Math 401 Paper 1, Side note 2: Page’s lemma

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.

Basic definitions

$SO(n)$

The special orthogonal group $SO(n)$ is the set of all distance preserving linear transformations on $\mathbb{R}^n$ .

It is the group of all $n\times n$ orthogonal matrices ( $A^\top A=I_n$ ) on $\mathbb{R}^n$ with determinant $1$ .

SO(n)=\{A\in \mathbb{R}^{n\times n}: A^\top A=I_n, \det(A)=1\}

Extensions

In The random Matrix Theory of the Classical Compact groups , the author gives a more general definition of the Haar measure on the compact group $SO(n)$ ,

$O(n)$ (the group of all $n\times n$ orthogonal matrices over $\mathbb{R}$ ),

O(n)=\{A\in \mathbb{R}^{n\times n}: AA^\top=A^\top A=I_n\}

$U(n)$ (the group of all $n\times n$ unitary matrices over $\mathbb{C}$ ),

U(n)=\{A\in \mathbb{C}^{n\times n}: A^*A=AA^*=I_n\}

Recall that $A^*$ is the complex conjugate transpose of $A$ .

$SU(n)$ (the group of all $n\times n$ unitary matrices over $\mathbb{C}$ with determinant $1$ ),

SU(n)=\{A\in \mathbb{C}^{n\times n}: A^*A=AA^*=I_n, \det(A)=1\}

$Sp(2n)$ (the group of all $2n\times 2n$ symplectic matrices over $\mathbb{C}$ ),

Sp(2n)=\{U\in U(2n): U^\top J U=UJU^\top=J\}

where $J=\begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}$ is the standard symplectic matrix.

Haar measure

Let $(SO(n), \| \cdot \|, \mu)$ be a metric measure space where $\| \cdot \|$ is the Hilbert-Schmidt norm and $\mu$ is the measure function.

The Haar measure on $SO(n)$ is the unique probability measure that is invariant under the action of $SO(n)$ on itself.

That is also called translation-invariant.

That is, fixing $B\in SO(n)$ , $\forall A\in SO(n)$ , $\mu(A\cdot B)=\mu(B\cdot A)=\mu(B)$ .

The Haar measure is the unique probability measure that is invariant under the action of $SO(n)$ on itself.

The existence and uniqueness of the Haar measure is a theorem in compact lie group theory. For this research topic, we will not prove it.

Random sampling on the $\mathbb{C}P^n$

Note that the space of pure state in bipartite system

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}