# $\lvert -\rangle$
**Minus 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 opposite signs. It is one of the six cardinal states on the [[bloch-sphere]], sitting at the negative $x$-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 minus state collapses to $\lvert 0\rangle$ or $\lvert 1\rangle$ with equal probability $1/2$ each, by the [[born-rule]]. The minus state is an eigenstate of the Pauli-X gate with eigenvalue $-1$, meaning $X\lvert -\rangle = -\lvert -\rangle$.

The minus state is obtained by applying the Hadamard gate $H$ to $\lvert 1\rangle$.

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

The minus state plays an important role in quantum algorithms as an ancilla. When used as the second qubit in a CNOT-based oracle (initialized to $\lvert -\rangle$), phase kickback writes the oracle's output as a phase on the first register rather than flipping a bit.