A colleague showed me a paper by L. Mandel "Is a photon amplifier always polarization-dependent?". It considers a state which can be written in simplified form as:
a |2v> + b |1v>|1h> (1)
that is, the number of photons in the given spatial mode is fixed at 2, and there is probability amplitude a for having both photons vertically polarized, and amplitude b for having one photon polarized vertically and the other horizontally. This raises a few questions:
1. By all accounts, the two superposed states in (1) do not constitute a complete set (the possible basis state |2h> is missing). If so, the sum |a|^2+|b|^2 remains indeterminate, and the only thing we can say is that |a|^2+|b|^2 <1, unless it is explicitly stated that c=0. Mandel does not consider such option.
2 The two states seem to be orthogonal. Then normalization condition must include only squares |a|^2 etc.
3. I think the two photons are not entangled ((1) is not an entangled superposition). If so, what is the correlation among the two photons - are they mutually independent, or not?
4. The first term in (1) describes a number (or Fock) state, and can be written as |2v> =|1v>|1v>. Then it must be possible to factor out the vertical photon in the above expression to write the state as the product
|1v> (a|1v> + b|1h>) (2)
If not, why?
5. We now want to measure the polarization of each photon, and pass the given 2-photon state through a vertical polarizer. What are the probabilities of each outcome: a) |1v>|1v>, b) |1h>|1h>, c) |1v>|1h> ? Outcome b) seems impossible if the photons act independently.
6. What are the probabilities of the 3 outcomes |1R>|1R>, |1L>|1L>, |1R>|1L> if we use the circular (R,L) basis to measure polarizations?