Security proof of the unbalanced phase-encoded Bennett-Brassard 1984 protocol

Abstract

In optical implementations of the phase-encoded BB84 protocol, the bit information is usually encoded in the phase of two consecutive photon pulses generated in a Mach-Zehnder interferometer. In the actual experimental realization, the loss in the arms of the Mach-Zehnder interferometer is not balanced, for example because only one arm contains a lossy phase modulator. Therefore, the amplitudes of the pulses is not balanced, and the structure of the signals and measurements no longer corresponds to the (balanced) ideal BB84 protocol. Hence, the BB84 security analysis no longer applies in this scenario. We provide a security proof of the unbalanced phase-encoded BB84. The resulting key rate turns out to be lower than the key rate of the ideal BB84 protocol. Therefore, in order to guarantee security, the loss due to the phase modulator cannot be ignored.

Squashing model for detectors and applications to quantum-key-distribution protocols

“…(16). Since the second term in the last equation is strictly positive, which is guaranteed by our choice of ρ f ix and Remark 6, there exists a value of p that is smaller than the trivial value p = 1 and for which we still find τ ′ ≥ 0 and therefore a nontrivial complete positive squashing map Λ ′ .…”

“…We choose ρ f ix to have a full rank, so that the eigenvalues of the Choi matrix in Eq. (16) are all strictly positive.…”

“…This phase usually takes one of four values φ 0 = 0, π/2, π, 3π/2, which is motivated by the corresponding QKD protocol (cf. [16]). The long arm of the interferometer has a lossy phase modulator, which can be adjusted to φ = 0, π/2 and whose loss is modelled by a beam splitter with transmittance t. The signals pass through the interferometer and then the modes are detected by two threshold detectors with equal efficiencies η (see Fig.…”

“…[16]). The input-output relations (up to an unimportant overall phase) for the modes of interest are…”

“…This refines the security analysis of the corresponding QKD protocol, which was performed in Refs. [16,17].…”