  Among various quantum key distribution (QKD) protocols, the round-robin
differential-phase-shift (RRDPS) protocol has a unique feature that its
security is guaranteed without monitoring any statistics. Moreover, this
protocol has a remarkable property of being robust against source imperfections
assuming that the emitted pulses are independent. Unfortunately, some
experiments confirmed the violation of the independence due to pulse
correlations, and therefore the lack of a security proof without taking into
account this effect is an obstacle for the security. In this paper, we prove
that the RRDPS protocol is secure against any source imperfections by
establishing a proof with the pulse correlations. Our proof is simple in the
sense that we make only three experimentally simple assumptions for the source.
Our numerical simulation based on the proof shows that the long-range pulse
correlation does not cause a significant impact on the key rate, which reveals
another striking feature of the RRDPS protocol. Our security proof is thus
effective and applicable to wide range of practical sources and paves the way
to realize truly secure QKD in high-speed systems.
