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Consider the set of probability vectors $$ \mathcal P_n=\Big\lbrace x\in[0,1]^n\,\Big|\,\sum_{i=1}^n x_i=1\Big\rbrace\subset\mathbb C^n $$ for any $n\in\mathbb N$ where $x_i$ is the $i$-th component of $x$. Since $\mathcal P_n$ is a non-empty convex and closed set, there exists a unique vector of minimal norm which, unsurprisingly, turns out to be the equilibrium $\frac1n(1,\ldots,1)$ as a simple consequence of Cauchy-Schwarz.

On the other hand given $x\in \mathcal P_n$ we can define the entropy as in quantum information via

$$ S:\mathcal P_n\to\mathbb R_0^+\qquad x\mapsto-\sum_{i=1}^n x_i\log(x_i). $$

It is well known that the entropy is maximal if and only if $x=\frac1n(1,\ldots,1)$.

Question: I wondered if the entropy being maximized and the norm being minimized by the same unique vector $\frac1n(1,\ldots,1)$ is somehow related? I don't see a direct connection since the logarithm obviously is not linear but I still feel like there might be some kind of link between those two results.

Thanks in advance for any answer or comment!

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    $\begingroup$ mathoverflow.net/questions/118169/… $\endgroup$
    – wlad
    Oct 25, 2017 at 9:46
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    $\begingroup$ Another viewpoint on this is related to schur-convex and schur-concave functions from the majorization theory. Essentially norm is a schur-convex and entropy is a schur-concave function. In most scenarios, the solution to minimizing ( maximizing) a schur-convex function (concave function) is the symmetrical solution where every co-ordinate is the same. $\endgroup$ Oct 25, 2017 at 12:40

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I noticed today that this old question of mine is on top of the list of unanswered quantum-information questions (despite not being an open question anymore). While both comments to this question offer valuable insight, the cited MO page provides a deep link between entropy and 2-norm which -- looking back -- I found most satisfying.

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