# $\lvert 1\rangle$ (One state)
The **one state** $\lvert 1\rangle$ is one of the two computational basis states of a qubit. It is the quantum analogue of a classical `1` bit. The other computational basis state is [[ket-0|$\lvert 0\rangle$]].

$$\lvert 1\rangle = \begin{pmatrix}0\\1\end{pmatrix}$$

On the [[bloch-sphere]], $\lvert 1\rangle$ corresponds to the south pole at coordinates $(0, 0, -1)$. It is an eigenstate of the Pauli-Z gate with eigenvalue $-1$, meaning $Z\lvert 1\rangle = -\lvert 1\rangle$. Applying the [[h-gate|Hadamard gate]] to $\lvert 1\rangle$ produces the equal superposition state $\lvert -\rangle = (\lvert 0\rangle - \lvert 1\rangle)/\sqrt{2}$.

## Qiskit

```python
# Prepare |1⟩ = X|0⟩ — flip the default |0⟩ with an X gate.
from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

qc = QuantumCircuit(1)
qc.x(0)

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

## Applying gates
^ Gate ^ Result ^ Comment ^
| [[i-gate]] | $I\lvert 1\rangle = \lvert 1\rangle$ | The identity gate leaves $\lvert 1\rangle$ unchanged. |
| [[x-gate]] | $X\lvert 1\rangle = \lvert 0\rangle$ | Flips $\lvert 1\rangle$ to $\lvert 0\rangle$; quantum analogue of classical NOT. |
| [[y-gate]] | $Y\lvert 1\rangle = -i\lvert 0\rangle$ | Bit flip with an imaginary phase factor of $-i$. |
| [[z-gate]] | $Z\lvert 1\rangle = -\lvert 1\rangle$ | $\lvert 1\rangle$ is an eigenstate of $Z$ with eigenvalue $-1$; the minus sign is a global phase in isolation but observable in superpositions. |
| [[h-gate]] | $H\lvert 1\rangle = \lvert -\rangle$ | Rotates the south pole to the $-x$ equatorial point of the Bloch sphere. |
| [[s-gate]] | $S\lvert 1\rangle = i\lvert 1\rangle$ | $S$ adds a phase of $i$ to the $\lvert 1\rangle$ component; acting on $\lvert 1\rangle$ alone this is a global phase. |
| [[t-gate]] | $T\lvert 1\rangle = e^{i\pi/4}\lvert 1\rangle$ | $T$ adds a phase of $e^{i\pi/4}$ to the $\lvert 1\rangle$ component; global phase when acting on $\lvert 1\rangle$ alone. |
| [[rx-gate]] | $R_x(\theta)\lvert 1\rangle = -i\sin\tfrac{\theta}{2}\lvert 0\rangle + \cos\tfrac{\theta}{2}\lvert 1\rangle$ | Tilts the state from $\lvert 1\rangle$ toward $\lvert 0\rangle$ with an imaginary phase on the $\lvert 0\rangle$ component. |
| [[ry-gate]] | $R_y(\theta)\lvert 1\rangle = -\sin\tfrac{\theta}{2}\lvert 0\rangle + \cos\tfrac{\theta}{2}\lvert 1\rangle$ | Real amplitudes; at $\theta=\pi/2$ gives $-\lvert -\rangle$, equivalent to $\lvert -\rangle$ up to global phase. |
| [[rz-gate]] | $R_z(\theta)\lvert 1\rangle = e^{i\theta/2}\lvert 1\rangle$ | Global phase only; $\lvert 1\rangle$ is on the $z$-axis so a $z$-rotation has no observable effect. |
| [[u-gate]] | $U\lvert 1\rangle = -e^{i\lambda}\sin\tfrac{\theta}{2}\lvert 0\rangle + e^{i(\phi+\lambda)}\cos\tfrac{\theta}{2}\lvert 1\rangle$ | Both $\phi$ and $\lambda$ contribute (unlike from $\lvert 0\rangle$ where $\lambda$ drops out). |

## Reaching other states
^ State ^ Gates ^ Comment ^
| [[ket-0|$\lvert 0\rangle$]] | $X\lvert 1\rangle = \lvert 0\rangle$ | X flips the qubit; the quantum analogue of classical NOT. $X$ is its own inverse: $X^2 = I$. |
| $\lvert 1\rangle$ | $I\lvert 1\rangle = \lvert 1\rangle$ | The identity gate leaves $\lvert 1\rangle$ unchanged. |
| [[ket-plus|$\lvert +\rangle$]] | $ZH\lvert 1\rangle = \lvert +\rangle$ | H first produces $\lvert -\rangle$, then Z flips its relative phase to give $\tfrac{1}{\sqrt{2}}(\lvert 0\rangle + \lvert 1\rangle)$. |
| [[ket-minus|$\lvert -\rangle$]] | $H\lvert 1\rangle = \lvert -\rangle$ | Hadamard rotates the south pole to the $-x$ equatorial point, creating the equal superposition $\tfrac{1}{\sqrt{2}}(\lvert 0\rangle - \lvert 1\rangle)$. |
| [[ket-plus-i|$\lvert +i\rangle$]] | $S^\dagger H\lvert 1\rangle = \lvert +i\rangle$ | H rotates to the $-x$ equatorial point, then $S^\dagger$ rotates $-90°$ around $z$ to land on the $+y$ pole. |
| [[ket-minus-i|$\lvert -i\rangle$]] | $SH\lvert 1\rangle = \lvert -i\rangle$ | H rotates to the $-x$ equatorial point, then S rotates $90°$ around $z$ to land on the $-y$ pole. |