  One of the simplest methods for implementing quantum key distribution over
fiber-optic communication is the Bennett-Brassard 1984 protocol with phase
encoding (PE-BB84 protocol), in which the sender uses phase modulation over
double pulses from a laser and the receiver uses a passive delayed
interferometer. Using essentially the same setup and by regarding a train of
many pulses as a single block, one can carry out the so-called differential
quadrature phase shift (DQPS) protocol, which is a variant of differential
phase shift (DPS) protocols. Here we prove the security of the DQPS protocol
based on an adaptation of proof techniques for the BB84 protocol, which
inherits the advantages arising from the simplicity of the protocol, such as
accommodating the use of threshold detectors and simple off-line calibration
methods for the light source. We show that the secure key rate of the DQPS
protocol in the proof is eight thirds as high as the rate of the PE-BB84
protocol.
