NoteNextra-origin/content/Math401/Math401_P1_2.md

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: