Galaxy image translation is an important application in galaxy physics and cosmology. With deep learning-based generative models, image translation has been performed for image generation, data quality enhancement, information extraction, and generalized for other tasks such as deblending and anomaly detection. However, most endeavors on image translation primarily focus on the pixel-level and morphology-level statistics of galaxy images. There is a lack of discussion on the preservation of complex high-order galaxy physical information, which would be more challenging but crucial for studies that rely on high-fidelity image translation. Therefore, we investigated the effectiveness of generative models in preserving high-order physical information (represented by spectroscopic redshift) along with pixel-level and morphology-level information. We tested four representative models, i.e. a Swin Transformer, an SRGAN, a capsule network, and a diffusion model, using the SDSS and CFHTLS galaxy images. We found that these models show different levels of incapabilities in retaining redshift information, even if the global structures of galaxies and morphology-level statistics can be roughly reproduced. In particular, the cross-band peak fluxes of galaxies were found to contain meaningful redshift information, whereas they are subject to noticeable uncertainties in the translation of images, which may substantially be due to the nature of many-to-many mapping. Nonetheless, imperfect translated images may still contain a considerable amount of information and thus hold promise for downstream applications for which high image fidelity is not strongly required. Our work can facilitate further research on how complex physical information is manifested on galaxy images, and it provides implications on the development of image translation models for scientific use.