This is a questions on 't Hooft's beable models (see here: Discreteness and Determinism in Superstrings?) for quantum mechanics, and the goal is to understand to what extent these succeed in reproducing quantum mechanics. To be precise, I will say an "'t Hooft beable model" consists of the following:
- A very large classical cellular automaton, whose states form a basis of a Hilbert space.
- A state which is imagined to be one of these basis elements.
- A unitary quantum time evolution operator which, for a series of discrete times, reproduces the cellular automaton evolution rules.
't Hooft's main argument (which is interesting and true) is that it is possible to reexpress many quantum systems in this form. The question is whether this rewriting automatically then allows you to consider the quantum system as classical.
The classical probabilisitic theory of a cellular automaton necessarily consists of data which is a probability distribution $\rho$ on CA states evolving according to two separate rules:
- Time evolution: $\rho'(B') = \rho(B)$, where prime means "next time step" and B is the automaton state. You can extend this to a probabilistic diffusion process without difficulty.
- Probabilistic reduction: if a bit of information becomes available to an observer through an experiment, the CA states are reduced to those compatible with the observation.
I should define probabilistic reduction—it's Bayes' rule: given an observation that we see produces a result $x$, but we don't know the exact value $x$, we know a the probability $p(x)$ that the result is $x$, the probabilistic reduction is
$$ \rho'(B) = C \rho(B) p(x(B)), $$
where $x(B)$ is the value of $x$ which would be produced if the automaton state is $B$, and $C$ is a normalization constant. This process is the reason that classical probability theory is distinguished over and above any other system—one can always interpret the Bayes' reduction process as reducing ignorance of hidden variables.
The bits of information that become available to a macroscopic observer internal to the CA through experiment are not microscopic CA values, but horrendously nonlocal and horrendously complex functions of gigantic chunks of the CA. Under certain circumstances, the probabilistic reduction plus the measurement process could conceivably approximately mimic quantum mechanics, I don't see a proof otherwise. But the devil is in the details.
In 't Hooft models, you also have two processes:
- Time evolution: $\psi \rightarrow U \psi$.
- Measurement reduction: the measurement of an observable corresponding to some subsystem at intermediate times, which, as in standard quantum mechanics, reduces the wavefunction by a projection.
The first process, time evolution, is guaranteed to keep you not superposed in the global variables, since this is just a permutation in 't Hooft's formulation, that's the whole point. But I have seen no convincing argument that the second process, learning a bit of information through quantum measurement, corresponds to learning something about the classical state and reducing the CA probabilistic state according to Bayes' rule.
Since 't Hooft's models are completely precise and calculable (this is the great virtue of his formulation), this can be asked precisely: is the reduction of the wavefunction in response to learning a bit of information about the CA state through an internal observation always mathematically equivalent to a Bayes reduction of the global wavefunction?
I will point out that if the answer is no, the 't Hooft models are not doing classical automata, they are doing quantum mechanics in a different basis. If the answer is yes, then the 't Hooft models could be completely rewritable as proper activities on the probability distribution $\rho$, rather than on quantum superposition states.