Z Gate and Relative Phase

Z Gate & Relative Phase

qc/gates qc/phase

The Z gate leaves |0⟩ unchanged but maps |1⟩ → −|1⟩:

$Z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$

This minus sign is a relative phase. It doesn’t change measurement probabilities in the standard basis (still 50/50) but is physically real — it affects how the state interferes in later computation. It’s what distinguishes |00⟩+|11⟩ from |00⟩−|11⟩.

The general phase gate

Z is one member of a whole family, the phase gate $P_{\theta }$:

$P_{\theta }=\left(\begin{array}{cc}1& 0\\ 0& e^{i\theta }\end{array}\right)$

Z is the special case $\theta =\pi$ (a half-turn about the Z-axis on the Bloch Sphere). See S and T Gates for the two other special cases that matter most in practice.

Related

• Pauli Operators
• Bell States — the two “minus” Bell states come from combining X and Z
• S and T Gates
• Bloch Sphere

Self-Check

• Why does the Z gate not change measurement probabilities in the standard basis, if it’s clearly doing something?
• What is a “relative phase,” and why does it matter even though it’s invisible to a Z-basis measurement?
• What’s the difference between $|00⟩+|11⟩$ and $|00⟩-|11⟩$, and how would you actually detect it?
• What angle $\theta$ makes $P_{\theta }$ equal to the Z gate?