Breaking Bell's Inequality with Monte Carlo Simulations in Python
In this article, we explore a game that challenges the very foundations of classical physics and introduces us to the strange world of quantum mechanics. The game involves three players — Alice, Bob, and Victor — who perform a series of experiments to test a fundamental principle known as Bell's theorem. By carefully selecting their measurement devices and analyzing the outcomes, Alice and Bob aim to determine whether the physical world can be explained by deterministic local hidden variables. Through Monte Carlo Python simulations, we will see how classical strategies adhere to Bell's inequality, and how to break it using non-local action-at-a-distance and entangled qubits. The code is up on Github.
Imagine a game — let's call it the Bell game — involving three players: Alice, Bob, and Victor. Alice and Bob are situated in two distant locations, far enough apart that no signal — not even light — could travel between them in the time it takes to perform a single round of the game. Their colleague, Victor, prepares a pair of particles, the left particle and the right particle, and sends them to Alice and Bob, respectively. The particles may be classical objects like coins, dice, apples, or, bits recorded on digital storage, or, intriguingly, quantum particles such as entangled photons, which can be modeled as qubits. Before the game, Victor tells Alice and Bob what kind of particles he will send, and Alice and Bob can agree between each other what measurement devices to use. Once the game starts, the choice of measurement devices is fixed.
Upon receiving their respective particles, Alice and Bob each perform one of two possible measurements. Alice chooses between two measurement devices, which we will call $A_H$ (for Heads) and $A_T$ (for Tails). Similarly, Bob also chooses between two measurement devices, $B_H$ and $B_T$. Both Alice and Bob flip a fair coin to pick their measurement devices, which means the 4 cases $HH$, $HT$, $TH$ and $TT$ are all equally like at $p=1/4$. Each measurement results in a binary outcome: either $+1$ or $-1$.
