  Measuring the CP symmetry in the Higgs sector is one of the key tasks of the
LHC and a crucial ingredient for precision studies, for example in the language
of effective Lagrangians. We systematically analyze which LHC signatures offer
dedicated CP measurements in the Higgs-gauge sector, and discuss the nature of
the information they provide. Based on the Fisher information measure, we
compare the maximal reach for CP-violating effects in weak boson fusion,
associated ZH production, and Higgs decays into four leptons. We find a subtle
balance between more theory-independent approaches and more powerful analysis
channels, indicating that rigorous evidence for CP violation in the Higgs-gauge
sector will likely require a multi-step process.
