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Math 401, Fall 2025: Thesis notes, R3, Page's lemma
Progress: 0/4=0% (denominator and enumerator may change)
The page's lemma is a fundamental result in quantum information theory that provides a lower bound on the probability of error in a quantum channel.
Statement
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:
\mathbb{E}[H(\rho_A)]\geq \ln d_A -\frac{1}{2\ln 2} \frac{d_A}{d_B}
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}\simeq \ln m-\frac{m}{2n}