The Theorem

Quantum information is protected by storing it in a subspace $C$ of a system’s state space, the code, with orthogonal projector $P$ . Noise acts on the system as a quantum operation $\mathcal E$ with operation elements $E_i$ . It can be undone on the code when there is a trace preserving quantum operation $\mathcal R$ , the recovery, that takes $\mathcal E(\rho)$ back to $\rho$ , up to a factor, for every state $\rho$ supported on $C$ . Theorem 10.1 gives a condition on $P$ and the $E_i$ alone that is necessary and sufficient for such a recovery to exist.

A Quantum System

The library states the theorem for operators on a finite dimensional complex Hilbert space $H$ , and again for operators on the state space of a quantum system. A quantum system $S$ is described by its state space $\mathcal H_S$ , a finite dimensional complex Hilbert space.^✦

Correcting Noise on a Code

On a finite dimensional complex Hilbert space $H$ , finitely many operators $E_i$ act on an operator $X$ as $\mathcal E(X) = \sum_i E_i X E_i^\dagger$ , and finitely many operators $R_k$ act as $\mathcal R(X) = \sum_k R_k X R_k^\dagger$ .

A finite family of operators $R_k$ on $H$ corrects a family $E_i$ on the code of an operator $P$ when, for every operator $\rho$ on $H$ with $P \rho P = \rho$ , there is a complex number $c$ with $\mathcal R\big(\mathcal E(\rho)\big) = c\,\rho .$ (1)^✦

The conditions of Knill and Laflamme concern a family of operators $E_i$ on $H$ , an operator $P$ on $H$ and a complex matrix $\alpha$ with rows and columns indexed like the $E_i$ : the $E_i$ satisfy them for $P$ with $\alpha$ when $P E_i^\dagger E_j P = \alpha_{ij} P \quad \text{for all } i \text{ and } j .$ (2)^✦

On the state space of a quantum system $S$ , an operator $P$ and a family of operators $E_i$ satisfy the quantum error correction conditions when they satisfy (2) with some Hermitian matrix $\alpha$ .^✦

The Claim

Theorem 1. Let $H$ be a finite dimensional complex Hilbert space, $P$ a nonzero orthogonal projector on $H$ and $E_i$ finitely many operators on $H$ . There are finitely many operators $R_k$ on $H$ with $\sum_k R_k^\dagger R_k = I$ that correct the $E_i$ on the code of $P$ if and only if the $E_i$ satisfy the conditions of Knill and Laflamme for $P$ with some Hermitian matrix $\alpha$ : $\sum_k R_k^\dagger R_k = I,\ \ \mathcal R\big(\mathcal E(\rho)\big) = c\,\rho \text{ if } P\rho P = \rho \quad\Longleftrightarrow\quad P E_i^\dagger E_j P = \alpha_{ij} P,\ \ \alpha^\dagger = \alpha .$ (3)^✦ For $P$ and the $E_i$ on the state space $\mathcal H_S$ of a quantum system $S$ , they satisfy the quantum error correction conditions if and only if such operators $R_k$ exist: $P E_i^\dagger E_j P = \alpha_{ij} P \text{ on } \mathcal H_S,\ \ \alpha^\dagger = \alpha \quad\Longleftrightarrow\quad \sum_k R_k^\dagger R_k = I,\ \ \mathcal R\big(\mathcal E(\rho)\big) = c\,\rho \text{ if } P\rho P = \rho .$ (4)^✦

With the operation elements of the noise for the $E_i$ and the code $C$ as the image of $P$ , this is the theorem.

What Is Checked

The library proves both equivalences, (3) and (4).