Eavesdropping a Quantum Key Distribution network using sequential quantum unsharp measurement attacks

We investigate the possibility of eavesdropping on a quantum key distribution network by local sequential quantum unsharp measurement attacks by the eavesdropper. In particular, we consider a pure two-qubit state shared between two parties Alice and Bob, sharing quantum steerable correlations that form the one-sided device-independent quantum key distribution network. One qubit of the shared state is with Alice and the other one while going to the Bob's place is intercepted by multiple sequential eavesdroppers who perform quantum unsharp measurement attacks thus gaining some positive key rate while preserving the quantum steerable correlations for the Bob. In this way, Bob will also have a positive secret key rate although reduced. However, this reduction is not that sharp and can be perceived due to decoherence and imperfection of the measurement devices. At the end, we show that an unbounded number of eavesdroppers can also get secret information in some specific scenario.