Bell 01 (|Ψ⁺⟩)

Bell 01 state $|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$ is a maximally entangled two-qubit state where the measurement outcomes of the two qubits are anti-correlated. One of the four Bell states, it is equally likely to yield outcomes 01 or 10 when measured.

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

• Maximally entangled: cannot be factored into independent qubit states
• Antisymmetric under qubit exchange: swapping qubits gives minus the state
• Anti-correlated measurements: if qubit 1 is 0, qubit 2 is 1, and vice versa
• Eigenstate of $Z_1 Z_2$ with eigenvalue -1: $(Z \otimes Z)|\Psi^+\rangle = -|\Psi^+\rangle$
• Eigenstate of $X_1 X_2$ with eigenvalue +1: $(X \otimes X)|\Psi^+\rangle = |\Psi^+\rangle$

Apply CNOT to $(H \otimes I)|01\rangle$:

1. Prepare $|01\rangle$
2. Apply Hadamard to first qubit: $H \otimes I$ gives $\frac{1}{\sqrt{2}}(|01\rangle + |11\rangle)$
3. Apply CNOT with first qubit as control: yields $|\Psi^+\rangle$

• Z basis measurement: always yields anti-correlated outcomes (01 or 10)
• Bell measurement distinguishes $|\Psi^+\rangle$ from other Bell states
• Both outcomes 01 and 10 equally likely: $P(01) = P(10) = 1/2$, $P(00) = P(11) = 0$

• Quantum teleportation: alternative Bell pair for teleporting quantum state
• Superdense coding: sender encodes two classical bits using $|\Psi^+\rangle$
• Bell inequality testing: demonstrates non-locality with anti-correlated measurements