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content/Math401/Math401_P1_1.md

@@ -145,8 +145,10 @@ QED
 
 ## Drawing the connection between the space $S^{2n+1}$, $CP^n$, and $\mathbb{R}$
 
-A pure quantum state of size $N$ can be identified with a Hopf circle on the sphere $S^{2N-1}$.
+A pure quantum state of size $N$ can be identified with a **Hopf circle** on the sphere $S^{2N-1}$.
 
 A random pure state $|\psi\rangle$ of a bipartite $N\times K$ system such that $K\geq N\geq 3$.
 
 The partial trace of such system produces a mixed state $\rho(\psi)=\operatorname{Tr}_K(|\psi\rangle\langle \psi|)$, with induced measure $\mu_K$. When $K=N$, the induced measure $\mu_K$ is the Hilbert-Schmidt measure.
+
+Consider the function $f:S^{2N-1}\to \mathbb{R}$ defined by $f(x)=S(\rho(\psi))$, where $S(\cdot)$ is the von Neumann entropy. The Lipschitz constant of $f$ is $\sim \ln N$.

content/Math401/Math401_P1_3.md

@@ -205,6 +205,22 @@ The goal is the same as the Gromov's version, first we bound the probability of
 
 Let $f:S^n\to \mathbb{R}$ be a 1-Lipschitz function.
 
+Consider the two sets of points on the sphere $S^n$ with radius $\sqrt{n}$:
+
+$$
+\Omega_+=\{x\in S^n: f(x)\leq a_0-\epsilon\}, \Omega_-=\{x\in S^n: f(x)\geq a_0+\epsilon\}
+$$
+
+Note that $\Omega_+\cup \Omega_-$ is the whole sphere $S^n(\sqrt{n})$.
+
+By the Levy's isoperimetric inequality, we have
+
+$$
+\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\epsilon))\leq \operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\Omega_+))+\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\Omega_-))
+$$
+
+
+
 We define $\kappa_n(\epsilon)$ as the following:
 
 $$