The Exercise

A quantum code stores a system of dimension $d$ in a subspace of a composite system $Q$ made of two parts, $Q_1$ and $Q_2$ . To keep track of the stored information, a reference system $R$ of the same dimension is maximally entangled with the code, so that $R$ and $Q$ together are in a pure state. Suppose $Q_1$ holds no correlation with $R$ : once $Q_2$ is discarded, the joint state of $R$ and $Q_1$ is a product. The exercise is to show that any error confined to $Q_1$ can then be undone by a recovery operation on $Q$ , for every state of the code.

The library states this with the errors written in a basis of $Q_1$ , and gives the recovery in two forms: as a quantum channel, and as a family of recovery operators.

A Quantum System

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ The composite system $S \otimes T$ of two quantum systems $S$ and $T$ is the quantum system whose state space is the tensor product of the state spaces of $S$ and $T$ .^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$ .^✦

A quantum channel $\mathcal{E}$ on $S$ is a map from the states of $S$ to the states of $S$ that is the restriction to states of a completely positive linear map $\Phi$ on the operators of the state space of $S$ , with $\operatorname{tr} \Phi(A) = \operatorname{tr} A \ \text{ for every operator } A, \qquad \Phi(\rho) = \mathcal{E}(\rho) \ \text{ for every state } \rho .$ (1)^✦

The Code and Its Errors

The code is spanned by orthonormal vectors $|w_1\rangle, \dots, |w_d\rangle$ of $Q_1 \otimes Q_2$ , and $P = \sum_x |w_x\rangle\langle w_x|$ is the projector onto it. A state $\rho$ of the code is a state of $Q_1 \otimes Q_2$ with $P \rho P = \rho$ .

The reference system $R$ is a finite dimensional complex Hilbert space with an orthonormal basis $|e_1\rangle, \dots, |e_d\rangle$ , labelled like the code vectors. The code purification is the vector of $R \otimes (Q_1 \otimes Q_2)$ $|RQ\rangle = \sum_{x=1}^{d} |e_x\rangle \otimes |w_x\rangle ,$ (2)^✦ and $|RQ\rangle\langle RQ| / d$ is the pure state in which $R$ is maximally entangled with the code. The reduced state $\rho^R$ of the reference is the partial trace of $|RQ\rangle\langle RQ|/d$ over $Q_1$ and $Q_2$ , and the reduced state $\rho^{RQ_1}$ of $R$ and $Q_1$ is its partial trace over $Q_2$ alone. The reference is decoupled from $Q_1$ when, for some operator $\rho_1$ on $Q_1$ , $\rho^{RQ_1} = \rho^R \otimes \rho_1 .$ (3)

An operator $A$ on $Q_1$ acts on the composite system as $A \otimes I$ , with the identity on $Q_2$ .^✦ With an orthonormal basis $|j\rangle$ of $Q_1$ , the errors on $Q_1$ are the operators $E_{jk} = |j\rangle\langle k| \otimes I ,$ (4)^✦ one for each pair of basis labels $j, k$ . Every operator acting on $Q_1$ alone is a linear combination of them.

The Claim

Theorem 1. Suppose the reference is decoupled from $Q_1$ , as in (3). Then there are a quantum channel $\mathcal{R}$ on $Q_1 \otimes Q_2$ , acting on every state as $\mathcal{R}(\rho) = \sum_o R_o \rho R_o^\dagger$ with finitely many Kraus operators $R_o$ , and a constant $\lambda \ge 0$ such that every state $\rho$ of the code satisfies $\sum_o R_o \Big( \sum_{j,k} E_{jk} \, \rho \, E_{jk}^\dagger \Big) R_o^\dagger = \lambda \, \rho .$ (5)^✦ These operators $R_o$ satisfy the completeness relation $\sum_o R_o^\dagger R_o = I .$ (6)^✦

What Is Checked

The claim is stated over the composite system $Q_1 \otimes Q_2$ of two quantum systems, its states and the quantum channels on them, the code purification (2) of the code vectors with the reference, and the errors (4).