Explore quantum entanglement: prepare four Bell states, measure Alice and Bob's qubits, observe perfect correlation, and run a CHSH inequality test that demonstrates quantum nonlocality.
Two entangled qubits share correlations that cannot be explained by any local hidden-variable theory (Bell's theorem). The CHSH parameter S measures these correlations: S ≤ 2 classically, but quantum mechanics predicts S = 2√2 ≈ 2.828.
Select a Bell state (Φ+, Φ−, Ψ+, Ψ−). Measure Alice's and Bob's qubits to see correlated outcomes. Run the CHSH test — accumulate statistics and watch S converge to 2√2, violating the classical bound.
John Bell proved in 1964 that no local hidden-variable theory can reproduce all quantum predictions. The 2022 Nobel Prize in Physics honoured Aspect, Clauser and Zeilinger for experiments confirming Bell inequality violations.
This simulation models a pair of entangled qubits prepared in one of the four maximally entangled Bell states, written as (|00⟩±|11⟩)/√2 and (|01⟩±|10⟩)/√2. Measuring one qubit (Alice) instantly fixes the correlated outcome of its distant partner (Bob). The key tool is the CHSH parameter S = |E(a,b) − E(a,b′) + E(a′,b) + E(a′,b′)|, built from correlation functions E that, for entangled spins, follow cos(2θ).
The toolbar lets you pick a Bell state, fire a single Measure to collapse both qubits, or run a CHSH Test that takes 100 shots and reports the outcome histogram alongside S. With the optimal measurement angles a=0, a′=π/4, b=π/8, b′=−π/8, quantum correlations reach S=2√2≈2.828, exceeding the classical bound of 2. This Bell-inequality violation underpins quantum cryptography and device-independent randomness.
What is quantum entanglement?
Entanglement is a quantum correlation in which two or more particles share a single joint state that cannot be described as separate individual states. Measuring one particle immediately determines properties of the other, no matter how far apart they are. This simulation shows two entangled qubits whose measurement results are perfectly correlated.
What are the four Bell states?
The Bell states are the four maximally entangled two-qubit states: Φ⁺ = (|00⟩+|11⟩)/√2, Φ⁻ = (|00⟩−|11⟩)/√2, Ψ⁺ = (|01⟩+|10⟩)/√2 and Ψ⁻ = (|01⟩−|10⟩)/√2. The Φ states give correlated outcomes (both up or both down) and the Ψ states give anti-correlated outcomes.
What does the CHSH test show?
The CHSH test combines four correlation measurements at chosen angles into a single number S. Any theory using local hidden variables must give S ≤ 2, but quantum entanglement pushes S up to 2√2 ≈ 2.828. Seeing S above 2 is direct evidence that nature is not locally realistic.
Measure performs a single measurement, collapsing both qubits and recording one correlated outcome (↑↑, ↑↓, ↓↑ or ↓↓). CHSH Test (×100) runs 100 measurements at once, builds the outcome histogram, and computes the CHSH parameter S so you can compare it against the classical limit of 2.
If each particle carried predetermined values set by local hidden variables, mathematical bounds force the CHSH combination to never exceed 2. This is Bell's inequality. Because entangled qubits routinely exceed it, no local pre-arranged plan can reproduce their statistics.
Tsirelson's bound limits quantum mechanics to S = 2√2 ≈ 2.828. It is achieved with the measurement angles used here: a=0 and a′=π/4 for Alice, b=π/8 and b′=−π/8 for Bob, where the correlation E(θ)=cos(2θ) is optimally combined.
The outcome probabilities follow the Born rule (probability equals amplitude squared), so measured frequencies match real quantum predictions in the long run. The CHSH value is computed from the ideal cos(2θ) correlation, giving exactly 2√2 for the optimal angles, consistent with experiment.
No. Although the partner's outcome is instantly correlated, each individual result is random, so no usable information travels between Alice and Bob. Comparing results requires a classical message limited by the speed of light, which is why entanglement does not violate relativity.
In 1935 Einstein, Podolsky and Rosen argued that quantum mechanics was incomplete because entanglement seemed to require “spooky action at a distance”. Bell later showed the disagreement is testable, and experiments confirm that quantum correlations are real rather than the result of hidden variables.
Entanglement powers quantum key distribution for secure communication, enables quantum teleportation of states, boosts precision in quantum metrology, and is essential for quantum computers. Bell-test violations also certify genuine randomness for cryptography in a device-independent way.