popular analogy of Schroedinger's cat then this means that if we already looked inside the box to find that the cat is dead (or alive), and we look again later (while the box has been left unchanged in the meantime), then we still expect to find the cat dead (or alive) and it is not suddenly resurrected. It should be understood of course, that researchers draw such analogies for explanatory purposes and that our postulate has nothing to do with the fact that the cat is conscious.The cat example should not be taken too literally.
This postulate is thus very well motivated from the perspective of information. Indeed, standard quantum mechanics itself satisfies this postulate in a much stronger form than we postulate it here. Namely, researchers know that in quantum theory gaining little information also causes little disturbance. This means that if we can predict the measurement outcome pretty well, then the output state of our measurement device is approximately equal to the input state. That is, there is a direct relation between the amount of information gain and the amount of disturbance.
It turns out that if we do accept this simple postulate, then the only discretization that is possible is classical theory. That is, the only discretization of quantum mechanics that would be compatible with this postulate brings us all the way down to the classical world, loosing all quantumness.