Measurement

Collapse isn't mystical — it's a rule. The rule has two parts, and the second one depends on which question you ask.

The standard story about quantum measurement goes like this: “the wavefunction collapses.” Then the storyteller waves a hand vaguely and you're supposed to nod along. This is a bad story. It doesn't tell you which outcome you get, how often, or — and this is the part that breaks most people's intuition — what counts as a measurement at all.

Here is the real rule, in two parts. (1) Born: the probability of measuring outcome k is |⟨k|ψ⟩|² — the squared magnitude of the amplitude sitting in front of the basis state you're asking about. (2) Projection: once you've gotten outcome k, the state is |k⟩. The other amplitudes vanish. Same particle, different state — you changed it by looking at it.

That's it. No consciousness, no observers, no philosophy. A measurement is any interaction that correlates a quantum amplitude with a classical record, and the rule above tells you what the record will say and what the state looks like afterwards. The only knobs are: which state you started in, and which basis you measure in.

Press a gate, then measure

Start from |0⟩. The probability bars show one outcome with probability 1. If you measured right now, you would get 0. Every single time. Forever.

P(outcome) — Born's rule rendered live.

P(|0⟩) = …, P(|1⟩) = …

The amplitudes the rule reads from.

…

Try this: press Run 100× without touching any gates. The little histogram below the button fills up with 100 zeros and zero ones. The simulator runs a real measurement 100 times against the current state, and the Born rule says P(0) = 1, so… every roll comes back 0. The randomness is mathematically there; it just isn't doing anything when one outcome has probability 1.

Re-runs the circuit you've built (current gates, then a measurement) 100 times against fresh randomness.

|0⟩

|1⟩

Try this: press Reset, then H, then Run 100×. The bars split close to 50/50 — not exactly, because 100 flips of a fair coin doesn't land on 50-50, but close. Press it a few more times and watch the numbers wobble in the 40–60 band. That's the Born rule meeting the law of large numbers. The wobble shrinks like 1/√N; at 10,000 shots you'd see numbers in the 49–51 band.

Same state, different question

Here's where most quantum-curious developers get tripped up. People talk about “measuring a qubit” as if there's one way to do it. There isn't. Every measurement happens in a basis — a choice of two orthogonal states you're asking the qubit to commit to. Our buttons measure in the computational (Z) basis: |0⟩ vs. |1⟩. But you could equally well measure in the X basis (|+⟩ vs. |−⟩), or any other.

The Born rule cares which basis you picked. So does the projection step. Same state, different basis, different distribution. This is not a paradox — it's the whole point of being a vector in a 2D complex space. There are infinitely many orthogonal axes you can decompose along.

We don't have a basis-switching button up there, but we have the next best thing: rotate the state instead of rotating the measurement axis. Measuring H|ψ⟩ in the Z basis is mathematically identical to measuring |ψ⟩ in the X basis — applying H swaps the two bases for you.

Try this: reset, press X (so you're in |1⟩), then Run 100×. You'll get 100 ones. Now press H and run again. The bars split — the deterministic |1⟩ answer in one basis is a fair coin flip in another. Nothing about the underlying state has “become random”; you just asked a different question of the same vector.

Open it in the sandbox

The Run-100 button above is a shortcut that recreates whatever gate sequence you've clicked on this page. The sandbox is the full thing: a real composer where you can drag ops around, swap the order of H and measure, or add an X in front and watch the histogram flip.

Open this circuit and try swapping the order of the two ops, or removing the H entirely:

Open [H, measure] in sandbox →

The headline observation: [measure] on |0⟩ is deterministic. [H, measure] on |0⟩ is a fair coin. Same hardware. Same act of measuring. The only difference is one gate sitting one column to the left.

Self-test

You apply H, measure, get the outcome 1, and then immediately apply H and measure again. What's the probability of getting 1 on the second measurement, and why? (Hint: the first measurement projected the state. The second H is acting on the collapsed state, not the original superposition.)
You're handed a qubit in an unknown state. You measure 100 times in the Z basis with no gate first and get 100 zeros. Can you conclude the state was |0⟩? (Hint: what would the superposition state |+⟩ = (|0⟩+|1⟩)/√2 give you? What about the same state with a phase flip on the |1⟩ component? The Z basis only sees probabilities, not phases — so which states does it actually distinguish?)

The rule, formally

Let $|\psi\rangle = \sum_k \alpha_k |k\rangle$ in some orthonormal basis $\{|k\rangle\}$ . The Born rule:

P(k) = |\langle k | \psi \rangle|^2 = |\alpha_k|^2

The projector onto outcome $k$ is $\Pi_k = |k\rangle\langle k|$ . The post-measurement state, given outcome $k$ , is the projection renormalized:

|\psi'\rangle = \frac{\Pi_k |\psi\rangle}{\sqrt{\langle \psi | \Pi_k | \psi \rangle}} = |k\rangle

That renormalization is doing the “collapse” work: after the measurement, every amplitude except the one you observed is zeroed out, and the surviving amplitude is rescaled so the new vector is still a unit vector. This is exactly what runCircuit's measurement path does in code — pick an outcome with weight |α_k|², zero the rest, divide by the surviving norm.

Basis dependence falls out for free. Measuring $|\psi\rangle$ in the X basis is the same calculation with $\{|+\rangle, |-\rangle\}$ in place of $\{|0\rangle, |1\rangle\}$ . Equivalently — and this is the identity the Run-100 demo leans on — applying $H$ first and then measuring in the Z basis gives the same statistics as measuring the original state in the X basis directly.

For the math nerds

A philosophical aside that often comes up in office hours: “is the collapse real, or is it just the observer updating their beliefs?” This essay takes the pragmatic line. The post-measurement state, whatever its ultimate ontology, is the right object to plug into your next gate. Every single working interpretation of quantum mechanics — Copenhagen, many-worlds, Bohmian, QBism — agrees on the Born rule and on what follow-up gates do to the projected state. That's the part we can simulate and the part this essay is teaching.

A keyboard-only note: the Run 100× button is reachable via Tab and activates with Enter or Space. The histogram updates in place — its readouts are inside <output> elements so assistive tech announces the new counts on each run.