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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Fannes–Audenaert inequality ： ウィキペディア英語版 Fannes–Audenaert inequality The Fannes–Audenaert inequality is a mathematical bound on the difference between the von Neumann entropies of two density matrices as a function of their trace distance. It was proved by Koenraad M. R. Audenaert in 2007〔Koenraad M. R. Audenaert, ("A sharp continuity estimate for the von Neumann entropy" ), J. Phys. A: Math. Theor. 40 8127 (2007). Preprint: (arXiv:quant-ph/0610146 ).〕 as an optimal refinement of Mark Fannes' original inequality, which was published in 1973.〔M. Fannes, ("A continuity property of the entropy density for spin lattice systems " ), Communications in Mathematical Physics 31 291–294 (1973).〕 Mark Fannes is a Belgian physicist specialised in mathematical quantum mechanics. He works at the KU Leuven. == Statement of inequality == For any two density matrices $\backslash rho$ and $\backslash sigma$ of dimensions $d$, :$|S(\backslash rho)-S(\backslash sigma)|\; \backslash le\; T\; \backslash log\; (d-1)\; +\; H()$ where :$H()\; =\; -\; \backslash sum\; p\_i\; \backslash log\; p\_i\; \backslash ,$ is the (Shannon) entropy of the probability distribution $\backslash$, :$S(\backslash rho)\; =\; H()\; \backslash ,$ is the (von Neumann) entropy of a matrix $\backslash rho$ with eigenvalues $\backslash lambda\_i$, and :$T(\backslash rho,\backslash sigma)\; =\; \backslash frac||\backslash rho-\backslash sigma||\_\; =\; \backslash frac\backslash mathrm\; \backslash left(\backslash sqrt\; \backslash right)$ is the trace distance between the two matrices. Note that the base for the logarithm is arbitrary, so long as the same base is used on both sides of the inequality. Audenaert also proved that—given only the trace distance ''T'' and the dimension ''d''—this is the ''optimal'' bound. He did this by directly exhibiting a pair of matrices which saturate the bound for any values of ''T'' and ''d''. The matrices (which are diagonal in the same basis, i.e. they commute) are :$\backslash rho\; =\; \backslash mathrm(1-T,\; T/(d-1),\backslash dots,T/(d-1))\; \backslash ,$ :$\backslash sigma\; =\; \backslash mathrm(1,0,\backslash dots,0)\; \backslash ,$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fannes–Audenaert inequality」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.