# Quantum codes#

We proceed as before by defining the message space, the codespace and finally the code. The quantum message space consists of states of qubits.

For a quantum code, the

message spaceis the state space of $k$-qubits, i.e. the $2^k$-dimensional Hilbert space $\vecs{H}_m$. Any $\ket{\psi} \in \vecs{H}^m$ is a valid message.

The quantum codespace is constructed from the state of $n$-qubits.

For a quantum code, the

codespaceis the state space of $n$-qubits, i.e. the $2^n$-dimensional Hilbert space $\vecs{H}^c$.

Finally, we can define the code.

A

qubit quantum code$\vecs{C}$ is a $2^k$-dimensional subspace of the codespace $\vecs{H}^c$. Such a code is labelled as $[[n,k]]$.

Elements of the code are codewords, and if we choses a basis for the code, then the basis elements are also called basis codewords.