We have examined Chandra observations of the recently discovered X-ray thread G0.13-0.11 in the Galactic center Radio Arc region. Part of the Chandra data was studied by Yusef-Zadeh, Law, & Wardle (2002), who reported the detection of 6.4-keV line emission in this region. We find, however, that this line emission is not associated with G0.13-0.11. The X-ray spectrum of G0.13-0.11 is well characterized by a simple power law with an energy slope of 1.8+0.7_-0.4 (90% confidence uncertainties). Similarly, the X-ray spectrum of the point-like source embedded in G0.13-0.11 has a power law energy slope of 0.9+0.9_-0.7. The 2 - 10 keV band luminosities of these two components are 3.2* 1033\rm ergs s-1 (G0.13-0.11) and 7.5 *1032 \rm ergs s-1 (point source) at the Galactic center distance of 8 kpc. The morphological, spectral, and luminosity properties strongly indicate that G0.13-0.11 represents the leading-edge of a pulsar wind nebula, produced by a pulsar (point source) moving in a strong magnetic field environment. The main body of this pulsar wind nebula is likely traced by a bow-shaped radio feature, which is apparently bordered by G0.13-0.11 and is possibly associated with the prominent nonthermal radio filaments of the Radio Arc. We speculate that young pulsars may be responsible for various unique nonthermal filamentary radio and X-ray features observed in the Galactic center region.