From joint convexity of quantum relative entropy to a concavity theorem of Lieb

TL;DR

This paper offers a concise proof of Lieb's 1973 theorem on the concavity of a trace function, utilizing quantum relative entropy's joint convexity and recent analytical techniques from quantum information theory.

Contribution

It presents a simplified proof of Lieb's concavity theorem, connecting quantum information concepts with classical matrix analysis.

Findings

Confirmed the concavity of the trace function using quantum relative entropy

Connected quantum information theory with matrix analysis techniques

Provided a more accessible proof of a classical theorem in quantum physics

Abstract

This note provides a succinct proof of a 1973 theorem of Lieb that establishes the concavity of a certain trace function. The development relies on a deep result from quantum information theory, the joint convexity of quantum relative entropy, as well as a recent argument due to Carlen and Lieb.