Spectral geometric mean and trace characterizations

We use nearly parallel pure states to characterize positive linear functionals $\phi$ on ${\mathbb{M}}_n$ as positive multiples of the trace if and only if $\phi (A\mathrm{♮}B)\le \sqrt{\phi (A)\phi (B)}$ for all positive definite matrices $A$ and $B$. Here $A\mathrm{♮}B=(A^{-1}\mathrm{\#}B{)}^{1\mathrm{/}2}A(A^{-1}\mathrm{\#}B{)}^{1\mathrm{/}2}$ represents the spectral geometric mean. For further clarification, we establish novel characterizations through the inequality $\phi (A\mathrm{♮}B)\le \phi ((A+B)\mathrm{/}2)$ 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.