Network fault management is crucial for a wireless sensor network(WSN) to maintain a normal running state because faults(e.g., link failures) often occur. The existing lossy link localization(LLL) approach usually infers the most probable failed link set first, and then gives the fault hypothesis set. However, the inferred failed link set contains many possible failures that do not actually occur. That quantity of redundant information in the inferred set can pose a high computational burden on fault hypothesis inference, and consequently decreases the evaluation accuracy and increases the failure localization time. To address the issue, we propose the conditional information entropy based redundancy elimination(CIERE), a redundant lossy link elimination approach, which can eliminate most redundant information while reserving the important information. Specifically, we develop a probabilistically correlated failure model that can accurately reflect the correlation between link failures and model the nondeterministic fault propagation. Through several rounds of mathematical derivations, the LLL problem is transformed to a set-covering problem. A heuristic algorithm is proposed to deduce the failure hypothesis set. We compare the performance of the proposed approach with those of existing LLL methods in simulation and on a real WSN, and validate the efficiency and effectiveness of the proposed approach. The existing lossy link localization (LLL) approach usually infers the most probable failed link set first, and then, the inferred failed link set contains many possible failures that do not actually occur. That quantity of redundant information in the inferred set can pose a high computational burden on fault hypothesis inference, and the decrease reduces the evaluation accuracy and increases the failure localization time. to address the issue, we propose the conditional information entropy based redundancy elimination (CIERE), a redundant lossy link elimination approach, which can may most most redundant information while reserving the important information. correlated failure model that can precisely reflect the correlation between link failures and model the nondeterministic fault propagation. Through several rounds of mathematical derivations, the LLL problem is transformed to a set-covering problem. A heuristic algorithm is proposed to deduce the failure hypothesis set. We compare the performance of the proposed approach with those of existing LLL methods in simulation and on a real WSN, and validate the efficiency and effectiveness of the proposed approach.