# Generality of Lieb's Concavity Theorem

## TL;DR

This paper extends Lieb's concavity theorem to a broader class of unitarily invariant, monotone, and concave matrix functions, demonstrating joint concavity under more general conditions.

## Contribution

It generalizes Lieb's concavity theorem to any unitarily invariant, monotone, and concave matrix function, broadening its applicability.

## Key findings

- Proves joint concavity of a generalized matrix function.
- Extends Lieb's theorem to broader matrix functions.
- Applicable for a wide range of parameters and functions.

## Abstract

We show that Lieb's concavity theorem holds more generally for any unitarily invariant matrix function $\phi:\mathbf{H}^n_+\rightarrow \mathbb{R}$ that is monotone and concave. Concretely, we prove the joint concavity of the function $(A,B) \mapsto\phi\big[(B^\frac{qs}{2}K^*A^{ps}KB^\frac{qs}{2})^{\frac{1}{s}}\big] $ on $\mathbf{H}_+^m\times\mathbf{H}_+^n$, for any $K\in \mathbb{C}^{m\times n},s\in(0,1],p,q\in[0,1], p+q\leq 1$.