Reply to "Comment on 'Semiquantum-key distribution using less than four quantum states' "

TL;DR

This paper addresses a correction to a previous quantum key distribution proof, clarifying which parts remain valid after an identified mistake, and affirms the correctness of subsequent proofs based on the corrected conclusion.

Contribution

The authors clarify the validity of their original proofs by distinguishing between two conclusions of a lemma, confirming the correctness of the second conclusion and related theorems.

Findings

The first conclusion of Lemma 1 is incorrect.

The second conclusion of Lemma 1 is correct.

Proofs of Lemmas 2, 3, and Theorems 3--6 are valid based on the second conclusion.

Abstract

Recently Boyer and Mor pointed out the first conclusion of Lemma 1 in our original paper is not correct, and therefore, the proof of Theorem 5 based on Lemma 1 is wrong. Furthermore, they gave a direct proof for Theorem 5 and affirmed the conclusions in our original paper. In this reply, we admit the first conclusion of Lemma 1 is not correct, but we need to point out the second conclusion of Lemma 1 is correct. Accordingly, all the proofs for Lemma 2, Lemma 3, and Theorems 3--6 are only based on the the second conclusion of Lemma 1 and therefore are correct.