diff --git a/presentation/ZheyuanWu_HonorThesis_Presentation.pdf b/presentation/ZheyuanWu_HonorThesis_Presentation.pdf
index 598e3a6..42f22d7 100644
Binary files a/presentation/ZheyuanWu_HonorThesis_Presentation.pdf and b/presentation/ZheyuanWu_HonorThesis_Presentation.pdf differ
diff --git a/presentation/ZheyuanWu_HonorThesis_Presentation.tex b/presentation/ZheyuanWu_HonorThesis_Presentation.tex
index ca98819..fa5a1cf 100644
--- a/presentation/ZheyuanWu_HonorThesis_Presentation.tex
+++ b/presentation/ZheyuanWu_HonorThesis_Presentation.tex
@@ -155,7 +155,7 @@
 \end{block}
 \end{frame}
 
-\input{./backgrounds.tex}
+% \input{./backgrounds.tex}
 
 \begin{frame}{Information theory in classical systems}
 
@@ -187,7 +187,7 @@ This measures the intrinsic uncertainty of the quantum state and is basis-indepe
 \begin{block}{Entanglement entropy}
 For a bipartite pure state $|\Psi\rangle\in \mathcal{H}_A\otimes\mathcal{H}_B$, define the reduced state $\rho_A=\operatorname{Tr}_B\bigl(|\Psi\rangle\langle\Psi|\bigr).$ Its entanglement entropy is
 $$
-E(|\Psi\rangle)=S(\rho_A).
+E(|\Psi\rangle)=H(\rho_A).
 $$
 Thus entanglement entropy is the von Neumann entropy of a subsystem, and it measures how entangled the bipartite pure state is.
 \end{block}
@@ -272,7 +272,7 @@ Thus entanglement entropy is the von Neumann entropy of a subsystem, and it meas
         $$
         \item A \textbf{mixed state} is represented by a density matrix
         $$    
-        \rho=\sum_{j=1}^n p_j|\psi_j\rangle\langle\psi_j|
+        \rho=\sum_{j=1}^n p_j|\psi_j\rangle\langle\psi_j|, \qquad \sum_{j=1}^n p_j=1, \qquad p_j\geq 0.
         $$
         \item Pure states describe maximal information; mixed states describe probabilistic mixtures or partial information.
     \end{itemize}