Trivial Two-Qubit Example

A quantum version of the two bit example from the previous section consists of two physical qubits, where the errors randomly apply the identity or one of the Pauli operators to the first qubit. The Pauli operators are defined by

$$\begin{array}{rcl} {\mathchoice {\rm 1\mskip-4mu l} {\rm 1\m... ...left(\begin{array}{cc}1&0\\ 0&-1\end{array}\right). \end{array}$$ (9)

Explicitly, the errors have the effect

$$\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{{\color... ...!\!{\mathsf {12}}}} & \mbox{Prob. $.25$} \end{array} \right.,$$ (10)

where the superscripts in parentheses specify the qubit that an operator acts on. This error model is called ``completely depolarizing'' on qubit $\mathsf {1}$. Obviously, a one-qubit state can be stored in the second physical qubit without being affected by the errors. An encoding operation that implements this observation is

$$\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi}\m... ...3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {2}}}},$$ (11)

which realizes an ideal qubit as a two-dimensional subspace of the physical qubits. This subspace is the ``quantum code'' for this encoding. To decode one can discard physical qubit $\mathsf {1}$ and return qubit $\mathsf {2}$, which is considered a natural subsystem of the physical system. In this case, the identification of syndrome and information-carrying subsystems is the obvious one associated with the two physical qubits.