Security proof for quantum cryptography against entanglement-measurement attack

TL;DR

The paper addresses the risk of entanglement-measurement attacks in quantum cryptography and develops security proofs for two eavesdropping-checking paradigms: decoy single-particle states from dual non-orthogonal bases $\{|k\rangle\}_{k=0}^{d-1}$ and $\{F|k\rangle\}_{k=0}^{d-1}$ (with $F|k\rangle=\frac{1}{\sqrt{d}}\sum_{r=0}^{d-1}\zeta^{kr}|r\rangle$, $\zeta=e^{2\pi i/d}$) and decoy entangled states leveraging entanglement correlations, extended from qubit to qudit protocols using maximally entangled carriers such as Bell states and GHZ states. The results show that any entangling attack must impose constraints on Eve’s unitary operators that force the information-carrier system and ancilla into a product state, thereby preventing information leakage. For decoy photons, constraints like $\lambda_{01}=\lambda_{10}=0$ and $\lambda_{00}|\epsilon_{00}\rangle=\lambda_{11}|\epsilon_{11}\rangle$ arise; for GHZ-based checks, similar equalities among ancilla states emerge, supported by rank arguments from the Fourier-transformed basis. These findings provide a basic analytical framework guiding security proofs across different quantum-state constructions and inform practical design choices for decoy strategies. The work thus strengthens the theoretical foundation for secure quantum cryptography against entanglement-based attacks in both low- and high-dimensional settings.

Abstract

Entanglement-measurement attack is one of the most famous attacks against quantum cryptography. In quantum cryptography protocols, eavesdropping checking is an effective means to resist this attack. There are currently two commonly used eavesdropping checking methods: one is to prepare two sets of non-orthogonal single-particle states as decoy states, and determine whether there are eavesdroppers in the quantum channel by comparing the states obtained by measurements with the original states; The other is to prepare two sets of non-orthogonal entangled states and use their entanglement correlations to judge whether there are eavesdroppers in the quantum channel. In this paper, we theoretically demonstrate how quantum cryptography can utilize these two eavesdropping checking methods to resist entanglement-measurement attacks. We take the quantum cryptography protocols based on maximally entangled states as examples to demonstrate the proof process, transitioning from qubit-based protocols to qudit-based ones.