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

## Properties

- 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: $P(0) = P(1) = 1/2$

## Bloch Sphere Position

On the [[quantum-state-bloch-sphere|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.

## Creation

The [[quantum-gate-h|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.

## Measurement

- 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

## Applications

- 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