Lorentz symmetry is one of the cornerstones of modern physics. However, a number of theories aiming at unifying gravity with other fundamental interactions including string field theory suggest violation of Lorentz symmetry. While the energy scale of such strongly Lorentz symmetry-violating physics is much higher than that currently attainable by particle accelerators, Lorentz violation (LV) may nevertheless be detectable via precision measurements at low energies. I will give an overview of such tests with atomic systems, describing the most recent experiment with trapped Ca+ ions in more detail. I will also discuss a systematic study of LV sensitivities in atomic systems that identified ytterbium ion as an idea system with high sensitivity as well as excellent experimental controllability. By applying quantum information inspired technology to Yb+, we expect tests of LLI violating physics in the electron-photon sector to reach levels of 10-23, five orders of magnitude more sensitive than the current best bounds.