The Photo Injector Test Facility at DESY, Zeuthen site (PITZ) was built to test and optimize high brightness electron sources for Free Electron Lasers (FELs) like FLASH and the European XFEL. Although the beam emittance has been optimized and experimentally demonstrated to meet the requirements of FLASH and XFEL, transverse beam asymmetries were observed during operation of the RF guns. Based on previous studies [1], the beam asymmetries most probably stem from beam transverse coupling by

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... pole field errors in the gun section. A pair of normal and skew gun quadrupoles was successfully used for reducing the beam asymmetries in experiment. In this paper, we discuss the beam transverse coupling between X and Y planes due to quadrupole field errors and its impact onto horizontal and vertical rms emittance. Multi-quads scan and two quads with rotated slits scan were proposed to measure the 4D beam matrix for PITZ and tested by simulation, which will give the residual beam coupling after gun quadrupoles compensation and would be helpful for minimizing the 2D rms emittance experimentally.