Table of Contents
$\lvert -\rangle$ (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
# 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. |
| Hadamard 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 $225°$. |
| Rotation-X 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. |
| Rotation-Y 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 $\lvert 1\rangle$). |
| Rotation-Z 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 $\lvert +i\rangle$. |
| Unitary gate | general rotation from $-x$ pole | Both $\phi$ and $\lambda$ contribute at all angles. |
Reaching other states
| State | Gates | Comment |
|---|---|---|
| $\lvert 0\rangle$ | $R_y(\pi/2)\lvert -\rangle = \lvert 0\rangle$ | Tilts the $-x$ equatorial point to the north pole with a real-amplitude rotation. |
| $\lvert 1\rangle$ | $H\lvert -\rangle = \lvert 1\rangle$ | Hadamard maps the $-x$ pole to the south pole; destructive interference on $\lvert 0\rangle$, constructive on $\lvert 1\rangle$. |
| $\lvert +\rangle$ | $Z\lvert -\rangle = \lvert +\rangle$ | 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. |
| $\lvert +i\rangle$ | $S^\dagger\lvert -\rangle = \lvert +i\rangle$ | $S^\dagger$ rotates $-90°$ around $z$, moving from the $-x$ pole to the $+y$ pole. |
| $\lvert -i\rangle$ | $S\lvert -\rangle = \lvert -i\rangle$ | S rotates $+90°$ around $z$, moving from the $-x$ pole to the $-y$ pole. |