  We prove several trace inequalities that extend the Golden-Thompson and the
Araki-Lieb-Thirring inequality to arbitrarily many matrices. In particular, we
strengthen Lieb's triple matrix inequality. As an example application of our
four matrix extension of the Golden-Thompson inequality, we prove remainder
terms for the monotonicity of the quantum relative entropy and strong
sub-additivity of the von Neumann entropy in terms of recoverability. We find
the first explicit remainder terms that are tight in the commutative case. Our
proofs rely on complex interpolation theory as well as asymptotic spectral
pinching, providing a transparent approach to treat generic multivariate trace
inequalities.
