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.

• Computational 00 (|00⟩): $|00\rangle$
  • Both qubits in ground state
  • Z eigenvalue: +1 on both qubits
  • Default initial state
• Computational 01 (|01⟩): $|01\rangle$
  • First qubit ground, second excited
  • Z eigenvalue: +1 on first, -1 on second
• Computational 10 (|10⟩): $|10\rangle$
  • First qubit excited, second ground
  • Z eigenvalue: -1 on first, +1 on second
• Computational 11 (|11⟩): $|11\rangle$
  • Both qubits in excited state
  • Z eigenvalue: -1 on both qubits

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

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.

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 single-qubit computational basis for the component states.

Superpositions of computational states create quantum phenomena:

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

• 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

Bell states are entangled superpositions of computational states:

• $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$: Bell 00
• $|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)$: Bell 11
• $|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$: Bell 01
• $|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)$: Bell 10

• 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