# 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.

## Definition

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

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

## Properties

- **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

## Role in Quantum Computing

- **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)

## Construction

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

## Relation to Other States

- Single-qubit basis: first qubit is [[quantum-state-0|zero state]], second is [[quantum-state-1|one state]]
- Symmetric partner: [[quantum-state-computational-10|Computational 10]] (reversed roles)
- No excitation: [[quantum-state-computational-00|Computational 00]]
- Both excited: [[quantum-state-computational-11|Computational 11]]
- Bell states: |01⟩ is a component of [[quantum-state-bell-01|Bell 01 (|Ψ⁺⟩)]] and [[quantum-state-bell-10|Bell 10 (|Ψ⁻⟩)]]

## Measurement

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