The Z-gate

The $Z$-gate reflects the state of a qubit on the $Z$-axis. It has a similar effect to the $X$-gate that reflects the state on the $X$-axis. A reflection on the $X$-axis affects the resulting measurement probabilities because it changes the proximities to the ends of the Z-axis ($|0\rangle$ and $|1\rangle$). However, it leaves the phase untouched. On the other hand, a reflection on the $Z$-axis flips the phase but leaves the measurement probabilities unchanged.

The following equation denotes the transformation matrix of the $Z$-gate.

$Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$

The $Z$-gate turns a qubit in-state $|+\rangle$ into state $|-\rangle$. The states $|+\rangle$ and $|-\rangle$ reside on the $X$-axis. Mathematically, the following equation describes this transformation.

$HZ|0\rangle=\tfrac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix}=\tfrac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ -1 & 1 \\ \end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix}=\tfrac{1}{\sqrt{2}}\begin{bmatrix} 1\\ -1 \end{bmatrix}=\frac{|0\rangle - |1\rangle}{\sqrt{2}}=|-\rangle$