Computational 11 (|11⟩)

Computational 11 state |11⟩ is the product of two excited state qubits. It is a separable (unentangled) state and an eigenstate of both Z₁ and Z₂ operators with eigenvalue -1.

$$|11\rangle = |1\rangle_1 \otimes |1\rangle_2 = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}$$

in the computational basis ordering (|00⟩, |01⟩, |10⟩, |11⟩).

• Separable: factorizes into independent single-qubit states
• Product state: no entanglement between qubits
• Z eigenstates: Z₁|11⟩ = -|11⟩, Z₂|11⟩ = -|11⟩ (both with eigenvalue -1)
• Pure state: purity = 1, zero entropy
• Excited state: both qubits in their highest energy level

• Fully excited: represents both qubits in their excited state
• Measurement outcome: measured in Z basis yields 11 with certainty
• Basis state: building block for superpositions and entangled states
• Endpoint: result of applying X gates to both qubits starting from |00⟩

Prepare by:

1. Initialize both qubits to ground state |0⟩
2. Apply X (Pauli-X) gate to both qubits
3. Measure both qubits in Z basis: outcomes are always 1, 1

• Single-qubit: tensor product of two one states
• One-qubit excitation: Computational 01 or Computational 10
• No excitation: Computational 00
• Bell states: |11⟩ is a component of Bell 00 (|Φ⁺⟩) and Bell 11 (|Φ⁻⟩)

Measuring |11⟩ in the computational (Z) basis always yields outcome 11. Measuring in other bases (X or Y) requires rotating both qubits first.