Compact entanglement concentration for three-electron-spin W states with error-detected parity-check gates

We present a compact entanglement concentration protocol (ECP) for unknown less-entangled three-electron-spin W states, resorting to the interaction rules between the circularly polarized photon and cavity-quantum-dot (QD) system. In the first step of our ECP, the parties utilize two less-entangled three-electron-spin systems not only to obtain one partially entangled three-electron-spin system with two unknown parameters if the odd-parity occurs with the parity-check gate (PCG) but also to get one entangled two-electron-spin system if the even-parity occurs. By exploiting the above three-electron-spin and two-electron-spin systems as the resource for the second step of our ECP, the parties can obtain a standard three-electron-spin W state if the odd parity occurs. Meanwhile, the systems in the even-parity instance can be used as the resource in the next round of our ECP. As the imperfect performances originated from the side leakage and the limited coupling strength of the cavity-QD system can be reflected by clicking the single-photon detectors, the fidelity of the PCG is unit, in principle, immune to strong coupling-strength restriction. Moreover, the success of our ECP not only is heralded by the detectors but also its efficiency further is improved by repeating the operation processes. Therefore, our ECP is useful in the quantum communication network.