Hadamard-basis states |+⟩ and |−⟩

The Hadamard-basis states are

$|+\rangle = \frac{|0\rangle+|1\rangle}{\sqrt2}, \qquad |-\rangle = \frac{|0\rangle-|1\rangle}{\sqrt2}.$

Eigenstates of $X$. Direct matrix multiplication gives $X|+\rangle = |+\rangle$ and $X|-\rangle = -|-\rangle$, so $\{|+\rangle,|-\rangle\}$ is the eigenbasis of the Pauli-$X$ operator, exactly as $\{|0\rangle,|1\rangle\}$ is the eigenbasis of $Z$. Because $HXH=Z$ and $HZH=X$ ($H$ conjugation swaps the two Pauli operators), the Hadamard gate is precisely the change-of-basis unitary between the $Z$ and $X$ eigenbases: $|+\rangle=H|0\rangle$, $|-\rangle=H|1\rangle$.

Mutual unbiasedness. $|\langle 0|+\rangle|^2 = |\langle 0|-\rangle|^2 = |\langle1|+\rangle|^2=|\langle1|-\rangle|^2 = 1/2$: every computational-basis state has equal overlap with every Hadamard-basis state. This is the defining property of mutually unbiased bases — measuring a $Z$-eigenstate in the $X$ basis (or vice versa) yields a uniformly random outcome, which is exactly the security mechanism behind the BB84 quantum key distribution protocol's second basis choice.

Phase kickback. Preparing an ancilla in $|-\rangle$ before a controlled-$U_f$ oracle is the standard trick that converts a classical bit-flip oracle $|x\rangle|y\rangle \mapsto |x\rangle|y\oplus f(x)\rangle$ into a phase oracle $|x\rangle|-\rangle \mapsto (-1)^{f(x)}|x\rangle|-\rangle$, because $|-\rangle$ is an eigenstate of the flip with eigenvalue $-1$. This is the mechanism used by Deutsch–Jozsa, Bernstein–Vazirani, Simon's algorithm, and Grover's diffusion oracle.

Stabilizer view. $|+\rangle$ is stabilized by $X$ (the unique $+1$ eigenstate) and $|-\rangle$ by $-X$; both are single-qubit stabilizer states, sitting alongside $|0\rangle,|1\rangle,|+i\rangle,|-i\rangle$ as the six single-qubit states reachable by Clifford operations from $|0\rangle$ — the vertices of the stabilizer octahedron on the Bloch sphere, in contrast to the off-axis magic states elsewhere in this catalog.