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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}