Spectral geometric mean and trace characterizations
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866916024525258752 |
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| author | Bikchentaev, Airat Dinh, Trung Hoa Le, Anh Vu Moslehian, Mohammad Sal |
| author_facet | Bikchentaev, Airat Dinh, Trung Hoa Le, Anh Vu Moslehian, Mohammad Sal |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_18888 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral geometric mean and trace characterizations Bikchentaev, Airat Dinh, Trung Hoa Le, Anh Vu Moslehian, Mohammad Sal Quantum Physics Functional Analysis Operator Algebras 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. |
| title | Spectral geometric mean and trace characterizations |
| topic | Quantum Physics Functional Analysis Operator Algebras |
| url | https://arxiv.org/abs/2605.18888 |