Low dissipation, tunable coupling to other quantum systems, and unique features of phonons in the aspects of propagation, detection and others suggest the applications of quantized mechanical resonators in phonon-based quantum information processing (QIP) in a way different from their photonic counterpart. In this paper, we propose the first protocol of entanglement concentration for nonlocal phonons from quantized mechanical vibration. We combine the optomechanical cross-Kerr interaction with the Mach-Zehnder interferometer and, by means of twice optomechanical interactions and the photon analysis with respect to the output of the interferometer, achieve ideal entanglement concentration about less-entangled nonlocal phonon Bell and Greenberger-Horne-Zeilinger states. Our protocol is useful for preserving the entangled phonons for the use of high quality phonon-based QIP in future.PACS numbers:
I. INTRODUCTIONQuantum entanglement is a quantum resource indispensable in the areas, such as quantum key distribution [1-3], quantum teleportation [4], quantum secure direct communication [5][6][7][8] and quantum dense coding [9,10]. In quantum communications, an entangled state is usually used for building a quantum channel between remote parties. However, the channel noise will induce decoherence and degrade the entanglement between the quantum systems. Entanglement concentration [11][12][13][14][15][16][17][18][19][20][21][22][23] is an operation which converts a partially entangled state to a more or maximally entangled state. Since the first entanglement concentration protocol (ECP), the well known Schmidt projection, was proposed by Bennett et al [11], its realization on the various physical systems has been reported. The examples include ECPs with linear optical elements in the principle of Schmidt projection [12,13], nonlocal-photon ECP with linear [17] or nonlinear optical elements [18,19], and ECP based on electron-spin systems [20] or on circuit quantum electrodynamics [21,22], etc.