fix katex

content/Math401/Math401_P1_2.md

@@ -4,8 +4,29 @@ The page's lemma is a fundamental result in quantum information theory that prov
 
 ## Statement
 
-Let $\mathcal{H}$ be a Hilbert space and $\mathcal{L}(\mathcal{H})$ be the set of linear operators on $\mathcal{H}$.
+Choosing a random pure quantum state $\rho$ from the bi-partite pure state space $\mathcal{H}_A\otimes\mathcal{H}_B$ with the uniform distribution, the expected entropy of the reduced state $\rho_A$ is:
 
-Let $\mathcal{E}: \mathcal{L}(\mathcal{H}) \to \mathcal{L}(\mathcal{H})$ be a quantum channel.
+$$
+\mathbb{E}[H(\rho_A)]\geq \ln d_A -\frac{1}{2\ln 2} \frac{d_A}{d_B}
+$$
 
-Then, for any $\rho \in \mathcal{L}(\mathcal{H})$, we have:
+## Page's conjecture
+
+A quantum system $AB$ with the Hilbert space dimension $mn$ in a pure state $\rho_{AB}$ has entropy $0$ but the entropy of the reduced state $\rho_A$, assume $m\leq n$, then entropy of the two subsystem $A$ and $B$ is greater than $0$.
+
+unless $A$ and $B$ are separable.
+
+In the original paper, the entropy of the average state taken under the unitary invariant Haar measure is:
+
+$$
+S_{m,n}=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n}
+$$
+
+
+
+
+## References
+
+- [Page's conjecture](https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.71.1291)
+
+- [Page's conjecture simple proof](https://journals.aps.org/pre/pdf/10.1103/PhysRevE.52.5653)