Uhlmann's theorem for measured divergences
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917303918002176 |
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| author | Fang, Kun Fawzi, Hamza Fawzi, Omar |
| author_facet | Fang, Kun Fawzi, Hamza Fawzi, Omar |
| contents | Uhlmann's theorem is a cornerstone of quantum information theory, stating that for any quantum state $ρ_{AB}$ and any state $σ_A$, there exists an extension $σ_{AB}$ of $σ_A$ such that the fidelity between $ρ_{AB}$ and $σ_{AB}$ equals the fidelity between their marginals $ρ_A$ and $σ_A$. This property underpins many results and applications in quantum information science. In this work, we generalize Uhlmann's theorem to a broad class of measured $f$-divergences, including the measured $α$-Rényi divergences for all $α\geq 0$. The well-known Uhlmann's theorem for the fidelity corresponds to the special case $α= \frac{1}{2}$. Since most commonly used quantum Rényi divergences, including the Petz and sandwiched Rényi divergences, cannot satisfy this property (except for degenerate cases). This fundamentally distinguishes measured $f$-divergences from other quantum divergences and highlights their unique mathematical structure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_07745 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uhlmann's theorem for measured divergences Fang, Kun Fawzi, Hamza Fawzi, Omar Quantum Physics Uhlmann's theorem is a cornerstone of quantum information theory, stating that for any quantum state $ρ_{AB}$ and any state $σ_A$, there exists an extension $σ_{AB}$ of $σ_A$ such that the fidelity between $ρ_{AB}$ and $σ_{AB}$ equals the fidelity between their marginals $ρ_A$ and $σ_A$. This property underpins many results and applications in quantum information science. In this work, we generalize Uhlmann's theorem to a broad class of measured $f$-divergences, including the measured $α$-Rényi divergences for all $α\geq 0$. The well-known Uhlmann's theorem for the fidelity corresponds to the special case $α= \frac{1}{2}$. Since most commonly used quantum Rényi divergences, including the Petz and sandwiched Rényi divergences, cannot satisfy this property (except for degenerate cases). This fundamentally distinguishes measured $f$-divergences from other quantum divergences and highlights their unique mathematical structure. |
| title | Uhlmann's theorem for measured divergences |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2502.07745 |