Entanglement
Two qubits, one shared state — and results that match more often than any independent pair could.
Entanglement visualization
Measure one orb and watch the other. The beam between them is the entanglement.
State amplitudes
| Basis state | Probability |
|---|---|
| |00⟩ | 50.0% |
| |01⟩ | 0.0% |
| |10⟩ | 0.0% |
| |11⟩ | 50.0% |
The state, as a matrix
Every entry ρij = ψiψj* is a bar: height is its size, colour its phase. Switch to one qubit alone — that is the partial trace, summing out the partner — and watch what survives.
Drag to walk around the bars; arrow keys also rotate. Full height is 1.
The pair is pure: Tr ρ² = 1. The tall off-diagonal bars are the coherences — the superposition itself, written into the matrix.
For the Bell state the whole pair has purity 1 and each half has purity ½ — and no other way of writing the state changes that. A pure whole with maximally mixed parts is not a side effect of entanglement; it is entanglement. The information is not in A and not in B. It is in the four corner bars, which belong to neither.
Compare the product state (H on A only): the whole and both parts are pure at once. The pair's four bars sit in the top-left block, A alone keeps its off-diagonal bars — it is |+⟩, a superposition in its own right — and B alone is a single bar at |0⟩⟨0|. Each half has a state of its own, so nothing was shared. Then measure A on the Bell state and look at the pair again: the corner bars are gone, and so is the entanglement.
Correlated, not connected
For the Bell state you only ever see 00 or 11. Yet look at P(A=1) on its own: it is 50%, a perfectly fair coin. Nothing about A alone reveals that B exists.
That is why entanglement cannot send a message: the correlation is only visible once the two result lists are brought together and compared.
Entropy measures how entangled
A single qubit pulled out of an entangled pair has no state of its own. Its Bloch vector shrinks to zero length and its entropy rises to 1 bit.
Drag θ on the partial preparation and watch the number sweep from 0 to 1 and back.
Why the product state is different
With H on each qubit separately, the two qubits are independent — all four outcomes appear equally, and knowing A tells you nothing about B. The link goes dark, and the joint outcomes spread across all four possibilities instead of two. Compare the histograms.
Matching outcomes are not yet proof
Everything above measures both qubits along the same axis, and a pair of gloves posted to opposite ends of the country would pass that test too — always matching, nothing spooky. The step that rules the gloves out is to measure the two halves along different axes, combine four such correlations into one number S, and check it against the ceiling that any pre-agreed plan is arithmetically stuck below.
The Bell Test runs that experiment: it races this entangled source against a real local hidden-variable simulation over the same trials, and you can watch the classical rival stall at exactly 2 while the quantum one keeps climbing to 2√2.
Keep going
Build this yourself
Every visualization on this page is computed by the same engine as the simulator. Open it to place the gates by hand, up to 16 qubits, free and without an account. Or install Qubit16 so the whole lab is one tap away and works offline.