Heralded high-efficiency entanglement concentration of cluster states for photon systems with linear optics via time-delay effect

Entanglement concentration stands as a pivotal technique to safeguard against degraded fidelity in long-distance quantum communication. In this manuscript, propose ultra-efficient entanglement concentration protocols (ECPs) tailored for less-entangled two-photon cluster states and four-photon cluster states with two unknown parameters, leveraging simple linear optical elements and acute single-photon detectors. Additionally, ancillary photons, post-selection techniques, or photon-number-resolving detectors are unnecessary for complete identification of the parity-check measurement. By coordinating auxiliary time degrees of freedom and implementing multiple recycling conditions, the success of two ECPs can be heralded by detection signatures without compromising the incident qubits, thereby allowing their efficiencies to be close to 0.75, in principle. The combination of heralded detection and basic linear optical elements renders our practical ECPs accessible for experimental exploration with current technology.