Spectral geometric mean and trace characterizations

Abstract

We use nearly parallel pure states to characterize positive linear functionals $\ddot{I}•$ on ${\mathbb{M}}_n$ as positive multiples of the trace if and only if $\ddot{I}•(A♮B)\le \sqrt{\ddot{I}•(A)\ddot{I}•(B)}$ for all positive definite matrices $A$ and $B$. Here $A♮B=(A^{-1}\#B{)}^{1/2}A(A^{-1}\#B{)}^{1/2}$ represents the spectral geometric mean. For further clarification, we establish novel characterizations through the inequality $\ddot{I}•(A♮B)\le \ddot{I}•((A+B)/2)$ for all positive definite matrices $A$ and $B$. We also present a trace inequality related to quantum fidelity that applies to all positive definite matrices, and demonstrate that it does not characterize the trace.