Pauli-Z Gate

Overview

Pauli-Z (often just “Z”) is a single-qubit gate that illustrates a key quantum theme:

• you can change a state in a way that does not change computational-basis probabilities,
• yet still changes future measurement statistics in other bases.

Z is the canonical phase flip. It is one of the simplest examples of something with no true classical analogue.

Intuition

In the computational basis, the Z gate:

• leaves $|0\rangle$ alone,
• multiplies $|1\rangle$ by a minus sign.

That minus sign is a relative phase between the two basis components.

If you measure immediately in the computational basis, probabilities only depend on magnitudes like $|\alpha|^2$ and $|\beta|^2$. So a sign change may appear to “do nothing.”

But relative phase matters when amplitudes later interfere (for example, when you measure in a different basis). So Z changes the state in a way that becomes visible only in the right context.

On the Bloch sphere, Z corresponds to a 180° rotation about the $z$ axis. That means it flips the equator point at +$x$ to −$x$ (and +$y$ to −$y$), while leaving the poles fixed.

Formal Description

Action on basis states

The defining action is

$$Z|0\rangle = |0\rangle,\qquad Z|1\rangle = -|1\rangle.$$

By linearity, for

$$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle,$$

we get

$$Z|\psi\rangle = \alpha|0\rangle - \beta|1\rangle.$$

So Z keeps the $|0\rangle$ amplitude and flips the sign (phase by $\pi$) of the $|1\rangle$ amplitude.

Minimal matrix form

In the computational basis,

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

Read this as:

• the first component (the $|0\rangle$ component) is unchanged,
• the second component (the $|1\rangle$ component) gets a minus sign.

Worked Example

Start with

$$|+\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle).$$

Apply Z:

$$Z|+\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle - |1\rangle) = | - \rangle.$$

Now notice:

• Measuring $|+\rangle$ in the computational basis gives $P(0)=P(1)=1/2$.
• Measuring $| - \rangle$ in the computational basis also gives $P(0)=P(1)=1/2$.

So Z does not change computational-basis probabilities for this input.

But $|+\rangle$ and $| - \rangle$ are perfectly distinguishable if you measure in the $\{|+\rangle,| - \rangle\}$ basis:

• $|+\rangle$ gives “+” with probability 1.
• $| - \rangle$ gives “−” with probability 1.

This is why Z has no clean classical analogue: it changes relative phase, which is only revealed through interference or a different measurement basis.

Turtle Tip

Common Pitfalls

Quick Check

What’s Next

We now have two concrete Pauli gates:

• X (bit-flip-like),
• Z (phase-flip-like).

Next we will introduce another foundational single-qubit gate that creates and reveals interference in a very direct way, building on the phase intuition from Z.