# 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.

## Definition

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

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

## Properties

- **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

## Role in Quantum Computing

- **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

## Construction

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

## Relation to Other States

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

## Measurement

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