# $\lvert +\rangle$
**Plus state** $\lvert +\rangle$ is a quantum state that is an equal superposition of the two computational basis states $\lvert 0\rangle$ and $\lvert 1\rangle$ with equal positive amplitudes. It is one of the six cardinal states on the [[bloch-sphere]], sitting at the positive $x$-axis.

$$\lvert +\rangle = \frac{1}{\sqrt{2}}\lvert 0\rangle + \frac{1}{\sqrt{2}}\lvert 1\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}1\\1\end{pmatrix}$$

When measured in the computational basis $\{\lvert 0\rangle, \lvert 1\rangle\}$, the plus state collapses to $\lvert 0\rangle$ or $\lvert 1\rangle$ with equal probability $1/2$ each, by the [[born-rule]]. The plus state is an eigenstate of the Pauli-X gate with eigenvalue $+1$, meaning $X\lvert +\rangle = \lvert +\rangle$.

The plus state is obtained by applying the Hadamard gate $H$ to $\lvert 0\rangle$.

$$\lvert +\rangle = H\lvert 0\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1\\ 1 & -1\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix} = \frac{1}{\sqrt{2}}\begin{pmatrix}1\\1\end{pmatrix}$$

Together with the [[minus-state]], it forms the Hadamard basis $\{\lvert +\rangle, \lvert -\rangle\}$, which is the eigenbasis of the Pauli-X operator.