Plus State (|+⟩)

Plus state $|+\rangle$ is an equal superposition of $|0\rangle$ and $|1\rangle$. The +1 eigenstate of the Pauli X operator, it is fundamental for creating superposition and is used in nearly every quantum algorithm.

Representation: $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix}$

• Eigenstate of X with eigenvalue +1: $X|+\rangle = |+\rangle$
• Orthogonal to $|-\rangle$: $\langle +|-\rangle = 0$
• Normalized: $\langle +|+\rangle = 1$
• Equal probability of measuring 0 or 1: $P(0) = P(1) = 1/2$

On the Bloch sphere, the plus state is at the positive x-axis (pointing right). It represents equal uncertainty between Z eigenstates but maximum certainty in the X measurement.

The Hadamard gate creates $|+\rangle$ from $|0\rangle$: $H|0\rangle = |+\rangle$. Applying H to any computational basis state yields a superposition in the X basis.

• Measuring in the computational (Z) basis: yields 0 or 1 with equal probability
• Measuring in the X basis: always yields outcome +1
• After Z measurement, the state collapses to either $|0\rangle$ or $|1\rangle$, destroying the superposition

• Quantum algorithms: starting state for many algorithms (Grover's search, phase estimation)
• Quantum error correction: code states are superpositions of basis states
• Quantum walks: uniform superposition of all positions