# Computational Basis States

**Computational basis states** $|0\rangle$ and $|1\rangle$ are the two orthogonal eigenstates of the Pauli Z operator. They represent the classical outcomes 0 and 1 and form the default measurement basis on most quantum computers.

## Overview

- **[[quantum-state-0|Zero state (|0⟩)]]**: eigenstate with $Z|0\rangle = |0\rangle$ (eigenvalue +1)
- **[[quantum-state-1|One state (|1⟩)]]**: eigenstate with $Z|1\rangle = -|1\rangle$ (eigenvalue -1)

## Properties

Both basis states are:
- Orthonormal: $\langle 0|1\rangle = 0$, $\langle 0|0\rangle = \langle 1|1\rangle = 1$
- Pure states: $\rho = |i\rangle\langle i|$ for $i \in \{0, 1\}$
- Eigenstates of Z: measurement in Z basis always yields definite outcome

## Role in Quantum Computing

- **Initial state**: qubits default to $|0\rangle$ on most platforms
- **Measurement outcome**: quantum states collapse to $|0\rangle$ or $|1\rangle$ when measured in the computational basis
- **Classical simulation**: any state without superposition of computational basis states is trivially classical
- **Basis decomposition**: any $n$-qubit state is $|\psi\rangle = \sum_{x \in \{0,1\}^n} \alpha_x |x\rangle$

## Superposition and Entanglement

The computational basis is the platform for understanding superposition:
- Single-qubit superposition: $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$
- Multi-qubit superposition: $|\psi\rangle = \sum_x \alpha_x |x_1 x_2 \ldots x_n\rangle$
- Entanglement: multi-qubit states that cannot factor into products of computational basis eigenstates

## Basis Choice

Measurement in the computational basis is the most direct; measurement in other bases (X or Y) requires rotating the state first via single-qubit gates before measurement.