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Table of Contents
Canonical states
Canonical states are the six single-qubit states corresponding to the six cardinal points of the Bloch sphere — the $\pm z$, $\pm x$, and $\pm y$ poles. They are the eigenstates of the three Pauli operators $Z$, $X$, $Y$ and are the most frequently encountered states in single-qubit quantum computing.
The $\pm z$ poles are the computational basis states $\lvert 0\rangle$ and $\lvert 1\rangle$:
$$\lvert 0\rangle = \begin{pmatrix}1\\0\end{pmatrix} \qquad \lvert 1\rangle = \begin{pmatrix}0\\1\end{pmatrix}$$
The $\pm x$ equatorial states $\lvert +\rangle$ and $\lvert -\rangle$ are equal superpositions that differ only in the relative sign of the $\lvert 1\rangle$ amplitude:
$$\lvert +\rangle = \frac{1}{\sqrt{2}}(\lvert 0\rangle + \lvert 1\rangle) \qquad \lvert -\rangle = \frac{1}{\sqrt{2}}(\lvert 0\rangle - \lvert 1\rangle)$$
The $\pm y$ equatorial states $\lvert +i\rangle$ and $\lvert -i\rangle$ are equal superpositions with a complex relative phase of $\pm i$:
$$\lvert +i\rangle = \frac{1}{\sqrt{2}}(\lvert 0\rangle + i\lvert 1\rangle) \qquad \lvert -i\rangle = \frac{1}{\sqrt{2}}(\lvert 0\rangle - i\lvert 1\rangle)$$
Pauli eigenstates
Each pair is the $+1$ and $-1$ eigenstate of one Pauli operator:
| State | Pauli | Eigenvalue |
| $\lvert 0\rangle$ | $Z$ | $+1$ |
| $\lvert 1\rangle$ | $Z$ | $-1$ |
| $\lvert +\rangle$ | $X$ | $+1$ |
| $\lvert -\rangle$ | $X$ | $-1$ |
| $\lvert +i\rangle$ | $Y$ | $+1$ |
| $\lvert -i\rangle$ | $Y$ | $-1$ |
Measuring in the $Z$ basis is the default. To measure in the $X$ basis, apply $H$ before measurement; to measure in the $Y$ basis, apply $S^\dagger H$.
Hadamard connections
The Hadamard gate exchanges the $Z$ and $X$ eigenstates:
$$H\lvert 0\rangle = \lvert +\rangle \qquad H\lvert 1\rangle = \lvert -\rangle \qquad H\lvert +\rangle = \lvert 0\rangle \qquad H\lvert -\rangle = \lvert 1\rangle$$
The $Y$ eigenstates require an additional phase gate: $HS^\dagger\lvert +i\rangle = \lvert 0\rangle$ and $HS^\dagger\lvert -i\rangle = \lvert 1\rangle$.