Security of passive entanglement-based key distribution protocols

Abstract

Entanglement-based key distribution protocols, such as the Bennett-Brassard-Mermin 1992 (BBM92) protocol and quantum conference key agreement (QCKA), are promising applications of quantum networks. In practical implementations, passive measurement setups are widely adopted because of their simplicity. However, the security analysis of passive protocols with biased basis choice is highly nontrivial, since standard proof techniques for threshold detectors are generally not applicable in this setting. In this work, we establish the security of passive entanglement-based key distribution protocols in the asymptotic regime. Specifically, we prove the security of passive BBM92 with biased basis choice and extend the proof to passive QCKA with an arbitrary number of parties. In addition, we numerically show that the key generation rate of passive BBM92 is almost identical to that of the corresponding active protocol. Our results provide a theoretical foundation for practical passive implementations of entanglement-based key distribution protocols.

Passive, biased basis selection with threshold detectors complicates entanglement-based QKD security because standard qubit reductions and uncertainty-based arguments do not directly apply. This paper derives asymptotic security bounds for passive BBM92 and extends the analysis to GHZ-based quantum conference key agreement with an arbitrary number of parties and an untrusted source. The proof moves basis-dependent optical loss into an effective splitting probability, introduces virtual photon-number measurements, and constructs a virtual qubit for complementarity-based phase-error estimation. Single-photon events reproduce active qubit measurements after filtering and a depolarizing channel; cross-click statistics bound the multiphoton contribution. Operator relations and Azuma’s inequality connect these bounds to observed frequencies without an independent-and-identically-distributed assumption. For BBM92, define $q_L=Q_Z-\alpha(Q_{\bot_A,Z_B}+Q_{Z_A,\bot_B})$, where $\alpha=\gamma\bar p_Z^2/(2\bar p_Z\bar p_X)$. The resulting bound is $R\ge q_L[1-h(\gamma(\bar p_Z/\bar p_X)^2E_X/q_L)]-Q_Zf_{\rm EC}$, with $E_X$ the error frequency per emitted round. It requires key-basis bias, positive $q_L$, and an entropy argument at most $1/2$. The multipartite expression replaces the cross-click sum by a sum over parties and the exponent by the party count. Optimized PDC-source simulations find nearly identical passive and active BBM92 rates for equal basis-line efficiencies, using detector efficiency $0.7$, dark-count probability $10^{-7}$, and misalignment error $0.03$. The treatment is asymptotic; finite-key security is left open. The detector model assumes matched efficiencies within each basis and identical dark-count probabilities.