Symmetric Incoherent Eavesdropping against MDI QKD

In this paper, we concentrate on the very recently proposed Measurement Device Independent Quantum Key Distribution (MDI QKD) protocol by Lo, Curty and Qi (PRL, 2012). In this protocol, a secret key is established between Alice and Bob with the help of an untrusted third-party called Eve. We study how one can suitably mount a symmetric incoherent eavesdropping strategy on MDI QKD considering that Eve will not be honest. Similar to the idea of Fuchs et al. (1997), we show that inducing a disturbance D on the Alice’s (or Bob’s) qubit, Eve can guess Alice’s (or Bob’s) bit with probability 12 + p D(1−D). If Eve likes to guess both the bits of Alice and Bob at the same time, then the success probability becomes 1 4 + ∆ 2 + q ∆ 2 with an equivalent disturbance of ∆ = 2D(1 − D) between Alice and Bob. We also study how well Eve can guess whether an error has been introduced between Alice and Bob due to her interaction. While in BB84, Eve can identify with certainty whether an error has been introduced because of eavesdropping, for MDI-QKD, she can only guess it with probability 1 2 +2D(1−D).