# Two-qubit computational basis states

**Two-qubit computational basis states** are the four orthonormal product states |00⟩, |01⟩, |10⟩, |11⟩ that form the default measurement basis on quantum computers. They represent definite, unentangled configurations of two qubits and are eigenstates of the Z operator on both qubits.

## The Four Computational States

- **[[quantum-state-computational-00|Computational 00 (|00⟩)]]**: $|00\rangle$
   - Both qubits in ground state
   - Z eigenvalue: +1 on both qubits
   - Default initial state

- **[[quantum-state-computational-01|Computational 01 (|01⟩)]]**: $|01\rangle$
   - First qubit ground, second excited
   - Z eigenvalue: +1 on first, -1 on second

- **[[quantum-state-computational-10|Computational 10 (|10⟩)]]**: $|10\rangle$
   - First qubit excited, second ground
   - Z eigenvalue: -1 on first, +1 on second

- **[[quantum-state-computational-11|Computational 11 (|11⟩)]]**: $|11\rangle$
   - Both qubits in excited state
   - Z eigenvalue: -1 on both qubits

## Properties

All four computational states are:

- **Separable**: factorize into single-qubit products (no entanglement)
- **Eigenstates of Z**: measurement in Z basis always yields definite outcome
- **Orthonormal**: $\langle ij | kl \rangle = \delta_{ik}\delta_{jl}$ for $i,j,k,l \in \{0,1\}$
- **Pure states**: density matrix is rank-1 projector
- **Complete basis**: span the entire two-qubit Hilbert space

## Measurement

Measuring any computational state in the Z basis yields a definite outcome (00, 01, 10, or 11) with probability 1. Measuring in other bases (X or Y) requires rotating both qubits before measurement.

## Composition from Single-Qubit Basis

Each two-qubit computational state is a tensor product of single-qubit states:

- $|00\rangle = |0\rangle \otimes |0\rangle$
- $|01\rangle = |0\rangle \otimes |1\rangle$
- $|10\rangle = |1\rangle \otimes |0\rangle$
- $|11\rangle = |1\rangle \otimes |1\rangle$

See [[quantum-state-computational|single-qubit computational basis]] for the component states.

## Superpositions and Entanglement

Superpositions of computational states create quantum phenomena:

- Equal superposition of all four states: $\frac{1}{2}(|00\rangle + |01\rangle + |10\rangle + |11\rangle)$
- [[quantum-state-bell|Bell states]] are maximally entangled superpositions of computational states
- Entanglement enables quantum speedup in algorithms

## Role in Quantum Computing

- **Initial state**: qubits default to $|0\rangle$ (computational 00) at startup
- **Measurement outcomes**: classical results of quantum computation
- **Algorithm initialization**: starting point before superposition is created
- **Reset operation**: returns qubits to computational 00 between circuit runs
- **Error basis**: single bit-flip errors swap between computational states

## Relation to Bell States

Bell states are entangled superpositions of computational states:

- $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ — [[quantum-state-bell-00|Bell 00]]
- $|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)$ — [[quantum-state-bell-11|Bell 11]]
- $|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$ — [[quantum-state-bell-01|Bell 01]]
- $|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)$ — [[quantum-state-bell-10|Bell 10]]

## Applications

- **State initialization**: starting point for quantum algorithms
- **Error correction**: detecting bit-flip errors via syndrome measurement
- **State preparation**: intermediate steps in quantum state engineering
- **Classical processing**: results of quantum computation read as binary strings
- **Quantum simulation**: basis for representing system states on quantum hardware