# Bell 10 (|Ψ⁻⟩)

**Bell 10 state** $|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)$ is a maximally entangled two-qubit state with anti-correlated measurements and a relative minus phase. One of the four [[quantum-state-bell|Bell states]].

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

## Properties

- Maximally entangled: cannot be factored into independent qubit states
- Antisymmetric under qubit exchange with relative phase
- Anti-correlated measurements: yields outcomes 01 or 10 when measured in Z basis
- 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: $(X \otimes X)|\Psi^-\rangle = -|\Psi^-\rangle$

## Creation

Apply CNOT after preparing $\frac{1}{\sqrt{2}}(|01\rangle - |11\rangle)$:
1. Apply Z to first qubit, then H: $(H Z) \otimes I$ on $|01\rangle$
2. Apply CNOT with first qubit as control
3. Yields $|\Psi^-\rangle$

Alternatively: $(X \otimes I) |\Psi^+\rangle$ converts the $|\Psi^+\rangle$ Bell state to $|\Psi^-\rangle$.

## Measurement

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

## Relation to Other Bell States

All four Bell states are related by single-qubit rotations:
- $|\Phi^+\rangle$ and $|\Phi^-\rangle$ are correlated (same outcomes)
- $|\Psi^+\rangle$ and $|\Psi^-\rangle$ are anti-correlated (opposite outcomes)
- Relative phases distinguish the four states