Free quantum tool · Level: First-year university
Superposition and the Bloch sphere
A qubit holds exactly two numbers, and this sphere shows you both. One of them changes the odds. The other one does not.
Try this on the tool above
- Set θ to 60°. The chance of 0 reads 75%, because cos²(30°) is 0,75. Halve the angle first, then square it.
- Leave θ alone and drag φ from 0° to 355°. The dot circles the sphere at a constant height while both percentages sit still.
- Switch on the global phase layer. The written state changes, the dot does not move, and 75% stays 75%.
Two angles describe a qubit completely. θ splits the probability between 0 and 1, so the chance of measuring 0 is cos²(θ/2). φ turns the phase around the equator and changes neither probability. Together they mark one point on the Bloch sphere, and every state a single qubit can be in is exactly one such point.
The usual mistake
It is really 0 or 1 already, and we just do not know which until we look.
If that were true, two Hadamard gates in a row could not turn a coin flip back into a certainty. They do. A coin that is secretly heads cannot be flipped twice into certainty, but two amplitudes can cancel, and that is what happens.
Only one of the two angles changes the odds
θ is the one a measurement can see. At 0° the qubit is |0⟩ and the answer is certain. At 180° it is |1⟩ and the answer is certain the other way. At 90° it is a genuine coin flip, and everything between is cos²(θ/2).
Mind the half. θ = 180° lands on the opposite pole, not a quarter turn to something halfway. Opposite points here are opposite states, so the picture is squeezed by a factor of two against the arrow you are probably imagining.
What φ does, and why you cannot see it yet
Spin φ a full turn and neither percentage moves. It is fair to wonder what the slider is for. The payoff arrives one page later: two states with identical odds behave completely differently the moment a gate touches them, and φ is the whole difference between them.
The name for it is relative phase, the angle between the two amplitudes. Invisible to the measurement you take now. Decisive for the one you take after a Hadamard.
Global phase, and why it is not there
Multiply both amplitudes by the same factor and every number in the written state changes. The dot does not move and no probability shifts by a percent, because the factor cancels the moment you square a modulus.
So two states that differ only by that factor are the same state, and no experiment anyone runs will separate them. Only the phase between the two amplitudes is real. The one out front is bookkeeping.
Six states you will keep meeting
Turn on the named states and six dots appear: |0⟩ and |1⟩ at the poles, |+⟩ and |−⟩ at either end of one equator axis, |i⟩ and |−i⟩ at either end of the other. Each opposite pair is a measurement basis, which is why these six turn up on every page after this one.
|+⟩ sits on the equator, the same distance from |0⟩ as from |1⟩. That is what an even superposition looks like as geometry. Measure it in the 0/1 basis and you get a coin flip; measure it along its own axis and you get the same answer every time.
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Frequently asked questions
- What is a qubit in superposition?
- A qubit carrying an amplitude on |0⟩ and an amplitude on |1⟩ at the same time. Amplitudes are complex numbers, and squaring one gives the chance of that outcome. The qubit is a direction, and the measurement you choose decides which question that direction answers.
- Why does the Bloch sphere use θ/2 instead of θ?
- Because opposite points on a sphere have to be orthogonal states, and orthogonal states are a half turn apart in the amplitudes. Writing cos(θ/2) puts |0⟩ at the north pole and |1⟩ at the south: 180° apart on the picture, perpendicular in the maths.
- What is the difference between global and relative phase?
- Relative phase is the angle between the two amplitudes. Change it and the point moves around the equator, and the next gate gives a different answer. Global phase multiplies both amplitudes at once and cancels in every probability, so nothing can detect it. Switch the layer on and watch the dot refuse to move.
- Can a Bloch sphere show two qubits?
- No, and that limit is the reason the entanglement page exists. Two qubits live in a bigger space, and an entangled pair has no pair of points on two spheres that describes it. Neither half has a state of its own to draw.