Minus-i State (|-i⟩)

Minus-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: negative imaginary unit $-i$ between $|0\rangle$ and $|1\rangle$ amplitudes

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

Apply $S^\dagger H$ to $|0\rangle$: $(S^\dagger H)|0\rangle = |-i\rangle$, where $S^\dagger$ is the conjugate phase gate. 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 minus-i state is orthogonal to plus-i: $\langle -i|+i\rangle = 0$. They form the Y basis eigenstates. Quantum error correction often uses Y measurements for syndrome extraction, which projects onto these eigenstates.