This lesson explores partial measurement and how selective observation changes only part of a composite quantum system.
Module 8 examines how partial measurement affects composite systems without fully collapsing every qubit involved. It highlights how local observation changes the structure of a larger shared state.
Understand the difference between full and partial measurement, reason about subsystem collapse, and interpret how measurements alter entangled or multi-qubit states.
Learners gain finer control over measurement intuition, a better understanding of subsystem behavior, and stronger preparation for circuit analysis involving selective observation and entanglement.
In multi-qubit systems, algorithms often measure only a subset of qubits instead of the full register. This is partial measurement.
To model partial measurement analytically, use projection operators with tensor products.
For computational $Z$-basis projections on one qubit:
$$\hat{P}(0) = |0\rangle\langle0| = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \quad \hat{P}(1) = |1\rangle\langle1| = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}$$
To measure qubit 0 while leaving qubit 1 unchanged:
$$\hat{P}_{q_0}(0) \otimes I$$
Start with two qubits and apply $H \otimes H$, creating an equal superposition:
$$|\psi_{\text{initial}}\rangle = \frac{1}{2}(|00\rangle + |01\rangle + |10\rangle + |11\rangle)$$
Now measure only qubit 0.
States with $q_0 = 1$ are removed, so $|10\rangle$ and $|11\rangle$ vanish.
Unnormalized collapsed state:
$$|\psi_{\text{collapsed}}\rangle = \frac{1}{2}|00\rangle + \frac{1}{2}|01\rangle$$
$$\text{New Length} = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{1}{4}} = \frac{1}{\sqrt{2}}$$
Final normalized state:
$$|\psi_{\text{final}}\rangle = \frac{1}{\sqrt{2}}|00\rangle + \frac{1}{\sqrt{2}}|01\rangle$$
| Physical Readout ($q_0$) | Surviving Basis States | Post-Collapse Renormalized State Function | Resulting State of Qubit 1 ($q_1$) |
|---|---|---|---|
| 0 | $|00\rangle, |01\rangle$ | $\frac{1}{\sqrt{2}}|00\rangle + \frac{1}{\sqrt{2}}|01\rangle$ | Remains in superposition over $0/1$ weights |
| 1 | $|10\rangle, |11\rangle$ | $\frac{1}{\sqrt{2}}|10\rangle + \frac{1}{\sqrt{2}}|11\rangle$ | Remains in superposition over $0/1$ weights |
Operational interpretation: For unentangled structures, partially measuring qubit 0 does not destroy relative weighting on qubit 1, which enables selective information extraction while preserving neighboring channels.