Exploring Quantum Entanglement and Information

Minh Duong-van

Dec 28 2024

The Setup (based on David Mermin)

Imagine you're at a magic show where three magicians, Alice, Bob, and Charlie, each receive a sealed box. Inside each box is a particle from a trio of quantum particles that are entangled. Quantum entanglement means that these particles share a quantum state in such a way that what happens to one particle instantly affects the others, regardless of the distance between them. This phenomenon defies classical physics where each particle would have a set, independent state.

The Game:

Questioning the Particles: Each magician can choose to "ask" their particle one of two questions, labeled "Q1" and "Q2". In quantum terms, these could represent measuring the spin of the particle in different directions (e.g., up/down or left/right). The answers to these questions are binary, either "yes" or "no" (or in quantum parlance, "1" or "-1").

Classical Expectation: Under local realism, where each particle's state is predetermined, the magicians' choices and outcomes should follow a predictable pattern. If Alice asks Q1 and gets 'yes', and Bob also asks Q1 getting 'yes', then Charlie, if he asks Q1, should theoretically get 'no' to maintain balance, assuming the particles are in oppositional states.

The Paradox:

Quantum Reality: However, quantum mechanics predicts something different. If the particles are entangled, the answers do not conform to classical logic. Instead, they can exhibit correlations where, for instance, all three magicians might get 'yes' for Q1, which contradicts local realism's expectations. This is because entanglement allows for a superposition of states where the outcome of one measurement can retroactively determine the states of all entangled partners.

Bell's Inequality:

Mermin's Game Analogy: This scenario can be thought of as a game where each magician tries to predict the others' outcomes based on their own.

Classical Inequality: Bell's inequality sets a boundary for how well these predictions can match if the world operates under local realism. Certain combinations of outcomes should statistically line up in a way that fits within classical limits.

Quantum Violation: Quantum mechanics, through entanglement, allows for correlations that exceed these classical boundaries. Experiments have repeatedly shown that the correlations between Alice, Bob, and Charlie's answers violate Bell's inequality, proving that quantum mechanics' predictions hold over those of local hidden variable theories.

Scientific Implications:

Entanglement as Information: This game illustrates how entanglement can be seen as a form of "information" where the state of one particle instantly informs about the others. This isn't classical information transfer but a correlation due to the shared quantum state.

Non-locality: The violation of Bell's inequality confirms the non-local nature of quantum mechanics, where the act of measurement on one particle can "communicate" the state of its entangled partners instantaneously, challenging our understanding of space, time, and causality.

Quantum Information Theory: This scenario underpins the principles of quantum information, where entangled states can represent and manipulate information in ways impossible with classical bits, leading to applications like quantum cryptography, quantum teleportation, and quantum computing.

In essence, this magical setup serves as a metaphor for the profound and counterintuitive aspects of quantum mechanics, where entanglement acts as a bridge for information correlation, transcending traditional views of how the physical world operates.