Noise and measurement errors in a practical two-state quantum bit commitment protocol

We present a two-state practical quantum bit commitment protocol, the security of which is based on the current technological limitations, namely the nonexistence of either stable long-term quantum memories or nondemolition measurements. For an optical realization of the protocol, we model the errors, which occur due to the noise and equipment (source, fibers, and detectors) imperfections, accumulated during emission, transmission, and measurement of photons. The optical part is modeled as a combination of a depolarizing channel (white noise), unitary evolution (e.g., systematic rotation of the polarization axis of photons), and two other basis-dependent channels, namely the phase- and bit-flip channels. We analyze quantitatively the effects of noise using two common information-theoretic measures of probability distribution distinguishability: the fidelity and the relative entropy. In particular, we discuss the optimal cheating strategy and show that it is always advantageous for a cheating agent to add some amount of white noise-the particular effect not being present in standard quantum security protocols. We also analyze the protocol's security when the use of (im)perfect nondemolition measurements and noisy or bounded quantum memories is allowed. Finally, we discuss errors occurring due to a finite detector efficiency, dark counts, and imperfect single-photon sources, and we show that the effects are the same as those of standard quantum cryptography.