More than a bit. Between its wavelength, its presence (or absence), polarization, and timing, a photon can carry a great deal more than one bit.
In most quantum information experiments, that observable is polarization (more precisely, polarization in some particular direction), which is an exact one-bit measurement result.
In radio telescope observations, it's typically amplitude and wavelength that are measured. I don't know that "the information content of a single photon" is really meaningful for such observations since they don't observe single photons, they observe the overall intensity at different wavelengths.
You can't transfer information by any of these means because you can't control the time or the location or the energy of the photon hitting the screen.
The reason polarization is used to carry information in quantum computers is that you can control it, so you can use it to transfer information. But, as I said, a single photon can only be used to read out one bit in this way.
The number of photons is itself a quantum observable (number operator) which is non-commuting with other observables. Does "each photon carries one bit" mean that if there are known N photons (eigenstate of number operator) there are N bits? Or is it saying something about distributions?
The strictly technically correct version applies (as I've said in response to others) to polarization in a setting like quantum information or quantum computing experiments, where the polarization is controlled in various ways, and where measuring the polarization of a single photon at the end of the experiment corresponds to "reading out" one bit of the result.
For cases like the radio telescope observations of light from the black hole in the article this discussion was originally about, the "one bit" more or less corresponds to a "pixel" of the image that gets constructed from the data collected by multiple telescopes and combined by computers.
> The number of photons is itself a quantum observable (number operator) which is non-commuting with other observables.
There isn't a single "number operator" for photons. There is a number operator for each "mode" (or Fock state in more technical language), which means for each possible combination (k, u) of momentum k and polarization u.
Eigenstates of the number operator for each mode are also eigenstates of the Hamiltonian and the polarization operator, so it is not true that the number operators do not commute with any other observables.
Actual states in laboratory experiments are not always Fock states, but AFAIK they are close to it in quantum information and quantum computing experiments. However, the states being detected by the radio telescopes are not Fock states or anything close to it; they are coherent states, which are basically classical EM field states and are not eigenstates of the number operator, or indeed of any observable.
Yes. Each photon carries one bit of information. (I don't know if Galison intended "a bit" to mean literally a bit of information, but it's true.)
> Does a photon have any information content other than its energy/wavelength?
Yes, its polarization. In laboratory experiments in quantum information, photon polarization is what encodes the information.
> how the AF are you supposed to be able to know which ones circled it once, twice, 42 times, whatever?
By putting together the images conveyed by very large numbers of photons and looking for patterns in them. Yes, the information can be hard to decode, but it's there.