Fast-spinning strongly magnetized newborn neutron stars, including nascent magnetars, are popularly implemented as the engine of luminous stellar explosions. Here, we consider the scenario that they power various stripped-envelope supernovae, not only super-luminous supernovae Ic but also broad-line supernovae Ibc and possibly some ordinary supernovae Ibc. This scenario is also motivated by the hypothesis that Galactic magnetars largely originate from fast-spinning neutron stars as remnants of stripped-envelope supernovae. By consistently modeling the energy injection from magnetized wind and Ni decay, we show that proto-neutron stars with >~ 10 ms rotation and B_dip >~ 5 x 10^14 G can be harbored in ordinary supernovae Ibc. On the other hand, millisecond proto-neuton stars can solely power broad-line supernovae Ibc if they are born with poloidal magnetic field of B_dip >~ 5 x 10^14 G, and superluminous supernovae Ic with B_dip >~ 10^13 G. Then, we study how multi-messenger emission can be used to discriminate such pulsar-driven supernova models from other competitive scenarios. First, high-energy x-ray and gamma-ray emission from embryonic pulsar wind nebulae is a promising smoking gun of the underlying newborn pulsar wind. Follow-up observations of stripped-envelope supernovae using NuSTAR ~ 50-100 days after the explosion is strongly encouraged for nearby objects. We also discuss possible effects of gravitational-waves on the spin-down of proto-neutron stars. If millisecond proto-neutron stars with B_dip <~ a few x 10^13 G emit gravitational waves through e.g., non-axisymmetric rotation deformed by the inner toroidal fields of B_t >~ 10^16 G, the gravitational wave signal can be detectable from ordinary supernova Ibc in the Virgo cluster by Advanced LIGO, Advanced Virgo, and KAGRA.
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