  Optical homodyne detection has been widely used in continuous-variable (CV)
quantum information processing for measuring field quadrature values. In this
paper we explore the possibility of operating a conjugate homodyne detection
system in "photon counting" mode to implement discrete-variable (DV) quantum
key distribution (QKD) protocols. A conjugate homodyne detection system, which
consists of a beam splitter followed by two optical homodyne detectors, can
simultaneously measure a pair of conjugate quadratures X and P of the incoming
quantum state. In classical electrodynamics, X^2 + P^2 is proportional to the
energy (the photon number) of the input light. In quantum optics, X and P do
not commute and thus the above photon-number measurement is intrinsically
noisy. This suggests that a blind application of the standard security proof
could result pessimistic QKD performance. We overcome this obstacle by taking
advantage of two special features of the proposed detection scheme. First, the
fundamental detection noise associated with vacuum fluctuation cannot be
manipulated by an external adversary. Second, the ability to reconstruct the
photon number statistics at the receiver's end can place additional constraints
on possible attacks from the adversary. As an example, we study the security of
the BB84 QKD using conjugate homodyne detection and evaluate its performance
through numerical simulations. This study may open the door to a new family of
QKD protocols, in complementary to the well-established DV-QKD based on single
photon detection and CV-QKD based on coherent detection.
