Quantum Mechanics Core

Measurement & the Born Rule

What actually happens when you 'look' at a qubit

By the end of this topic you'll be able to

State the Born rule and apply it to compute outcome probabilities

Explain wavefunction collapse and why measurement is different from unitary evolution

Describe what measuring in a different basis means

Equation |ψ⟩ = Σᵢ aᵢ|i⟩ expresses a quantum state as a sum over states in which some particular measurement is certain to yield a specified value. The coefficients in that expansion yield, as their modulus-squares, the probabilities of the possible results — so as long as more than one term appears in the sum, the outcome of the measurement is genuinely in doubt.

Prepare |ψ⟩ = cos(θ/2)|0⟩ + sin(θ/2)|1⟩θ = 60°

Before measuring, $P (0)$ and $P (1)$ above are just probabilities — press Measure to collapse the state to one definite outcome.

This uncertainty is not a shortcoming of the measuring apparatus. Any real defects in the apparatus only add to the uncertainty above the irreducible minimum already implied by the expansion coefficients — the amplitudes themselves set a floor that no amount of engineering can remove.

Binney and Skinner restrict attention, as we do here, to ideal measurements, which are reproducible: making a second measurement immediately after the first returns the same result. It follows that the first measurement changes the state from Σᵢaᵢ|i⟩ to |I⟩, the state in which the observed value is now certain.

This abrupt change is what's meant by the collapse of the wavefunction. It is tempting to treat collapse as nothing more than updating our knowledge — as though the system was secretly in |I⟩ all along, and we merely became aware of it — but this reading turns out to be untenable: collapse is bound up with a real physical disturbance of the system, not just a change in our information about it.

'Basis' matters enormously to what a measurement even asks. Measuring a spin-half particle's spin along z distinguishes |+⟩ from |−⟩; measuring instead along x asks an entirely different question, one for which |+⟩ and |−⟩ are not eigenstates at all.

Concretely, if a spin is prepared with sz certain to be +½, a measurement of sx is equally likely to return either of its two possible values. Certainty in one basis is compatible with maximal uncertainty in another — the same basis-dependence that lets a wrong-basis measurement in quantum key distribution disturb a qubit enough to expose an eavesdropper.

None of this is classical probability wearing a disguise. In classical probability, ignorance of a coin's exact trajectory is what makes its landing uncertain, and a sufficiently detailed measurement could in principle remove that ignorance entirely.

The Born rule offers no such escape hatch. Experimental tests of Bell's inequality have ruled out any theory in which a hidden variable fixes the outcome in advance — as far as these experiments can tell, the result of a quantum measurement is undetermined until the measurement is actually made.

Try It Yourself

Worked Example

A qubit is in state |+⟩ = (|0⟩+|1⟩)/√2. What's the probability of each outcome if you measure it in the computational basis {|0⟩,|1⟩}? What if you instead measure it in the Hadamard basis {|+⟩,|−⟩}?

1In the computational basis: |+⟩ = (1/√2)|0⟩ + (1/√2)|1⟩, so P(0) = |1/√2|² = 1/2 and P(1) = |1/√2|² = 1/2 — a fair coin.
2In the Hadamard basis, first rewrite |+⟩ in terms of |+⟩ and |−⟩ themselves: trivially, |+⟩ = 1·|+⟩ + 0·|−⟩, since |+⟩ already is one of this basis's two vectors.
3So P(+) = |1|² = 1 and P(−) = |0|² = 0 — measuring in this basis returns '+' with certainty.
4This is the essence of basis-dependence: the exact same physical qubit is maximally uncertain (50/50) in one basis and perfectly certain (100/0) in another, purely depending on what question the measurement asks.

Answer

Computational basis: 50% |0⟩, 50% |1⟩. Hadamard basis: 100% |+⟩, 0% |−⟩ — the state is 'certain' in the basis that matches how it was prepared, and 'maximally uncertain' in the basis rotated 45° away from it.

Reference

Born rule	$P (λ_{i}) = ∣ ⟨ e_{i} ∣ ψ ⟩ ∣^{2}$	Probability of measuring eigenvalue λᵢ
Collapse	$∣ ψ ⟩ \to ∣ e_{i} ⟩$	Post-measurement state snaps to the observed eigenstate
Computational basis	$∣0 ⟩, ∣1 ⟩$	The standard measurement basis for a qubit
Hadamard basis	$∣ + ⟩, ∣ - ⟩ = (∣0 ⟩ \pm ∣1 ⟩) /\sqrt2$	A rotated measurement basis, used e.g. in BB84 QKD

Quick Check

If you measure a qubit and immediately measure it again in the same basis, what happens?

Why does measuring in the 'wrong' basis matter for quantum cryptography?

Further Resources

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Density Matrices & Mixed States