How many bits of information does an elementary particle have? I've been studying information theory, and I'm curious about how many bits of information an elementary particle, like an electron, could have. If we take only the spin, of course it has 1 qubit (2 quantum states), but in the context of QFT, we could give it a qubit by just "being there" or not. So, my question is exactly this: how many bits of information does a particle carry?
 A: Potentially infinite. You can encode information into continuous degrees of freedom of a particle, and then the question becomes only of how much information are you able to reliably encode and decode with your technical capabilities.
As an example, you can take a single photon and encode information in the transverse spatial profile of its wavefunction (for example using the orbital angular momentum of the photon). Experimental demonstrations of these kinds of things have been done showing single photons carrying information in Hilbert spaces with hundreds of dimensions. A couple of relevant references that come to mind are Krenn et al. (2013), where they show high entanglement dimensionality of the state of a pair of photons (in their spatial, continuous degrees of freedom), and Fickler et al. (2012), where they demonstrate entanglement in the OAM degrees of freedom, and in particular generate and verify entanglement in a Hilbert space of dimension more than 600. You can find many other such papers in the literature. Note that what they do is slightly different than what you are talking about: they do not demonstrate writing and reading of information into the photon(s) (that is, they do not use the photons as quantum memories), but the entanglement they demonstrate wouldn't even make sense if the photons weren't able to carry that much information. There are surely more directly relevant references around but I would need to look for them.
So again, in principle, you can cram as much information as you want into a single particle, using some continuous degree of freedom of the particle such as position or frequency/momentum.
The problem is only how much of such information are you then able to reliably use for a given protocol, as it is of course generally highly nontrivial to exploit information encoded/compressed in such ways.
Note that in all of the above I'm talking about the number of qubits/qudits that can be carried by a single particle. The question of the correspondence between bits and qubits is a different matter entirely, and is probably already answered in some other question. Roughly speaking, however, it is a standard result that you cannot reliably store and use more bits than qubits in a given system. You won't be able to use a two-qubit system as a 4-bit memory for example, even though a two-qubit system lives in a space of dimension 4. You can do things like using a single qubit to communicate more than one bit, with the aid of a classical communication channel (this is called superdense coding), but this is different than using a single qubit to store two bits of information.
Finally, note that this is not a "quantum" property. One could say the same of a continuous classical system, which you can in principle use to encode an arbitrary amount of information (and in fact, we do encode pretty big amounts of information in the frequency and amplitude modulations of light for communication purposes).
