Redefining classicality

I’m in a terrible mood. Maybe it’s just the relentless blackness of Austrian winter, but I do have rational reasons to be annoyed. First is the firehose of nonsense coming from the wormhole-in-a-quantum-computer people, that I wrote about in my previous post. Second are two talks that I attended to here in Vienna in the past couple of weeks. One by Spekkens, claiming that he can explain interference phenomena classically, and another by Perche, claiming that a classical field can transmit entanglement, and therefore that the Bose-Marletto-Vedral proposed experiment wouldn’t demonstrate that the gravitational field must be quantized.

These talks were about very different subjects, but both were based on redefining “classical” to be something completely divorced from our understanding of classicality in order to reach their absurd conclusions. One might object that this is just semantics, you can define “classical” to be whatever you want, but I’d like to emphasize that semantics was the whole point of these talks. They were not trying to propose a physically plausible model, they only wanted to claim that some effect previously understood as quantum was actually classical.

The problem is that “classical” is not well-defined, so each author has a lot of freedom in adapting the notion to their purposes. One could define “classical” to strictly mean classical physics, in the sense of Newtonian mechanics, Maxwell’s equations, or general relativity. That’s not an interesting definition, though, first because you can’t explain even a rock with classical physics, and secondly because the context of these discussion is whether one could explain some specific physical effect with a new, classical-like theory, not whether current classical physics explains it (as the answer is always no).

One then needs to choose the features one wishes this classical-like theory to have. Popular choices are to have local dynamics, deterministic evolution, and trivial measurements (i.e., you can just read off the entire state without complications).

Spekkens’s “classical” theory violates two of these desiderata, it’s not local and you can’t measure the state. The entire theory is based on an “epistemic restriction”, that you have some incompatible variables that by fiat you can’t measure simultaneously. For me that already kills the motivation for studying such a theory: you’re copying the least appealing feature of quantum mechanics! And quantum mechanics at least has an elegant theory of measurement to determine what you can or can’t measure simultaneously, here you have just a bare postulate. But what makes the whole thing farcical is the nonlocality of the theory. In the model of the Mach-Zehnder interferometer, the “classical” state must pick up the phase of the right arm of the interferometer even if it actually went through the left arm. This makes the cure worse than the disease, quantum mechanics is local and if the particle went through the left it won’t pick up any phase from the right.

When I complained to Spekkens about this, he replied that one couldn’t interpret the vacuum state as implying that the particle was not there, that we should interpret the occupation number as just an abstract degree of freedom without consequence to whether the mode is occupied or not. Yeah, you can do that, but can you seriously call that classical? And again, this makes the theory stranger than quantum mechanics.

Let’s turn to Perche’s theory now. Here the situation is more subtle: we’re not trying to define what a classical theory is, but what a hybrid quantum-classical theory is. In a nutshell, the Bose-Marletto-Vedral proposal is that if we entangle two particles via the gravitational interaction, this implies that the gravitational field must be quantized, because classical fields cannot transmit entanglement.

The difficulty with this argument is that there’s no such thing as a hybrid quantum-classical theory where everything is quantum but the gravitational field is classical (except in the case of a fixed background gravitational field). Some such Frankesteins have been proposed, but always as strawmen that fail spectacularly. To get around this, what people always do is abstract away from the physics and examine the scenario with quantum information theory. Then it’s easy to prove that it’s not possible to create entanglement with local operations and classical communication (LOCC). The classical gravitational field plays the role of classical communication, and we’re done.

Perche wanted to do a theory with more meat, including all the physical degrees of freedom and their dynamics. A commendable goal. What he did was to calculate the Green function from the classical gravitational interaction (which subsumes the fields), and postulate that it should also be the Green function when everything else is quantum. The problem is that you don’t have a gravitational field anymore, and no direct way to determine whether it is quantum or classical. The result he got, however, was that this classical Green function was better at producing entanglement than the quantum one. I think that’s a dead giveaway that his (implicit) field was not classical.

The audience would have none of that, and complained several times that his classical field was anything but. Perche would agree that “quantum-controlled classical” would better describe his gravitational field, but would defend anyway calling it just “classical field” as an informal description.

If you want a theory with more meat, my humble proposal is to not treat classical systems as fundamentally classical, but accept reality: the world is quantum, and “classical” systems are quantum systems that are in a state that is invariant under decoherence. And to make them invariant under decoherence we simply decohere them. In this way we can start with a well-motivated and non-pathological quantum theory for the whole system, and simply decohere the “classical” subsystems as often as needed.

It’s easy to prove that the classical subsystems cannot transmit entanglement in such a theory. Let’s say you have a quantum system $|\psi\rangle$ and a classical mediator $|C\rangle$. After letting them interact via any unitary whatsoever, you end up in the state
\[ \sum_{ij} \alpha_{ij}|\psi_i\rangle|C_j\rangle. \] Now we decohere the classical subsystem (in the $\{|C_j\rangle\}$ basis, without loss of generality), obtaining
\[ \sum_{ijk} \alpha_{ij}\alpha_{kj}^*|\psi_i\rangle\langle\psi_k|\otimes|C_j\rangle\langle C_j|. \] This is equal to
\[ \sum_j p_j \rho_j \otimes |C_j\rangle\langle C_j|,\] where $p_j := \sum_i |\alpha_{ij}|^2$ and $\rho_j := \frac1{p_j}\sum_{ik} \alpha_{ij}\alpha_{kj}^*|\psi_i\rangle\langle\psi_k|$, which is an explicitly separable state, which therefore has no entanglement to transmit to anyone.