At θ=90°, what is P(heads)? Why is this the 'most quantum' state?

P = cos²(45°) = 0.5; equal superposition is maximally indeterminate.

At θ=0°, what happens before and after measurement?

Θ=0° → state is |0⟩; P(0)=1, P(1)=0; measurement always gives 0.

Why can't quantum randomness be predicted even with perfect knowledge of the state?

The Copenhagen interpretation: superposition is real; there is no hidden variable determining the outcome.

Pro 🔒~20 min

Quantum Coin Toss

Explore quantum superposition and measurement

How it works

A quantum system can exist in a superposition of states, unlike a classical coin which is definitively heads or tails while in flight. The quantum state encodes probabilities: measuring 'collapses' the superposition to a definite outcome. This is not classical ignorance — the system genuinely has no definite value before measurement. The Bloch sphere represents all possible qubit states as points on a unit sphere.

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Step-by-step

Set the superposition angle (90° = equal probability).
Toss the coin once to observe collapse.
Toss many times to see the probability distribution emerge.
Compare with classical random coin flips.
Rotate the Bloch sphere to change the quantum state.

Key formulas

$|\psi\rangle = \cos(\theta/2)|0\rangle + \sin(\theta/2)|1\rangle$ Qubit superposition state
$P(0) = \cos^2(\theta/2)$ Probability of measuring |0⟩

Frequently asked questions

At θ=90°, what is P(heads)? Why is this the 'most quantum' state?: P = cos²(45°) = 0.5; equal superposition is maximally indeterminate.
At θ=0°, what happens before and after measurement?: Θ=0° → state is |0⟩; P(0)=1, P(1)=0; measurement always gives 0.
Why can't quantum randomness be predicted even with perfect knowledge of the state?: The Copenhagen interpretation: superposition is real; there is no hidden variable determining the outcome.