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Efficient $N$-particle $W$ state concentration with different parity check gates

We present an universal way to concentrate an arbitrary $N$-particle less-entangled $W$ state into a maximally entangled $W$ state with different parity check gates. It comprises two protocols. The first protocol is based on the linear optical elements say the partial parity check gate and the second one uses the quantum nondemolition (QND) to construct the complete parity check gate. Both of which can achieve the concentration task. These protocols have several advantages. First, it can obtain a maximally entangled W state only with the help of some single photons, which greatly reduces the number of entanglement resources. Second, in the first protocol, only linear optical elements are required which is feasible with current techniques. Third, in the second protocol, it can be repeated to perform the concentration step and get a higher success probability. All these advantages make it be useful in current quantum communication and computation applications.