Plus-i State (|+i⟩)

Plus-i state $|+i\rangle$ is the +1 eigenstate of the Pauli Y operator. An equal superposition of $|0\rangle$ and $|1\rangle$ with a relative phase of $+i$, it is one of the four Pauli eigenstates.

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

• Eigenstate of Y with eigenvalue +1: $Y|+i\rangle = |+i\rangle$
• Orthogonal to $|-i\rangle$: $\langle +i|-i\rangle = 0$
• Normalized: $\langle +i|+i\rangle = 1$
• Equal probability of measuring 0 or 1 in Z basis: $P(0) = P(1) = 1/2$
• Relative phase: imaginary unit $i$ between $|0\rangle$ and $|1\rangle$ amplitudes

On the Bloch sphere, the plus-i state is at the positive y-axis. It represents equal uncertainty in the Z basis but maximum certainty in the Y measurement (+1 eigenvalue).

Apply $S H$ to $|0\rangle$: $(S H)|0\rangle = |+i\rangle$, where S is the phase gate and H is Hadamard. Alternatively, create $|+\rangle$ then apply a $\pi/2$ rotation around the Z-axis.

• Y basis measurement: always yields outcome +1
• Z basis measurement: yields 0 or 1 with equal probability
• After measurement: superposition collapses to either $|0\rangle$ or $|1\rangle$

The imaginary phase $i$ distinguishes $|+i\rangle$ from $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$. In interference experiments, replacing $|+\rangle$ with $|+i\rangle$ can alter measurement probabilities due to quantum interference.