  In the standard homodyne configuration, an unknown optical state is combined
with a local oscillator (LO) on a beam splitter (BS). Good quadrature
measurements require a high-amplitude LO and two high-efficiency photodiodes
whose signals are subtracted and normalized. By changing the LO phase, it is
then possible to infer the optical state in the mode matching the LO. For
quantum information processing, the states of interest are in well-separated
modes, corresponding to a pulsed configuration with one relevant LO mode per
measurement. We theoretically investigate what can be learned about the unknown
optical state by counting photons in one or both outgoing paths after the BS,
keeping the LO mode fixed but choosing its phase and magnitude. We consider
measurement configurations where the BS acts differently on different sets of
matching modes. When the BS acts identically on all matching modes it is
possible to determine the content of the unknown optical state in the mode
matching the LO conditional on each number of photons in the orthogonal modes
on the same path. In particular, if both the phase and the intensity of the LO
can be varied, then the statistics of just one of the counters is enough to
infer these parameters, while in the case of an LO with fixed intensity both
detectors are needed to accomplish this. Our results are derived by
demonstrating a bijection, or lack thereof, between the probability
distributions over the space of outcomes of the counter(s) and certain
parameters of the unknown state for different measurement configuration. We
report an experiment that was conducted to demonstrate the theory in the case
where the BS acts differently depending on the polarization.
