Bell 11 (|Φ⁻⟩)

Bell 11 state $|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)$ is a maximally entangled two-qubit state with perfectly correlated measurements and a relative minus phase. 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
• Symmetric under qubit exchange with relative phase
• Correlated measurements: yields outcomes 00 or 11 when measured in Z basis
• Eigenstate of $Z_1 Z_2$ with eigenvalue +1: $(Z \otimes Z)|\Phi^-\rangle = |\Phi^-\rangle$
• Eigenstate of $X_1 X_2$ with eigenvalue -1: $(X \otimes X)|\Phi^-\rangle = -|\Phi^-\rangle$

Apply Z to second qubit then create Φ⁺:

1. Apply $|01\rangle$ or $|10\rangle$ then apply Z to one qubit to induce phase
2. Or: $(I \otimes Z) |\Phi^+\rangle$ converts the $|\Phi^+\rangle$ Bell state to $|\Phi^-\rangle$
3. Or: $(H \otimes I)$ applied to $|11\rangle$, then CNOT

• Z basis measurement: yields correlated outcomes (00 or 11)
• Both outcomes equally likely: $P(00) = P(11) = 1/2$, $P(01) = P(10) = 0$
• Distinguishable from $|\Phi^+\rangle$ via relative phase

The four Bell states differ in symmetry and measurement outcomes:

• $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ — correlated, symmetric, $Z_1 Z_2 = +1$
• $|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)$ — correlated, symmetric, $Z_1 Z_2 = +1$
• $|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$ — anti-correlated, antisymmetric, $Z_1 Z_2 = -1$
• $|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)$ — anti-correlated, antisymmetric, $Z_1 Z_2 = -1$

Bell measurements distinguish all four states and are central to quantum key distribution and quantum error correction.