View source for Fannes–Audenaert inequality

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: [https://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 name=":0">M. Fannes, [https://doi.org/10.1007/BF01646490 "A continuity property of the entropy density for spin lattice systems "], Communications in Mathematical Physics '''31''' 291&ndash;294 (1973).</ref> Mark Fannes is a Belgian physicist specialised in mathematical quantum mechanics, and he works at the [[Katholieke Universiteit Leuven|KU Leuven]]. Koenraad M. R. Audenaert is a Belgian physicist and civil engineer. He currently works at [[University of Ulm]].

== Statement of inequality ==

For any two density matrices <math>\rho</math> and <math>\sigma</math> of dimensions <math>d</math>,

:<math>|S(\rho)-S(\sigma)| \le T \log (d-1) + H[\{T,1-T\}] </math>

where

:<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 (mathematics)|continuity]] 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|author2=Chuang, Isaac L|year= 2000|publisher= [[Cambridge University Press]]|location= [[Cambridge]]; [[New York City|New York]]|isbn= 978-0-521-63235-5|oclc= 43641333|title-link= Quantum Computation and Quantum Information (book)}}</ref>  Audenaert's proof of the optimal inequality, on the other hand, is significantly more complicated, and can be found in.<ref name=":0" /><ref>{{Cite book |last=Watrous |first=John |url=http://dx.doi.org/10.1017/9781316848142 |title=The Theory of Quantum Information |date=2018-04-26 |publisher=Cambridge University Press |isbn=978-1-316-84814-2}}</ref>

==References==
<references/>

{{DEFAULTSORT:Fannes-Audenaert inequality}}
[[Category:Inequalities]]