Have you ever wondered how quantum mechanics defies everyday intuition? This project demonstrates one of the most striking phenomena in quantum physics: quantum entanglement and its ability to violate classical physics constraints through a playable cooperative game.
🎮 Try the CHSH Game Simulator |
📂 View on GitHub
1 · What is the CHSH Game?
The CHSH (Clauser-Horne-Shimony-Holt) game is a cooperative thought experiment that reveals the strange power of quantum entanglement. Two players, Alice and Bob, cannot communicate during the game but may share a pre-agreed strategy or, crucially, a pair of entangled qubits.
| Player | Receives from referee | Outputs |
|---|---|---|
| Alice | Random bit x | Bit a |
| Bob | Random bit y | Bit b |
They win if and only if:
(a + b) mod 2 = x × y
With classical strategies, Alice and Bob can win at most 75% of the time. With a quantum strategy using entangled qubits, they can reach approximately 85.4%, violating the classical bound.
2 · Bell’s Inequality and the CHSH Value
In 1964, John Bell proved that no local hidden variable theory can reproduce all predictions of quantum mechanics. The CHSH inequality formalises Bell’s theorem as a single testable number S:
| Theory | CHSH Value (S) | Win Rate |
|---|---|---|
| Classical (any local strategy) | S ≤ 2 |
75% |
| Quantum mechanics | S = 2√2 ≈ 2.828 |
~85.4% |
This violation is not a loophole or a trick. It proves that the correlations produced by entangled qubits cannot be explained by any shared classical information, even with pre-agreed randomness. Loophole-free Bell test experiments confirm that quantum entanglement is a genuine physical phenomenon with no classical analogue.
3 · Measurement Angles and Quantum Correlation
The quantum advantage arises from choosing measurement bases at specific angles. For the standard |Φ+⟩ Bell state, the optimal angles are:
| Player | Input bit | Measurement basis |
|---|---|---|
| Alice | x = 0 | 0° |
| Alice | x = 1 | 45° |
| Bob | y = 0 | 22.5° |
| Bob | y = 1 | −22.5° |
The probability that Alice and Bob produce the same outcome is:
P(same) = cos²(δ)
where δ is the angle between their chosen measurement bases. This quantum correlation is what enables them to exceed the 75% classical ceiling. Different Bell states exhibit either parallel correlation (|Φ+⟩, |Ψ+⟩: qubits tend to give the same outcome) or orthogonal correlation (|Φ−⟩, |Ψ−⟩: one qubit effectively rotated 90° relative to the other). The simulator adjusts its probability calculations accordingly for each state.
🧪 4 · The Four Bell States
The simulator supports all four maximally entangled Bell states. Each uses optimised measurement angles to achieve the theoretical quantum win rate of ~85%.
| State | Definition | Correlation |
|---|---|---|
| |Φ+⟩ | (|00⟩ + |11⟩) / √2 | Parallel |
| |Φ−⟩ | (|00⟩ − |11⟩) / √2 | Orthogonal |
| |Ψ+⟩ | (|01⟩ + |10⟩) / √2 | Parallel |
| |Ψ−⟩ | (|01⟩ − |10⟩) / √2 | Orthogonal |
🎯 5 · Simulator Features
| Feature | Details |
|---|---|
| 🎯 Interactive Visualization | Real-time p5.js visualization showing the entangled state, Alice’s measurement collapse, and Bob’s final measurement. Coloured basis quadrants and correlation indicators included. |
| 🎮 Strategy Comparison | Switch between Classical (always outputs 0, max 75%) and Quantum (entangled qubits, ~85.4%) strategies in real time. |
| 🎲 Flexible Input Controls | Set Alice’s x and Bob’s y to Random or Fixed (0 or 1) to test specific measurement configurations. |
| 📊 Real-Time Statistics | Running totals for rounds played, wins, losses, and win percentage converging toward theoretical predictions. |
| 🔄 Round History Navigation | Step through previous rounds to review specific outcomes and trace the quantum measurement process. |
Built with: React, p5.js, and Claude Code
6 · Try It Yourself!
A suggested sequence to build intuition:
- Open the simulator and select the Classical strategy. Run 100 rounds and watch the win rate converge toward 75%.
- Switch to the Quantum strategy with the |Φ+⟩ Bell state. Run another 100 rounds and observe the rate climb toward 85%.
- Try the other three Bell states. Note how each state’s correlation type (parallel vs orthogonal) affects the outcome distribution.
- Set Alice and Bob’s inputs to Fixed values to test individual measurement configurations and trace the correlation manually.
- Use Round History to step back through specific rounds and verify the win condition
(a + b) mod 2 = x × yby hand.
7 · Open Source
The complete source code is on GitHub. Contributions welcome: explore the code, report issues, suggest improvements, or fork it to build your own quantum visualisations.
Planned enhancements include a 3D Bloch sphere visualisation (Three.js), step-by-step animated transitions, an in-app educational tutorial, a mathematical deep-dive panel, and improved mobile layout.
The CHSH game demonstrates that quantum entanglement is not a mathematical abstraction but a measurable, observable phenomenon with no classical analogue. This simulator makes that phenomenon interactive and accessible to anyone curious about quantum mechanics.
Part of the Quantum Series 2026. Originally published on Techucation.
