Measurement

January 4, 2025

Measurement connects quantum formalism to observable outcomes. In the simplest picture, measuring a qubit in the computational basis yields 0 or 1 with probabilities given by the squared magnitudes of the amplitudes.

What makes quantum measurement different from classical measurement?

In classical physics, you often model measurement as revealing a pre-existing value (up to noise). In quantum mechanics, measurement is a physical process that generally changes the state.

If a qubit is in

\lvert \psi \rangle = \alpha \lvert 0 \rangle + \beta \lvert 1 \rangle,

then a computational-basis measurement returns:

0 with probability $\lvert \alpha \rvert^2$
1 with probability $\lvert \beta \rvert^2$

Collapse (operational view)

After a projective measurement whose outcome you record, the post-measurement state lies in the subspace consistent with that outcome. This is why intermediate measurements change the rest of a circuit’s behavior.

“Collapse” is a useful operational description: it tells you how to update the state for future predictions after you learn a measurement outcome. It is not a license to treat measurement as a purely passive readout.

Wave function collapse (what people mean)

In many textbooks and popular articles, you will see a stronger phrase: wave function collapse (or “state vector reduction”). The informal story is:

before measurement, the system can be in a superposition of outcomes
after measurement, you observe one definite outcome

In equations, if an observable has eigenstates $\{\lvert \phi_i\rangle\}$ and your state is expanded as

\lvert\psi\rangle = \sum_i c_i \lvert \phi_i\rangle,

then a projective measurement yields one outcome $i$ with probability $|c_i|^2$ (Born rule), and the post-measurement state updates to the corresponding eigenstate (or eigenspace):

\lvert\psi\rangle \mapsto \lvert\phi_i\rangle.

What does (and doesn’t) travel faster than light?

Collapse can sound instantaneous and non-local, especially for entangled states. But what you can access locally is still limited:

each party’s local outcomes look random
correlations only become visible after comparing results using classical communication

So “collapse” does not give a controllable faster-than-light messaging channel.

Why this becomes the “measurement problem”

Unitary time evolution (the Schrödinger equation) is smooth and deterministic, while the collapse rule is discontinuous and probabilistic. Putting both into one story raises the question:

If everything is quantum, why do we observe a single outcome in each run?

Different interpretations answer this differently:

Copenhagen-like views treat collapse as part of the measurement update rule (a rule for predictions once you learn an outcome).
Many-worlds / decoherence-based views keep unitary evolution always, and explain why outcomes appear classical via entanglement with the environment.
Objective-collapse models postulate additional physical dynamics that truly collapses the state.

Decoherence explains why quantum superpositions become effectively indistinguishable from classical mixtures in practice (preferred “pointer” states, loss of interference). But decoherence by itself does not pick a single outcome; it explains why alternatives stop interfering.

Types of quantum measurements (high-level)

You will see different measurement models in quantum engineering and research. For a beginner roadmap, it is enough to know the intent behind each category:

Projective (von Neumann) measurement: the standard model in most circuits; outcomes correspond to eigenvalues of an observable, and the state updates to a compatible eigenstate/subspace.
Weak measurement: extracts a small amount of information per shot, typically trading off information gain against disturbance; used in metrology, control, and certain diagnostic protocols.
Quantum non-demolition (QND) measurement: designed so that repeated measurements of the same observable return consistent results (it preserves that observable), enabling repeated readout and feedback.

Be careful with wording: QND does not “preserve the entire quantum state.” It preserves the measured observable in a way that allows repeatable readout.

Why measurement matters in quantum computing

Measurement is not only the final step of a computation:

Algorithm output: many algorithms end with a measurement that extracts classical information from the quantum state (for example, Fourier-based routines and period-finding subroutines).
Error correction: quantum error correction relies on repeated syndrome measurements that reveal error information without directly measuring the logical state.
Cryptography (QKD): security proofs rely on the fact that measurement disturbs quantum states, so eavesdropping leaves detectable traces.

Practical challenges

In real devices, measurement is imperfect and can introduce:

readout errors (misclassifying 0 vs 1)
measurement-induced back-action on nearby qubits (cross-talk)
extra decoherence due to control and amplification hardware

This is one reason measurement calibration and error mitigation matter in experiments.

In Qiskit, measure_all() records outcomes into classical bits. Always think about when you measure relative to gates you apply afterward.

Exercise: Consider a qubit in $\lvert \psi \rangle = \sqrt{0.9}\lvert 0 \rangle + \sqrt{0.1}\lvert 1 \rangle$ . If you measure in the computational basis, what distribution do you expect over 1,000 shots? How would that change if you applied an H gate right before measuring?

Move on to Linear algebra for the mathematical toolkit.

Entanglement Types of Qubits (Superconducting, Ions, Photons, and More)