# 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.

## Definition

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

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

## Properties

- **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

## Role in Quantum Computing

- **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⟩

## Construction

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

## Relation to Other States

- Single-qubit: tensor product of two [[quantum-state-1|one states]]
- One-qubit excitation: [[quantum-state-computational-01|Computational 01]] or [[quantum-state-computational-10|Computational 10]]
- No excitation: [[quantum-state-computational-00|Computational 00]]
- Bell states: |11⟩ is a component of [[quantum-state-bell-00|Bell 00 (|Φ⁺⟩)]] and [[quantum-state-bell-11|Bell 11 (|Φ⁻⟩)]]

## Measurement

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