Quantum Entanglement

When two qubits are entangled, measuring one instantly determines the state of the other — no matter how far apart they are. Einstein called this "spooky action at a distance" and believed it meant quantum mechanics was incomplete. He was wrong.

Entanglement is not just a curiosity — it's a resource that powers quantum teleportation, superdense coding, quantum key distribution, and every quantum algorithm that outperforms classical computers.

Alice and Bob each hold one qubit from an entangled pair. Neither qubit has a definite value — but the moment either person measures, both qubits collapse. In the |Φ⁺⟩ state, they always get the same result. In |Ψ⁺⟩, they always get opposite results. Try different Bell states and watch the correlations build up.

There are exactly four maximally entangled two-qubit states, called the Bell basis. Each is created by a simple circuit: Hadamard on one qubit followed by CNOT. They form a complete basis for the 4D two-qubit Hilbert space — any two-qubit state can be written as a combination of Bell states.

|Φ⁺⟩ and |Φ⁻⟩: Same-outcome states. Both qubits always agree in the Z basis. The minus sign changes correlations in other measurement bases.

|Ψ⁺⟩ and |Ψ⁻⟩: Opposite-outcome states. Qubits always disagree. |Ψ⁻⟩ (the singlet) is special: anti-correlated in every measurement basis.

Could entanglement just be a pre-arranged classical agreement? The CHSH game proves it can't be. With the best classical strategy, Alice and Bob can win at most 75% of rounds. But with entangled qubits, they win ~85%. No amount of shared classical information can replicate quantum correlations. This is Bell's theorem in action.

What this proves: Run 100+ rounds and compare classical vs quantum win rates. The quantum strategy consistently exceeds 75%. This means entanglement cannot be explained by hidden variables — quantum correlations are genuinely non-classical. This was experimentally confirmed and won the 2022 Nobel Prize in Physics.

When Alice and Bob measure their entangled qubits at different angles, the probability of getting the same result follows a smooth cos² law. The heatmap shows P(same outcome) for all combinations of measurement angles. This sinusoidal dependence is the fingerprint of quantum entanglement — no classical model can reproduce it for all angle choices simultaneously.

Using one entangled pair and two classical bits, Alice can "teleport" an arbitrary qubit state to Bob — without physically sending the qubit. The original is destroyed in the process (no-cloning theorem), and Bob needs Alice's classical message to reconstruct the state. Step through the protocol below.