Bell 00 (|Φ⁺⟩)

Bell 00 state $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ is the maximally entangled two-qubit state where both qubits have perfectly correlated measurement outcomes. It cannot be written as a product of single-qubit states and is one of the four Bell states.

Representation: $|\Phi^+\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix}$ (in the basis $|00\rangle, |01\rangle, |10\rangle, |11\rangle$)

• Maximally entangled: cannot be factored into independent qubit states
• Equal superposition: each basis state $|00\rangle$ and $|11\rangle$ has amplitude $1/\sqrt{2}$
• Measurement correlation: measuring qubit 1 determines qubit 2's outcome (always the same)
• Symmetric: $|\Phi^+\rangle$ is unchanged by swapping qubits 1 and 2
• +1 eigenstate of $Z_1 Z_2$ (product of Z operators): $(Z \otimes Z)|\Phi^+\rangle = |\Phi^+\rangle$

Apply Hadamard to first qubit, then CNOT with first qubit as control:

$$\text{CNOT}(H \otimes I)|00\rangle = |\Phi^+\rangle$$

Alternatively: $H \otimes I$ applied to $|0\rangle$ gives $(|0\rangle + |1\rangle)/\sqrt{2} \otimes |0\rangle$, then CNOT creates the correlation.

• Measuring qubit 1 in Z basis: gives 0 or 1 with equal probability
• Measuring qubit 2 in Z basis: always matches qubit 1's outcome (0+0 or 1+1)
• Both outcomes equally likely: $P(00) = P(11) = 1/2$, $P(01) = P(10) = 0$

• Quantum teleportation: a Bell pair (Φ⁺) is shared between sender and receiver
• Quantum superdense coding: sender encodes two classical bits into one shared qubit using Φ⁺
• Entanglement swapping: combining Bell states extends entanglement across larger networks
• Quantum error correction: Bell measurements extract syndrome information