Computational 10 (|10⟩)

Computational 10 state |10⟩ is a product state with the first qubit in excited state and the second qubit in ground state. It is separable (unentangled) and an eigenstate of both Z₁ and Z₂ operators with eigenvalues -1 and +1 respectively.

$$|10\rangle = |1\rangle_1 \otimes |0\rangle_2 = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}$$

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

• Separable: factorizes into independent single-qubit states
• Product state: no entanglement between qubits
• Z eigenstate: Z₁|10⟩ = -|10⟩ (eigenvalue -1), Z₂|10⟩ = |10⟩ (eigenvalue +1)
• Pure state: purity = 1, zero entropy
• Asymmetric: first qubit excited, second qubit ground

• Mixed excitation: one qubit in excited state, one in ground
• Measurement outcome: measuring in Z basis yields 10 with certainty
• Basis state: component of many superpositions and entangled states
• Asymmetric operations: distinguishes between qubit roles in controlled gates

Prepare by:

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

• Single-qubit basis: first qubit is one state, second is zero state
• Symmetric partner: Computational 01 (reversed roles)
• No excitation: Computational 00
• Both excited: Computational 11
• Bell states: |10⟩ is a component of Bell 01 (|Ψ⁺⟩) and Bell 10 (|Ψ⁻⟩)

Measuring |10⟩ in the computational (Z) basis always yields outcome 10. Measuring either qubit in X or Y basis requires rotation before measurement.