Measurement Postulate

Overview

Up to now, we have talked about what a qubit is (a state vector) and how to picture it (the Bloch sphere). This page introduces the first rule about what we can observe: the measurement postulate.

The key idea is that measurement is a mathematical rule that takes a state and produces:

1. a probabilistic outcome, and
2. a post-measurement state.

We will focus on measurement in the computational basis and keep the framing rule-based rather than mystical.

Intuition

A qubit state is like a direction in a space. A measurement is like asking a yes/no question tied to reference directions.

When you measure in the computational basis, you are asking:

• “Is the state aligned with $|0\rangle$?”
• “Or is it aligned with $|1\rangle$?”

The state does not return a deterministic answer in general. Instead, it returns an outcome randomly—but with probabilities predicted by the state.

The second part is equally important: after you receive an outcome, the state you should use for future predictions is updated. The update rule is not an optional interpretation; it is part of the mathematical model.

Formal Description

Fix a measurement basis. For now, use the computational basis $\{|0\rangle,|1\rangle\}$.

Let the pre-measurement state be

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

with normalization $|\alpha|^2 + |\beta|^2 = 1$.

Outcomes and probabilities

Measurement in the computational basis has two possible outcomes, labeled 0 and 1.

• Outcome 0 occurs with probability $P(0)=|\alpha|^2$.
• Outcome 1 occurs with probability $P(1)=|\beta|^2$.

This is the probabilistic part.

State update (projection)

If the measurement outcome is 0, the post-measurement state becomes $|0\rangle$.

If the measurement outcome is 1, the post-measurement state becomes $|1\rangle$.

This “collapse” is not a claim about what physically happens inside a device; it is the rule for updating the state in the mathematical model after conditioning on the observed outcome.

More generally, measurement can be described using projectors. For computational-basis measurement, the projectors are:

$$\Pi_0 = |0\rangle\langle 0|,\quad \Pi_1 = |1\rangle\langle 1|.$$

• $\langle 0|$ is the dual (bra) of $|0\rangle$.
• $|0\rangle\langle 0|$ is an operator that “projects onto the $|0\rangle$ direction.”

The probability of outcome 0 is

$$P(0)=\langle\psi|\Pi_0|\psi\rangle,$$

and similarly for outcome 1.

The post-measurement state (conditioned on getting 0) is

$$\frac{\Pi_0|\psi\rangle}{\sqrt{P(0)}}.$$

For a single qubit in the computational basis, this reduces exactly to “the state becomes $|0\rangle$” (or $|1\rangle$) when the corresponding outcome is observed.

Worked Example

Let

$$|\psi\rangle = \tfrac{3}{5}|0\rangle + \tfrac{4}{5}|1\rangle.$$

Measurement in the computational basis yields:

• Outcome 0 with probability $P(0)=|3/5|^2=9/25$.
• Outcome 1 with probability $P(1)=|4/5|^2=16/25$.

Suppose you measure and observe outcome 1.

• The post-measurement state is $|1\rangle$.
• If you immediately measure again in the computational basis, you will get outcome 1 with probability 1.

This is a simple but crucial consequence of the update rule: a basis measurement prepares the system in the corresponding basis state.

Turtle Tip

Common Pitfalls

Quick Check

What’s Next

The measurement postulate tells you how outcomes and post-measurement states work. Next we will zoom in on the probability rule itself—Born’s rule—and practice applying it carefully with multiple examples.