Generalizing Liebs Concavity Theorem via Operator Interpolation

We introduce the notion of $k$-trace and use interpolation of operators to prove the joint concavity of the function $(A,B)mapstotext{Tr}_kbig[(B^frac{qs}{2}K^*A^{ps}KB^frac{qs}{2})^{frac{1}{s}}big]^frac{1}{k}$, which generalizes Liebs concavity theorem from trace to a class of homogeneous functions $text{Tr}_k[cdot]^frac{1}{k}$. Here $text{Tr}_k[A]$ denotes the $k_{text{th}}$ elementary symmetric polynomial of the eigenvalues of $A$. This result gives an alternative proof for the concavity of $Amapstotext{Tr}_kbig[exp(H+log A)big]^frac{1}{k}$ that was obtained and used in a recent work to derive expectation estimates and tail bounds on partial spectral sums of random matrices.