The space of positive semi-definite $n\times n$ matrices with unit trace, or more generally, the space of states on a $C^*$ algebra can be made into a metric space using constructions that a analogous to the transportation-theoretic constructions that make spaces of probability measures into metric spaces. That the analogy is natural is seen in the fact that gradient flows with respect to these metrics give rise to interesting classes of evolution equations and lead to the proof of new functional inequalities for $C^*$-algebra states, just as the classical theory of gradient flows with respect to transport metrics leads to functional inequalities for probability measures. We present joint work with Jan Maas in this area concerning both linear (Lindblad type) flows and non-linear flows. These flows and functional inequalities arise naturally in quantum information theory and quantum statistical mechanics, and some of the new results obtained this way have solved conjectures arising in these fields. All of the main ideas are already interesting in the case of $n\times n$ matrices, and so the exposition will require no background in operator algebras, though in the final part, some discussion of the extension of the results to the infinite dimensional setting will be provided, if only to make it clear that the constructions are valid beyond the matrix setting.

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