Free quantum tool · Level: First-year university
Measurement and the Born rule
Point the state one way and the detector another. Only the angle between them decides the odds, and you can make a thousand real runs check it.
Try this on the tool above
- Set the state to 0° and leave the detector at 0°. Every run gives +1. Now turn the detector to 90° without touching the state: the same qubit is a 50/50 coin flip.
- Set the state to 60° and the detector to 0°. The gap is 60°, cos²(30°) is 0,75, and the blue bar reads 75%.
- Switch on the shots layer at 10 shots and press run again five times. The amber bars jump around. Drag the shots up to 2000 and they stop jumping.
The Born rule turns an amplitude into a probability by squaring its size. Measure a qubit along an axis Δ away from where it points and the chance of the first outcome is cos²(Δ/2), the same law that governs light through two polarising filters. Measuring also moves the qubit: afterwards it sits on the axis you just used, so asking the same question again gives the same answer every time.
The usual mistake
The measurement just reads off a value the qubit already had.
Put the state at 0° and the detector at 90° and the result is a coin flip. Turn the detector back to 0° and it is certain again. The qubit never moved, only the question did, and a property that was already sitting there would not care which way you asked for it.
Turning the detector changes the question, not the qubit
Put the state at 0° and the detector at 0°: certain. Leave the state exactly where it is and turn the detector to 90°: coin flip. Nothing about the qubit changed between those two sentences.
Classical physics trains you to expect the opposite, that measuring uncovers a number already sitting there. Here the question is part of the experiment. No single question has an answer waiting for all the others at once.
A prediction is a distribution, so count
The blue bars are what the theory says. Turn on the shots layer and the amber bars are what one real run of the experiment gave. At ten shots they visibly disagree. At two thousand you can barely tell them apart.
Press run again a few times and watch the amber bars jump while the blue ones sit still. That gap is where the maths meets a lab bench, and it is why nobody publishes a single run.
Why cos² of half the angle
Turning the detector a full 180° has to take you from certain yes to certain no, and cos² of half the angle is what does that. At 0° it gives 1, at 180° it gives 0, and at 90° it gives a half. The same half turns up on the Bloch sphere for the same reason.
If you have already met the law of Malus for light through two polarising filters, this is that curve with that square. The two are the same statement, which makes optics the cheapest way in if it came first for you.
What collapse does, and what it does not
After a result the qubit sits on whichever axis you just measured along. Ask again at the same angle and the answer repeats, every time, with no randomness left in it.
It would be easy to read that as the qubit finally admitting what it always was. Turn the detector somewhere new, though, and a fresh 50/50 appears, built out of a state that was certain a moment ago. Collapse resets the question.
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Frequently asked questions
- What is the Born rule?
- The rule that turns amplitudes into probabilities: the chance of an outcome is the squared size of its amplitude. It is why amplitudes can be negative or complex while probabilities never are, and it is the only place randomness enters the theory at all.
- Why is it cos²(Δ/2) and not cos²(Δ)?
- Because a 180° turn of the detector has to run from certainty to impossibility, and only the half-angle version does that. It is the same half that puts |0⟩ and |1⟩ at opposite poles of the Bloch sphere while keeping them perpendicular in the maths.
- Does measuring really change the state?
- Yes, and you can check it here. Measure at one angle, then at another, and the second measurement starts from the collapsed state rather than the original. That is why the order of two measurements matters in quantum mechanics and never does in classical physics.
- How many shots before the numbers settle?
- The error shrinks roughly as one over the square root of the shot count, so ten times more shots buys about three times more precision. Drag the slider from 10 to 2000 and watch how stubbornly that last wobble hangs on.
More interactive quantum tools
- Superposition and the Bloch sphere →Two angles, one point on a sphere. See which of them a measurement can actually detect.
- Entanglement and the Bell test →Turn two detectors and watch the CHSH value climb past the classical limit of 2.
- Quantum gates and interference →Stack gates on a qubit and watch +0,5 and −0,5 cancel to nothing.