update test

chapters/chap1.tex

@@ -505,10 +505,7 @@ Then we have bound for Lipschitz constant $\eta$ of the map $S(\varphi_A): \math
 \end{lemma}
 
 \begin{proof}
-    The proof use lagrange multiplier method to find the maximum of the gradient of $S(\varphi_A)$.
-    %
-    TODO: use lagrange multiplier method to find the maximum of the gradient of $S(\varphi_A)$.
-    %
+    Consider the Lipschitz constant of the function $g:A\otimes B\to \R$ defined as $g(\varphi)=H(M(\varphi_A))$, where $M:A\otimes B\to \mathcal{P}(A)$ is the complete von Neumann measurement and $H: \mathcal{P}(A)\otimes \mathcal{P}(B)\to \R$ is the Shannon entropy.
 \end{proof}
 
 From Levy's lemma, we have