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:

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:

References to begin with

• The random Matrix Theory of the Classical Compact groups 

• Page’s conjecture 

• Page’s conjecture simple proof 

• Geometry of quantum states an introduction to quantum entanglement second edition