Distillation of Multipartite Gaussian EPR Steering Based on Measurement-Based Noiseless Linear Amplification

Multipartite Gaussian Einstein–Podolsky–Rosen (EPR) steering is a key resource for quantum networks, but in practice it is strongly degraded by channel loss and excess noise. This motivates the need to distill multipartite Gaussian EPR steering across all relevant mode partitions. Here we propose and analyze a measurement-based noiseless linear amplification (NLA) protocol that distills Gaussian EPR steering in a four-mode square cluster state transmitted through lossy and noisy channels. Starting from a CV cluster shared by a transmitted node A and three local nodes (B, C, and D), we reconstruct the covariance matrix of the Gaussian cluster state and evaluate Gaussian steering monotones for all (1+1), (1+2), and (1+3) bipartitions before and after applying measurement-based NLA. We show that appropriate conditioning on the noisy mode or on selected relay nodes systematically restores and enhances directional steering, extends both one-way and two-way steerable regions, and preserves the monogamy constraints characteristic of Gaussian graph states. Taken together, these results show that measurement-based NLA provides a practical route to distributing robust multipartite steering in CV cluster architectures, thereby strengthening the foundations for continuous-variable quantum information processing.