Minus State (|-⟩)

Minus state $|-\rangle$ is an equal superposition of $|0\rangle$ and $|1\rangle$ with a relative phase of -1. The -1 eigenstate of the Pauli X operator, it is used in phase-dependent quantum protocols and error correction.

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: $P(0) = P(1) = 1/2$
• Relative phase difference: amplitudes are $+1/\sqrt{2}$ for $|0\rangle$ and $-1/\sqrt{2}$ for $|1\rangle$

On the Bloch sphere, the minus state is at the negative x-axis (pointing left). Like $|+\rangle$, it represents equal uncertainty in Z but maximum certainty in X (toward the -1 eigenvalue).

Apply Hadamard to $|1\rangle$: $H|1\rangle = |-\rangle$. Alternatively, create $|+\rangle$ then apply a $\pi$ phase rotation around the X-axis (or equivalently, apply Z first then H).

• Z basis measurement: yields 0 or 1 with equal probability
• X basis measurement: always yields outcome -1
• Phase sensitivity: quantum interference uses the relative phase; replacing $|-\rangle$ with $|+\rangle$ in a superposition can flip measurement probabilities

The minus state appears in quantum error correction as part of syndrome extraction. Entangling data qubits with ancillas in the $|-\rangle$ state allows measurement of error syndromes without directly measuring the data.