Petz-Rényi Relative Entropy of Thermal States and their Displacements
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911842404663296 |
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| author | Androulakis, George John, Tiju Cherian |
| author_facet | Androulakis, George John, Tiju Cherian |
| contents | In this article, we obtain the precise range of the values of the parameter $α$ such that Petz-Rényi $α$-relative entropy $D_α(ρ||σ)$ of two displaced thermal states is finite. More precisely, we prove that, given two displaced thermal states $ρ$ and $σ$ with inverse temperature parameters $r_1, r_2,\dots, r_n$ and $s_1,s_2, \dots, s_n$, respectively, we have
\[
D_α(ρ||σ)<\infty \Leftrightarrow α< \min \left\{ \frac{s_j}{s_j-r_j}: j \in \{ 1, \ldots , n \} \text{ such that } r_j<s_j \right\},
\]
where we adopt the convention that the minimum of an empty set is equal to infinity. Along the way, we prove a special case of a conjecture of Seshdreesan, Lami and Wilde (J. Math. Phys. 59, 072204 (2018)). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_03380 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Petz-Rényi Relative Entropy of Thermal States and their Displacements Androulakis, George John, Tiju Cherian Quantum Physics Mathematical Physics Primary 81P17, Secondary 81P99 In this article, we obtain the precise range of the values of the parameter $α$ such that Petz-Rényi $α$-relative entropy $D_α(ρ||σ)$ of two displaced thermal states is finite. More precisely, we prove that, given two displaced thermal states $ρ$ and $σ$ with inverse temperature parameters $r_1, r_2,\dots, r_n$ and $s_1,s_2, \dots, s_n$, respectively, we have \[ D_α(ρ||σ)<\infty \Leftrightarrow α< \min \left\{ \frac{s_j}{s_j-r_j}: j \in \{ 1, \ldots , n \} \text{ such that } r_j<s_j \right\}, \] where we adopt the convention that the minimum of an empty set is equal to infinity. Along the way, we prove a special case of a conjecture of Seshdreesan, Lami and Wilde (J. Math. Phys. 59, 072204 (2018)). |
| title | Petz-Rényi Relative Entropy of Thermal States and their Displacements |
| topic | Quantum Physics Mathematical Physics Primary 81P17, Secondary 81P99 |
| url | https://arxiv.org/abs/2303.03380 |