# 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 $1$.

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

<details>
<summary>Extensions</summary>

In [The random Matrix Theory of the Classical Compact groups](https://case.edu/artsci/math/esmeckes/Haar_book.pdf), 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.

</details>

### Haar measure

Let $(SO(n), \| \cdot \|, \mu)$ be a metric measure space where $\| \cdot \|$ is the [Hilbert-Schmidt norm](https://notenextra.trance-0.com/Math401/Math401_T2#definition-of-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:

$$
S_{m,n}=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n}\simeq \ln m-\frac{m}{2n}
$$

## References

- [The random Matrix Theory of the Classical Compact groups](https://case.edu/artsci/math/esmeckes/Haar_book.pdf)

- [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)

- [Geometry of quantum states an introduction to quantum entanglement second edition](https://www.cambridge.org/core/books/geometry-of-quantum-states/46B62FE3F9DA6E0B4EDDAE653F61ED8C)