Computational 01 (|01⟩)

Computational 01 state |01⟩ is a product state with the first qubit in ground state and the second qubit in excited state. It is separable (unentangled) and an eigenstate of both Z₁ and Z₂ operators with eigenvalues +1 and -1 respectively.

$$|01\rangle = |0\rangle_1 \otimes |1\rangle_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}$$

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|01⟩ = |01⟩ (eigenvalue +1), Z₂|01⟩ = -|01⟩ (eigenvalue -1)
• Pure state: purity = 1, zero entropy
• Asymmetric: first qubit ground, second qubit excited

• Mixed excitation: one qubit in excited state, one in ground
• Measurement outcome: measuring in Z basis yields 01 with certainty
• Basis state: component of many superpositions and entangled states
• Asymmetric operations: distinguishes between qubit roles (control vs. target in gates)

Prepare by:

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

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

Measuring |01⟩ in the computational (Z) basis always yields outcome 01. Measuring either qubit in X or Y basis requires rotation before measurement.