dorsal/arxiv
View SchemaLieb's simple proof of concavity of Tr A^p K^* B^(1-p) K and remarks on related inequalities
| Authors | Mary Beth Ruskai |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0404126 |
| URL | https://arxiv.org/abs/quant-ph/0404126 |
| Journal | International Jour. Quant. Info 3, 570--590 (2005); erratum 4, 747-748 (2006) |
Abstract
A simple, self-contained proof is presented for the concavity of the map (A,B) --> Tr(A^p K^* B^(1-p) K). The author makes no claim to originality; this note gives Lieb's original argument in its simplest, rather than its most general, form. A sketch of the chain of implications from this result to concavity of A --> Tr e^[K + log(A)] is then presented. An independent elementary proof is given for the joint convexity of the map (A,B,X) --> Tr \int X^* (A+ uI)^{-1} X (B+ uI)^{-1} du which plays a key role in entropy inequalities.
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"abstract": "A simple, self-contained proof is presented for the concavity of the map\n(A,B) --\u003e Tr(A^p K^* B^(1-p) K). The author makes no claim to originality; this\nnote gives Lieb\u0027s original argument in its simplest, rather than its most\ngeneral, form. A sketch of the chain of implications from this result to\nconcavity of A --\u003e Tr e^[K + log(A)] is then presented. An independent\nelementary proof is given for the joint convexity of the map (A,B,X) --\u003e Tr\n\\int X^* (A+ uI)^{-1} X (B+ uI)^{-1} du which plays a key role in entropy\ninequalities.",
"arxiv_id": "quant-ph/0404126",
"authors": [
"Mary Beth Ruskai"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP",
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],
"journal_ref": "International Jour. Quant. Info 3, 570--590 (2005); erratum 4,\n 747-748 (2006)",
"title": "Lieb\u0027s simple proof of concavity of Tr A^p K^* B^(1-p) K and remarks on related inequalities",
"url": "https://arxiv.org/abs/quant-ph/0404126"
},
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