Superposition, Born rule, and measurement

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2Lesson 2 of 6inQuantum Principles, Theorems, and Laws

Superposition, Born rule, and measurement

The core principles say that quantum states add as amplitudes, probabilities come from squared amplitudes, and measurement turns a state into an observed outcome.

Every later theorem in this module depends on linearity and measurement. No-cloning is a consequence of linear evolution. Bell tests depend on measurement choices. Hardware control pulses matter because they must preserve the intended state evolution before measurement.

Quantum Brain

Follow this lesson into the surrounding principles, theorems, tools, and modules.

SuperpositionPrinciple Born RulePrinciple MeasurementPrinciple

The intuition

Start with the plain-language idea

Superposition is the structural rule: if two states are available, quantum mechanics allows a linear combination of them. The Born rule is the bridge to data: square the magnitude of an amplitude to get a probability. Measurement is the interface with the lab: it returns one outcome and changes the state relative to the measurement performed. These three ideas explain why the simulator needs amplitudes internally but shows histograms externally.

See it concretely

A real example before the abstraction

Run H on $∣0 ⟩$ . The state becomes an equal superposition, but a single shot is still just 0 or 1. Run many shots and the histogram estimates the Born probabilities. Add a second H and the same amplitudes recombine, turning a superposition back into a definite result.

Tempting but wrong

The mistake most people make

Tempting but wrong

It is tempting to say a qubit is literally both outcomes and measurement simply reveals both. The simulator makes the correction visible: each shot gives one classical result, while repeated shots expose the probability pattern predicted by the state.

Also watch out for

✕Treating amplitudes as ordinary probabilities before squaring them.

✕Forgetting that a measurement basis is part of the prediction.

The precise version

Now with the formal detail

∣ ψ ⟩ = \sum_{i} c_{i} ∣ i ⟩, P (i) = ∣ c_{i} ∣^{2}

For a state written in a measurement basis as $∣ ψ ⟩ = \sum_{i} c_{i} ∣ i ⟩$ , the Born rule assigns probability $∣ c_{i} ∣^{2}$ to outcome $i$ . If the measurement is projective and outcome $i$ is observed, the post-measurement state is the corresponding eigenspace projection, normalized. This is why basis choice matters: the same vector can have different coefficient lists in different measurement bases, producing different probability distributions.