  Since its discovery at the Large Hadron Collider in 2012 the Higgs boson has
arguably become the most famous of the Standard Model particles and many
measurements have been performed in order to assess its properties. Among
others, these include measurements of the Higgs boson's ${\cal CP}$ state which
is predicted to be ${\cal CP}$-even. Even though a pure ${\cal CP}$-odd state
has been ruled out, a possible admixture of a ${\cal CP}$-odd Higgs state has
yet to be excluded. In this work we present predictions for the associated
production of a leptonically decaying top quark pair and a stable Higgs boson
$p p \to e^+ \nu_e\, \mu^- \bar{\nu}_\mu\,b\bar{b}\,H$ with possible mixing
between ${\cal CP}$-even and ${\cal CP}$-odd states at NLO in QCD for the LHC
with $\sqrt{s} = 13$ TeV. Finite top-quark and gauge boson width effects as
well as all double-, single- and non-resonant Feynman diagrams including their
interference effects are taken into account. We compare the behaviour of the
${\cal CP}$-even, -odd and -mixed scenarios for the integrated fiducial
cross-sections as well as several key differential distributions. In addition,
we show that both NLO corrections and off-shell effects play an important role
even at the level of integrated fiducial cross-sections and that these are
further enhanced in differential distributions. Even though we focus here on
the Standard Model Higgs boson, the calculations could be straightforwardly
applied to models that have an extended Higgs-boson sector and predict the
existence of ${\cal CP}$-odd Higgs-like particles, such as the
two-Higgs-doublet model.
