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| The '''Fannes–Audenaert inequality''' is a mathematical bound on the difference between the [[von Neumann entropy|von Neumann entropies]] of two [[density matrix|density matrices]] as a function of their [[trace distance]]. It was proved by Koenraad M. R. Audenaert in 2007<ref>Koenraad M. R. Audenaert, [http://iopscience.iop.org/1751-8121/40/28/S18/ "A sharp continuity estimate for the von Neumann entropy"], J. Phys. A: Math. Theor. '''40''' 8127 (2007). Preprint: [http://arxiv.org/abs/quant-ph/0610146 arXiv:quant-ph/0610146].</ref> as an optimal refinement of Mark Fannes' original inequality, which was published in 1973.<ref>M. Fannes, [http://www.springerlink.com/content/k7254x6466633837/ "A continuity property of the entropy density for spin lattice systems "], Communications in Mathematical Physics '''31''' 291–294 (1973).</ref>
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| == Statement of inequality ==
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| For any two density matrices <math>\rho</math> and <math>\sigma</math> of dimensions <math>d</math>,
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| :<math>|S(\rho)-S(\sigma)| \le T \log (d-1) + H[\{T,1-T\}] </math>
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| where
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| :<math>H[\{p_i\}] = - \sum p_i \log p_i \,</math>
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| is the ([[Shannon entropy|Shannon]]) entropy of the probability distribution <math>\{p_i\}</math>,
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| :<math>S(\rho) = H[\{\lambda_i\}] \,</math>
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| is the (von Neumann) entropy of a matrix <math>\rho</math> with eigenvalues <math>\lambda_i</math>, and
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| :<math>T(\rho,\sigma) = \frac{1}{2}||\rho-\sigma||_{1} = \frac{1}{2}\mathrm{Tr} \left[ \sqrt{(\rho-\sigma)^\dagger (\rho-\sigma)} \right]</math>
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| 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.
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| 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
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| :<math>\rho = \mathrm{Diag}(1-T, T/(d-1),\dots,T/(d-1)) \,</math>
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| :<math>\sigma = \mathrm{Diag}(1,0,\dots,0) \,</math>
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| ==Fannes' inequality and Audenaert's refinement==
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| The original inequality proved by Fannes was
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| :<math>|S(\rho)-S(\sigma)| \le 2T \log (d) -2T \log 2T </math>
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| when <math>T \le 1/2e</math>. He also proved the weaker inequality
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| :<math>|S(\rho)-S(\sigma)| \le 2T \log (d) + 1 / (e \log 2) </math>
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| which can be used for larger ''T''.
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| Fannes proved this inequality as a means to prove the [[continuity]]{{disambiguation needed|date=October 2012}} of the [[von Neumann entropy]], which did not require an optimal bound. The proof is very compact, and can be found in the textbook by Nielsen and Chuang.<ref>{{cite book |title= Quantum Computation and Quantum Information|last= Nielsen|first= Michael A|authorlink= |coauthors= Chuang, Isaac L|year= 2000|publisher= [[Cambridge University Press]]|location= [[Cambridge]]; [[New York City|New York]]|isbn= 978-0-521-63235-5|oclc= 43641333|pages= }}</ref> Audenaert's proof of the optimal inequality, on the other hand, is significantly more complicated.
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Fannes-Audenaert inequality}}
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| [[Category:Inequalities]]
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Land Economist Anton from Regina, likes to spend time pinochle, bandage celebrity dresses and cosplay. Previous year just completed a journey Athens.
Feel free to surf to my homepage :: french connection pink bandage dress