Security of the differential-quadrature-phase-shift quantum key distribution

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.

Figure 1

Setup for the $L$-pulse DQPS protocol. At Alice's site, pulse trains are generated by a laser followed by phase randomization as well as phase modulation (PM) with $\{0,\pi \}$ or $\{\frac{\pi }{2},\frac{3\pi }{2}\}$ according to her random bits and basis choice. At Bob's site, each pulse train is fed to a delayed Mach-Zehnder interferometer with phase shift 0 or $\frac{\pi }{2}$ according to his basis choice. The trains leaving the interferometer are measured by two photon detectors corresponding to bit values “0” and “1.” Valid timings of detection are labeled by integers $1,2,...,L-1$, according to the index of the pulse from the short arm of the interferometer. Detection from interference between pulses from different blocks is regarded as invalid and ignored.

Figure 2

Secure key rate per pulse $R_L$ as a function of the overall channel transmission $\eta$. The solid curves represent the key rate under the assumption that a dark count probability is $p_{\mathrm{dark}}=0.5\times {10}^{-5}$ per pulse per detector, and the dotted curves represent the key rate with $p_{\mathrm{dark}}=0$. For both solid and dotted curves, the top, the middle, and the bottom curve (at a high dB loss) represent the rates for $L=20, L=4$, and $L=2$, respectively. The bit error rate of the sifted key depends on $p_{\mathrm{dark}}$ and $\eta$ in addition to a 3% loss-independent error. The block size $L$ is chosen to be 2, 4, and 20, where $L=2$ corresponds to the PE-BB84 protocol and the other values to the DQPS protocol.

Figure 3

Off-line calibration setup to determine an upper bound on $r_{\mathrm{tag}}$ for a general light source, when the dead time of detectors is shorter than pulse interval $\mathrm{\Delta }\tau$. $R$ and $T$ represent reflectance and transmittance of the beam splitter, respectively. ${\eta }_{\det }^{\left(1\right)}$ and ${\eta }_{\det }^{\left(2\right)}$ represent detection efficiencies of detector 1 and 2, respectively.

Figure 4

Off-line calibration setup to determine an upper bound on $r_{\mathrm{tag}}$ for a general light source, when the dead time of detectors is longer than pulse interval $\mathrm{\Delta }\tau$. An optical linear absorber with transmittance ${\eta }_{\mathrm{abs}}$ is set in front of beam splitters. $R^{\left(1\right)}, R^{\left(2\right)}, T^{\left(1\right)}$, and $T^{\left(2\right)}$ represent reflectance and transmittance of the two beam splitters. ${\eta }_{\det }^{\left(1\right)}, {\eta }_{\det }^{\left(2\right)}$, and ${\eta }_{\det }^{\left(3\right)}$ represent detection efficiencies of threshold detector 1, 2, and 3, respectively.