Measurement in Different Bases

Overview

So far we have mostly measured in the computational basis $\{|0\rangle,|1\rangle\}$. But quantum measurement is more general: you can choose a different basis, and that choice changes what information you extract from the state.

This page explains what a measurement basis is, how to compute probabilities in any basis using overlaps, and why basis choice has a clear physical meaning.

Intuition

A basis is a choice of “reference directions.” Measuring in a basis means you are asking: which of these directions does the state align with?

An everyday analogy is rotating coordinate axes. A vector in the plane is the same physical vector, but its coordinates change depending on how you rotate the axes.

In quantum mechanics, the state is the same object, but different measurement bases ask different questions about it. This is not about changing the qubit; it is about changing what you decide to observe.

On the Bloch sphere, changing measurement basis corresponds to choosing a different pair of opposite points to call “outcomes.” The computational basis uses the north/south poles. Other bases tilt that measurement axis.

Formal Description

A measurement basis for a qubit is a pair of orthonormal states

$$\{|b_0\rangle,|b_1\rangle\}$$

such that:

$$\langle b_0|b_0\rangle=1,\quad \langle b_1|b_1\rangle=1,\quad \langle b_0|b_1\rangle=0.$$

If the system is in state $|\psi\rangle$, then Born’s rule says:

$$P(b_0)=|\langle b_0|\psi\rangle|^2,\quad P(b_1)=|\langle b_1|\psi\rangle|^2.$$

After observing outcome $b_k$, the post-measurement state becomes $|b_k\rangle$.

A concrete non-computational basis

A widely used alternative basis is the “plus/minus” basis, defined by

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

These two states are orthonormal and form a valid measurement basis.

Worked Example

Let

$$|\psi\rangle = |0\rangle.$$

If you measure in the computational basis, the outcome is 0 with probability 1.

Now measure in the $\{|+\rangle,|-\rangle\}$ basis.

Compute overlaps:

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

So

$$P(+)=|\langle +|0\rangle|^2=\tfrac{1}{2}.$$

Similarly,

$$\langle -|0\rangle = \tfrac{1}{\sqrt{2}}(\langle 0|-\langle 1|)|0\rangle = \tfrac{1}{\sqrt{2}}(1-0)=\tfrac{1}{\sqrt{2}},$$

so $P(-)=1/2$.

Interpretation: the state $|0\rangle$ is perfectly definite along the $z$ measurement axis (computational basis), but it is maximally uncertain along the $x$ measurement axis (the $\{|+\rangle,|-\rangle\}$ basis).

Turtle Tip

Common Pitfalls

Quick Check

What’s Next

You now have the full single-qubit measurement toolkit: the measurement postulate, Born’s rule, and basis choice. Next we’ll start using these rules to analyze simple single-qubit experiments and, eventually, formalize transformations as gates.