# $\lvert\Phi^+\rangle$ (Phi-plus state)
The **phi-plus state** $\lvert\Phi^+\rangle$ is one of the four [[bell-states|Bell states]] — the maximally entangled two-qubit states. It is an equal superposition of $\lvert 00\rangle$ and $\lvert 11\rangle$ with the same sign, meaning both qubits are always found in the same state when measured in the computational basis. The other $\Phi$ Bell state is [[ket-phi-minus|$\lvert\Phi^-\rangle$]].

$$\lvert\Phi^+\rangle = \frac{1}{\sqrt{2}}(\lvert 00\rangle + \lvert 11\rangle) = \frac{1}{\sqrt{2}}\begin{pmatrix}1\\0\\0\\1\end{pmatrix}$$

It is prepared from $\lvert 00\rangle$ by applying $H\otimes I$ then CX — the simplest Bell state preparation circuit. The other three Bell states are reached from $\lvert\Phi^+\rangle$ by a single Pauli gate on qubit 1: $X_1\lvert\Phi^+\rangle = \lvert\Psi^+\rangle$, $Z_1\lvert\Phi^+\rangle = \lvert\Phi^-\rangle$, $Y_1\lvert\Phi^+\rangle = -i\lvert\Psi^-\rangle$. This one-to-one correspondence underlies the four Pauli corrections in quantum teleportation.

## Qiskit

```python
# Prepare |Φ+⟩ = (|00⟩ + |11⟩)/√2 — H on qubit 0, then CX(0→1).
from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

qc = QuantumCircuit(2)
qc.h(0)      # |00⟩ → (|00⟩ + |10⟩)/√2
qc.cx(0, 1)  # CX: → (|00⟩ + |11⟩)/√2 = |Φ+⟩

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

## Applying gates

Single-qubit gates act on qubit 1. By symmetry of $\lvert\Phi^+\rangle$, applying $X$, $Y$, or $Z$ to qubit 2 gives the same result.

^ Gate ^ Result ^ Comment ^
| [[i-gate]] | $I_1\lvert\Phi^+\rangle = \lvert\Phi^+\rangle$ | Identity leaves the state unchanged. |
| [[x-gate]] | $X_1\lvert\Phi^+\rangle = \lvert\Psi^+\rangle$ | Flips qubit 1; $\lvert 00\rangle\to\lvert 10\rangle$ and $\lvert 11\rangle\to\lvert 01\rangle$, switching from same-value to anti-correlated Z-basis correlations. |
| [[y-gate]] | $Y_1\lvert\Phi^+\rangle = -i\lvert\Psi^-\rangle$ | Bit-flip with phase; the $-i$ on the $\lvert 01\rangle$ term introduces a relative minus sign, giving $\lvert\Psi^-\rangle$ up to global phase. |
| [[z-gate]] | $Z_1\lvert\Phi^+\rangle = \lvert\Phi^-\rangle$ | Negates the $\lvert 11\rangle$ term; same-value Z correlations are preserved but the relative sign flips. |
| [[h-gate]] | $H_1\lvert\Phi^+\rangle$ (not a Bell state) | Bell measurement circuit: CX then $H_1$ maps $\lvert\Phi^+\rangle\to\lvert 00\rangle$. |
| [[s-gate]] | $S_1\lvert\Phi^+\rangle = \tfrac{1}{\sqrt{2}}(\lvert 00\rangle + i\lvert 11\rangle)$ | Adds a phase of $i$ to the $\lvert 11\rangle$ term; not a standard Bell state. |
| [[t-gate]] | $T_1\lvert\Phi^+\rangle = \tfrac{1}{\sqrt{2}}(\lvert 00\rangle + e^{i\pi/4}\lvert 11\rangle)$ | Adds a phase of $e^{i\pi/4}$ to the $\lvert 11\rangle$ term; not a standard Bell state. |
| [[rx-gate]] | Mixes $\lvert\Phi^+\rangle$ with $\lvert\Psi^+\rangle$ | X-type rotations couple the $\lvert\Phi^+\rangle$ and $\lvert\Psi^+\rangle$ Bell states. |
| [[ry-gate]] | Mixes $\lvert\Phi^+\rangle$ with $\lvert\Psi^-\rangle$ | Y-type rotations couple the $\lvert\Phi^+\rangle$ and $\lvert\Psi^-\rangle$ Bell states. |
| [[rz-gate]] | $R_z(\theta)_1\lvert\Phi^+\rangle = \tfrac{e^{-i\theta/2}}{\sqrt{2}}(\lvert 00\rangle + e^{i\theta}\lvert 11\rangle)$ | Modifies relative phase only; at $\theta=\pi$ gives $\lvert\Phi^-\rangle$ up to global phase. |
| [[cx-gate]] | $\text{CX}\lvert\Phi^+\rangle = \lvert +\rangle\lvert 0\rangle$ | Disentangles $\lvert\Phi^+\rangle$ back to the product state used to prepare it. |
| [[swap-gate]] | $\text{SWAP}\lvert\Phi^+\rangle = \lvert\Phi^+\rangle$ | Symmetric under qubit exchange; eigenstate of SWAP with eigenvalue $+1$. |
| [[iswap-gate]] | $\text{iSWAP}\lvert\Phi^+\rangle = \lvert\Phi^+\rangle$ | $\lvert 00\rangle$ and $\lvert 11\rangle$ are unchanged by iSWAP; eigenstate with eigenvalue $+1$. |

## Reaching other Bell states

^ State ^ Gates ^ Comment ^
| $\lvert\Phi^+\rangle$ | $I\lvert\Phi^+\rangle = \lvert\Phi^+\rangle$ | The identity gate leaves $\lvert\Phi^+\rangle$ unchanged. |
| [[ket-phi-minus|$\lvert\Phi^-\rangle$]] | $Z_1\lvert\Phi^+\rangle = \lvert\Phi^-\rangle$ | Z on qubit 1 puts a minus sign on the $\lvert 11\rangle$ term; same-value Z correlations are preserved. |
| [[ket-psi-plus|$\lvert\Psi^+\rangle$]] | $X_1\lvert\Phi^+\rangle = \lvert\Psi^+\rangle$ | X on qubit 1 (or qubit 2) switches from same-value to anti-correlated Z-basis correlations. |
| [[ket-psi-minus|$\lvert\Psi^-\rangle$]] | $Y_1\lvert\Phi^+\rangle = -i\lvert\Psi^-\rangle$ | Y on qubit 1 reaches the antisymmetric singlet state up to global phase $-i$. |