# 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.

## Statement

Let $\mathcal{H}$ be a Hilbert space and $\mathcal{L}(\mathcal{H})$ be the set of linear operators on $\mathcal{H}$.

Let $\mathcal{E}: \mathcal{L}(\mathcal{H}) \to \mathcal{L}(\mathcal{H})$ be a quantum channel.

Then, for any $\rho \in \mathcal{L}(\mathcal{H})$, we have: