The production of a highly-polarized positron beam via nonlinear Breit-Wheeler processes during the interaction of an ultraintense circularly polarized laser pulse with a longitudinally spin-polarized ultrarelativistic electron beam is investigated theoretically. A new Monte Carlo method employing fully spin-resolved quantum probabilities is developed under the local constant field approximation to include three-dimensional polarizations effects in strong laser fields. The produced positrons are longitudinally polarized through polarization transferred from the polarized electrons by the medium of high-energy photons. The polarization transfer efficiency can approach 100% for the energetic positrons moving at smaller deflection angles. This method simplifies the post-selection procedure to generate high-quality positrons in further applications. In a feasible scenario, a highly polarized ($40%-65%$), intense ($10^5$/bunch$-10^6 $/bunch), collimated ($5$mrad$-70$ mrad) positron beam can be obtained in a femtosecond timescale. The longitudinally polarized positron sources are desirable for applications in high-energy physics and material science .
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