Zero State (|0⟩)

Zero state (or ground state) $|0\rangle$ is the computational basis state corresponding to a classical bit value 0. It is the standard initial state of qubits in most quantum computers and one of the two eigenstates of the Pauli Z operator.

Representation: $|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$

• Eigenstate of Z with eigenvalue +1: $Z|0\rangle = |0\rangle$
• Orthogonal to $|1\rangle$: $\langle 0|1\rangle = 0$
• Normalized: $\langle 0|0\rangle = 1$
• Pure state with density matrix $\rho = |0\rangle\langle 0| = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$

On the Bloch sphere, the zero state is at the north pole (pointing up along the z-axis). It represents maximum certainty in the Z measurement.

• Initial state: qubits are prepared in $|0\rangle$ by default in most platforms
• Measurement: measuring a qubit that is definitively in $|0\rangle$ always yields outcome 0
• From superposition: applying $R_X(\pi)$ to $|1\rangle$ produces $|0\rangle$ (X gate flips $|1\rangle$ to $|0\rangle$)

• Superposition with $|1\rangle$: $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$
• Bell state: $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ has $|0\rangle|0\rangle$ component