  We consider a family of multivariate trace inequalities recently derived by
Sutter, Berta and Tomamichel. These inequalities generalize the Golden-Thompson
inequality and Lieb's three-matrix inequality to an arbitrary number of
matrices in a way that features complex matrix powers. We show that their
inequalities can be rewritten as an $n$-matrix generalization of Lieb's
original three-matrix inequality. The complex matrix powers are replaced by
resolvents and appropriate maximally entangled states. We expect that the
technically advantageous properties of resolvents, in particular for
perturbation theory, can be of use in applications of the $n$-matrix
inequalities, e.g., for analyzing the rotated Petz recovery map in quantum
information theory.
