When people generally speak of quantum information in the context of continuous variables, what is generally meant is that observables, like position/momentum or the field quadratures of quantum optics, can take a continuous range of values and that information processing tasks are performed using operators on these variables.
However, most work in this area restricts attention to a countable number of modes of the radiation field (in quantum optics). In this case, the state of the system can be described using a countably infinite number of variables, like the number eigenstates in the field. But this is still called 'continuous variable' quantum information because the relevant observables still take continuous values. Is that so?
If so, is the boson sampling problem in the domain of continuous variables? I don't know what exactly the relevant variables are in this case. A preliminary reading seems to suggest to me that the observable is the photon number, since what is observed is the probability after measuring in some basis. This would now mean that in the sense I think of continuous variables, boson sampling is not a problem in this domain.
Am I right/wrong? Which of my statements here are false?