One State (|1⟩)

One state (or excited state) $|1\rangle$ is the computational basis state corresponding to a classical bit value 1. It is the second basis state and one of the two eigenstates of the Pauli Z operator.

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

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

On the Bloch sphere, the one state is at the south pole (pointing down along the negative z-axis). It represents maximum certainty in the Z measurement toward the -1 eigenvalue.

• Preparation: apply the Pauli X gate to $|0\rangle$ to flip to $|1\rangle$
• Measurement: measuring a qubit that is definitively in $|1\rangle$ always yields outcome 1
• From superposition: preparing $|1\rangle$ from superposition requires measuring and conditioning on outcome 1

• Superposition with $|0\rangle$: $|-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$
• Bell state: $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ has $|1\rangle|1\rangle$ component
• Pauli eigenstate: $|1\rangle$ is the -1 eigenstate of Z, while $|0\rangle$ is the +1 eigenstate