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The '''Fannes&ndash;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&ndash;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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:<math>H[\{p_i\}] = - \sum p_i \log p_i \,</math>
 
is the ([[Shannon entropy|Shannon]]) entropy of the probability distribution <math>\{p_i\}</math>,
 
:<math>S(\rho) = H[\{\lambda_i\}] \,</math>
 
is the (von Neumann) entropy of a matrix <math>\rho</math> with eigenvalues <math>\lambda_i</math>, and
 
:<math>T(\rho,\sigma) = \frac{1}{2}||\rho-\sigma||_{1} = \frac{1}{2}\mathrm{Tr} \left[ \sqrt{(\rho-\sigma)^\dagger (\rho-\sigma)} \right]</math>
 
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&mdash;given only the trace distance ''T'' and the dimension&nbsp;''d''&mdash;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&nbsp;''d''. The matrices (which are diagonal in the same basis, i.e. they commute) are
 
:<math>\rho = \mathrm{Diag}(1-T, T/(d-1),\dots,T/(d-1)) \,</math>
:<math>\sigma = \mathrm{Diag}(1,0,\dots,0) \,</math>
 
==Fannes' inequality and Audenaert's refinement==
 
The original inequality proved by Fannes was
 
:<math>|S(\rho)-S(\sigma)| \le 2T \log (d) -2T \log 2T </math>
 
when <math>T \le 1/2e</math>.  He also proved the weaker inequality
 
:<math>|S(\rho)-S(\sigma)| \le 2T \log (d) + 1 / (e \log 2) </math>
 
which can be used for larger&nbsp;''T''.
 
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.
 
==References==
<references/>
 
{{DEFAULTSORT:Fannes-Audenaert inequality}}
[[Category:Inequalities]]

Latest revision as of 20:21, 20 December 2014

There's no dubiousness that Google is becoming a full-mature computer hardware company, just the in style Google X see is a quite a little different that tablets and smartphones ??�the society simply proclaimed that it's construction a "smart contact lens." However, it's not meant to be a miniaturized rendering of Deoxyephedrine ?? it's meant to helper diabetes patients preserve racecourse of their glucose levels. Indoors the lens is a miniaturized radiocommunication scrap and glucose reminder that volition evaluate the glucose levels of the wearer's crying. 

Google's hoping that it'll be a less atrocious and trespassing way of life to Monitor glucose levels than the distinctive method acting of prick a fingerbreadth and examination line droplets multiple multiplication a mean solar day. Ideally, the detector would be capable to render a meter reading at one time per second, and Google wants it to enactment as an early on exemplary twist for when glucose levels bulge falling ?? the company imagines putt a small letter LED idle in the genus Lens that could betoken levels falling to a higher place or on a lower floor a rig room access.

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