# $\lvert -\rangle$ (Minus state)
The **minus state** $\lvert -\rangle$ is an equal superposition of $\lvert 0\rangle$ and $\lvert 1\rangle$ with a relative minus sign. It is one of the six cardinal states on the [[bloch-sphere]], sitting at the negative $x$-axis at coordinates $(-1, 0, 0)$.

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

On the [[bloch-sphere]], $\lvert -\rangle$ is an eigenstate of the Pauli-X gate with eigenvalue $-1$, meaning $X\lvert -\rangle = -\lvert -\rangle$. It is prepared from $\lvert 0\rangle$ by $ZH\lvert 0\rangle$, or equivalently from $\lvert 1\rangle$ by $H\lvert 1\rangle = \lvert -\rangle$. Because $X\lvert -\rangle = -\lvert -\rangle$, it is the standard ancilla qubit for phase kickback: a CX with $\lvert -\rangle$ as target writes the oracle's output as a phase on the control rather than flipping the target.

## Qiskit

```python
# Prepare |−⟩ = ZH|0⟩ — Hadamard then Z to flip the relative phase, or equivalently H|1⟩.
from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

qc = QuantumCircuit(1)
qc.h(0)
qc.z(0)

print(Statevector(qc).data)
```

## Applying gates
^ Gate ^ Result ^ Comment ^
| [[i-gate]] | $I\lvert -\rangle = \lvert -\rangle$ | The identity gate leaves $\lvert -\rangle$ unchanged. |
| [[x-gate]] | $X\lvert -\rangle = -\lvert -\rangle$ | $\lvert -\rangle$ is an eigenstate of $X$ with eigenvalue $-1$; the state acquires a global minus sign, unobservable in isolation. |
| [[y-gate]] | $Y\lvert -\rangle = i\lvert +\rangle$ | Swaps and phase-shifts; the result is $\lvert +\rangle$ up to global phase $i$. |
| [[z-gate]] | $Z\lvert -\rangle = \lvert +\rangle$ | Z negates the $\lvert 1\rangle$ component, turning $-1/\sqrt{2}$ into $+1/\sqrt{2}$ and flipping the $-x$ pole to the $+x$ pole. |
| [[h-gate]] | $H\lvert -\rangle = \lvert 1\rangle$ | The amplitudes cancel for $\lvert 0\rangle$ (destructive) and add for $\lvert 1\rangle$ (constructive); collapses to the south pole. |
| [[s-gate]] | $S\lvert -\rangle = \lvert -i\rangle$ | Rotates $+90°$ around $z$, taking the $-x$ pole to the $-y$ pole. |
| [[t-gate]] | $T\lvert -\rangle = \tfrac{1}{\sqrt{2}}(1,\, -e^{i\pi/4})^T$ | Adds $\pi/4$ to the azimuthal angle; lands midway between $\lvert -\rangle$ and $\lvert -i\rangle$ at $225°$. |
| [[rx-gate]] | $R_x(\theta)\lvert -\rangle = e^{i\theta/2}\lvert -\rangle$ | Global phase only; $\lvert -\rangle$ is on the $-x$ axis so an $x$-rotation has no observable effect. |
| [[ry-gate]] | $R_y(\theta)\lvert -\rangle = \tfrac{1}{\sqrt{2}}(\cos\tfrac{\theta}{2} + \sin\tfrac{\theta}{2},\; \sin\tfrac{\theta}{2} - \cos\tfrac{\theta}{2})^T$ | Tilts off the $-x$ pole toward the poles; at $\theta=\pi/2$ gives $\lvert 0\rangle$, at $\theta=-\pi/2$ gives $-\lvert 1\rangle$ (equivalent to $\lvert 1\rangle$). |
| [[rz-gate]] | $R_z(\theta)\lvert -\rangle = \tfrac{e^{-i\theta/2}}{\sqrt{2}}(1,\, -e^{i\theta})^T$ | Sweeps the azimuthal angle; at $\theta=\pi/2$ gives $\lvert -i\rangle$, at $\theta=\pi$ gives $\lvert +\rangle$, at $\theta=3\pi/2$ gives $\lvert +i\rangle$. |
| [[u-gate]] | general rotation from $-x$ pole | Both $\phi$ and $\lambda$ contribute at all angles. |

## Reaching other states
^ State ^ Gates ^ Comment ^
| [[ket-0|$\lvert 0\rangle$]] | $R_y(\pi/2)\lvert -\rangle = \lvert 0\rangle$ | Tilts the $-x$ equatorial point to the north pole with a real-amplitude rotation. |
| [[ket-1|$\lvert 1\rangle$]] | $H\lvert -\rangle = \lvert 1\rangle$ | Hadamard maps the $-x$ pole to the south pole; destructive interference on $\lvert 0\rangle$, constructive on $\lvert 1\rangle$. |
| [[ket-plus|$\lvert +\rangle$]] | $Z\lvert -\rangle = \lvert +\rangle$ | Z flips the $\lvert 1\rangle$ phase, taking the $-x$ pole to the $+x$ pole. |
| $\lvert -\rangle$ | $I\lvert -\rangle = \lvert -\rangle$ | The identity gate leaves $\lvert -\rangle$ unchanged. |
| [[ket-plus-i|$\lvert +i\rangle$]] | $S^\dagger\lvert -\rangle = \lvert +i\rangle$ | $S^\dagger$ rotates $-90°$ around $z$, moving from the $-x$ pole to the $+y$ pole. |
| [[ket-minus-i|$\lvert -i\rangle$]] | $S\lvert -\rangle = \lvert -i\rangle$ | S rotates $+90°$ around $z$, moving from the $-x$ pole to the $-y$ pole. |