Single-Qubit Measurements

In classical computing, reading a bit is a passive operation—the bit is 0 or 1, and looking at it doesn't change it. Quantum mechanics is fundamentally different.

A qubit can exist in a superposition of the basis states $$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$$ where \(\alpha\) and \(\beta\) are complex amplitudes satisfying \(|\alpha|^2 + |\beta|^2 = 1\).

When we measure a qubit, the superposition is destroyed and we obtain a definite classical outcome. The probability of each outcome is given by the squared magnitude of the corresponding amplitude.

Any single-qubit pure state can be parameterized by two angles and visualized as a point on a unit sphere called the Bloch sphere:

$$|\psi\rangle = \cos\frac{\theta}{2}|0\rangle + e^{i\varphi}\sin\frac{\theta}{2}|1\rangle$$

Drag the sliders to explore. Click on the special states in the table, or rotate the sphere by dragging it.

The most common measurement is in the computational basis (Z-basis). The measurement is described by the projection operators:

$$P_0 = |0\rangle\langle 0| = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \qquad P_1 = |1\rangle\langle 1| = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$$

The probabilities are \(P(0) = \cos^2(\theta/2)\) and \(P(1) = \sin^2(\theta/2)\). Click Measure to see a probabilistic outcome and watch the state collapse.

We can measure along any axis on the Bloch sphere. The X-basis eigenstates are:

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

The Y-basis eigenstates are:

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

For an arbitrary measurement axis \(\hat{n}\), the probability of the \(+\) outcome is: $$P(+) = \tfrac{1}{2}(1 + \hat{n}\cdot\vec{r})$$

The orange arrow shows the measurement axis and the green dots mark the eigenstates.

The Born rule connects the quantum state to measurement probabilities:

$$P(m) = |\langle m|\psi\rangle|^2 = \langle\psi|P_m|\psi\rangle$$

where \(P_m = |m\rangle\langle m|\) is the projector onto eigenstate \(|m\rangle\). The projectors satisfy completeness: \(\sum_m P_m = I\).

Set a state below, choose a measurement basis, then run 1000 simulated measurements and compare the observed frequency against the Born rule prediction.

After obtaining outcome \(m\), the post-measurement state is:

$$|\psi\rangle \;\to\; |\psi'\rangle = \frac{P_m|\psi\rangle}{\sqrt{\langle\psi|P_m|\psi\rangle}}$$

Since \(P_m^2 = P_m\), measuring again immediately gives the same outcome with certainty. Watch the collapse in slow motion below.

Quantum mechanics is inherently probabilistic, but the law of large numbers ensures convergence:

$$f_N(m) = \frac{N_m}{N} \;\xrightarrow{N\to\infty}\; P(m)$$

Run measurements below and watch the histogram converge to the Born rule prediction (dashed lines).