Spectral geometric mean and trace characterizations

• URL: http://arxiv.org/abs/2605.18888v1
• Date: Sat, 16 May 2026 21:33:16 GMT
• Title: Spectral geometric mean and trace characterizations
• Abstract summary: We characterize positive linear functionals $$ on $mathbbM_n$ as positive multiples of the trace if and only if $(A natural B) leq sqrt(A) (B)$ for all positive definite matrices $A$ and $B$.<n>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.
• Score: 2.85114441694432
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: We use nearly parallel pure states to characterize positive linear functionals $φ$ on $\mathbb{M}_n$ as positive multiples of the trace if and only if $φ(A \natural B) \leq \sqrt{φ(A) φ(B)}$ for all positive definite matrices $A$ and $B$. Here $A \natural B = (A^{-1} \# B)^{1/2} A (A^{-1} \# B)^{1/2}$ represents the spectral geometric mean. For further clarification, we establish novel characterizations through the inequality $φ(A \natural B) \leq φ((A+B)/2)$ 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.

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