**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.