Pauli-Z gate

Pauli-$Z$ is the phase-flip operator $Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$. On the computational basis, $Z|0\rangle=|0\rangle$ and $Z|1\rangle=-|1\rangle$, exactly the identity checked here by comparing the matrix action against both basis states directly. Because a measurement in the computational basis cannot distinguish $|1\rangle$ from $-|1\rangle$ (both give outcome "1" with probability 1), $Z$ is invisible on a bare basis state; its effect only shows up once the qubit is in a superposition, where the flipped sign changes how later amplitudes interfere — the point the entry's own explanation makes. $Z$ is Hermitian and unitary with $Z^2=I$, so it is its own inverse, and it is a $\pi$ rotation of the Bloch sphere about the $z$-axis. $Z$ underlies the phase-kickback trick used in Deutsch–Jozsa-style oracles and phase-flip error models in stabilizer codes.