PEPSI (Postham Echelle Polarimetric and Spectroscopic Instrument) is to use the unique feature of the LBT and its powerful double mirror configuration to provide high and extremely high spectral resolution full-Stokes four-vector spectra in the wavelength range 450-1100nm. For the given aperture of 8.4m in single mirror mode and 11.8m in double mirror mode, and at a spectral resolution of 40,000-300,000 as designed for the fiber-fed Echelle spectrograph, a polarimetric accuracy between 10$^-4$ and 10$^-2$ can be reached for targets with visual magnitudes of up to 17th magnitude. A polarimetric accuracy better than 10$^-4$ can only be reached for either targets brighter than approximately 10th magnitude together wiht a substantial trade- off wiht the spectral resolution or with spectrum deconvolution techniques. At 10$^-2$, however, we will be able to observe the brightest AGNs down to 17th magnitude.