# Bell states
**Bell states** are the four maximally entangled two-qubit states, named after physicist John Bell. They form an orthonormal basis for the two-qubit Hilbert space $\mathbb{C}^4$ and are the canonical examples of quantum entanglement. No product state can approximate them without a dramatic increase in entropy.

$$\lvert\Phi^+\rangle = \frac{1}{\sqrt{2}}(\lvert 00\rangle + \lvert 11\rangle) \qquad \lvert\Phi^-\rangle = \frac{1}{\sqrt{2}}(\lvert 00\rangle - \lvert 11\rangle)$$
$$\lvert\Psi^+\rangle = \frac{1}{\sqrt{2}}(\lvert 01\rangle + \lvert 10\rangle) \qquad \lvert\Psi^-\rangle = \frac{1}{\sqrt{2}}(\lvert 01\rangle - \lvert 10\rangle)$$

The $\Phi$ states have same-value Z-basis correlations — measuring either qubit gives the same bit on the other. The $\Psi$ states have opposite-value Z-basis correlations. The sign ($+$ or $-$) distinguishes whether the X-basis or Y-basis correlations are same-value or anti-correlated.

## Preparation

All four Bell states are prepared from computational basis states using a Hadamard gate on the first qubit followed by a CX:

$$\lvert 00\rangle \xrightarrow{H\otimes I,\;\text{CX}} \lvert\Phi^+\rangle \qquad \lvert 01\rangle \xrightarrow{H\otimes I,\;\text{CX}} \lvert\Psi^+\rangle$$
$$\lvert 10\rangle \xrightarrow{H\otimes I,\;\text{CX}} \lvert\Phi^-\rangle \qquad \lvert 11\rangle \xrightarrow{H\otimes I,\;\text{CX}} \lvert\Psi^-\rangle$$

The reverse circuit — CX then $H\otimes I$ — is the Bell measurement. It maps each Bell state back to a unique two-bit string in the computational basis.

## Navigation between Bell states

All four Bell states are connected by single Pauli gates on one qubit. Using $\lvert\Phi^+\rangle$ as the starting point:

$$Z_1\lvert\Phi^+\rangle = \lvert\Phi^-\rangle \qquad X_1\lvert\Phi^+\rangle = \lvert\Psi^+\rangle \qquad iY_1\lvert\Phi^+\rangle = \lvert\Psi^-\rangle$$

This one-to-one mapping between Bell states and Pauli operations is the core mechanism behind quantum teleportation: Alice performs a Bell measurement and sends a 2-bit classical result to Bob, who applies the corresponding Pauli correction.

## Correlation structure

Each Bell state has a definite same-value or anti-value correlation in each of the X, Y, Z measurement bases:

| State | Z-basis | X-basis | Y-basis |
|-------|---------|---------|---------|
| $\lvert\Phi^+\rangle$ | same | same | anti |
| $\lvert\Phi^-\rangle$ | same | anti | same |
| $\lvert\Psi^+\rangle$ | anti | same | same |
| $\lvert\Psi^-\rangle$ | anti | anti | anti |

$\lvert\Psi^-\rangle$ is the singlet state — rotationally invariant and anti-correlated in every measurement basis. It is the unique state (up to global phase) with SWAP eigenvalue $-1$; the other three Bell states are SWAP eigenstates with eigenvalue $+1$.

## Symmetry and iSWAP

The Bell states are also eigenstates of iSWAP:
$$\text{iSWAP}\lvert\Phi^\pm\rangle = \lvert\Phi^\pm\rangle \qquad \text{iSWAP}\lvert\Psi^+\rangle = i\lvert\Psi^+\rangle \qquad \text{iSWAP}\lvert\Psi^-\rangle = -i\lvert\Psi^-\rangle$$

## Individual Bell state articles

 - [[ket-phi-plus|$\lvert\Phi^+\rangle$ (Phi-plus state)]]
 - [[ket-phi-minus|$\lvert\Phi^-\rangle$ (Phi-minus state)]]
 - [[ket-psi-plus|$\lvert\Psi^+\rangle$ (Psi-plus state)]]
 - [[ket-psi-minus|$\lvert\Psi^-\rangle$ (Psi-minus state)]]