Quantum Entanglement

Entangled states

A two-particle system in entangled state can't be written as product of individual states:

|Ψ⟩ = (|0⟩_A · |1⟩_B − |1⟩_A · |0⟩_B) / √2  (singlet state)

Both particles are in superposition; measurement of A's spin instantly determines B's (anti-correlated). Distance doesn't matter.

Bell states

Bell state	Form	Properties
|Φ⁺⟩	(|00⟩ + |11⟩) / √2	Same (correlated)
|Φ⁻⟩	(|00⟩ − |11⟩) / √2	Same with phase
|Ψ⁺⟩	(|01⟩ + |10⟩) / √2	Anti-correlated
|Ψ⁻⟩ (singlet)	(|01⟩ − |10⟩) / √2	Maximally anti-correlated

These four are a complete basis for two-qubit entangled states.

Bell test experiments

Standard setup — entangled pair generated centrally; sent to Alice and Bob far apart. Each chooses random measurement direction. Correlations measured.

Bell inequality (CHSH form): for any local hidden-variable theory, |S| ≤ 2, where S is a specific combination of correlation measurements.

Quantum mechanics predicts |S| = 2√2 ≈ 2.83 for entangled states.

Experiments (Aspect 1982, Hensen 2015 loophole-free, etc.) consistently observe S ≈ 2.7-2.8, violating Bell's inequality — confirming quantum non-locality.

JavaScript — entanglement basics

// Two-qubit Bell state correlation simulation
function bellState_singlet() {
  // Singlet: random outcomes, anti-correlated when measured in same basis
  const outcome_A = Math.random() < 0.5 ? 0 : 1;
  const outcome_B = 1 - outcome_A;  // anti-correlated
  return [outcome_A, outcome_B];
}

// Confirm anti-correlation
let same = 0, opposite = 0;
for (let i = 0; i < 10000; i++) {
  const [a, b] = bellState_singlet();
  if (a === b) same++;
  else opposite++;
}
console.log(`Same: ${same}, Opposite: ${opposite}`);  // 0, 10000 (perfect anti-correlation)

// CHSH inequality: classical hidden vars give ≤ 2; quantum can give up to 2√2 ≈ 2.83
function bellCorrelation(theta_A, theta_B) {
  // For singlet, P(same) = sin²((θ_A - θ_B)/2)
  // E(A,B) = -cos(θ_A - θ_B)
  return -Math.cos(theta_A - theta_B);
}

// Compute S = E(a,b) - E(a,b') + E(a',b) + E(a',b')
function chshValue(theta_a, theta_aPrime, theta_b, theta_bPrime) {
  return bellCorrelation(theta_a, theta_b)
       - bellCorrelation(theta_a, theta_bPrime)
       + bellCorrelation(theta_aPrime, theta_b)
       + bellCorrelation(theta_aPrime, theta_bPrime);
}

// Optimal angles for max violation
const S = chshValue(0, Math.PI/2, Math.PI/4, 3*Math.PI/4);
console.log(`CHSH S = ${S.toFixed(4)}`);  // 2.828 = 2√2
console.log(`Classical bound: 2; Quantum prediction: 2√2 = ${(2*Math.sqrt(2)).toFixed(4)}`);

// Quantum teleportation — high level (skip math)
// 1. Alice & Bob share entangled pair
// 2. Alice has qubit |Ψ⟩ to teleport
// 3. Alice does Bell measurement on her two qubits → 2 classical bits
// 4. Sends 2 bits to Bob
// 5. Bob applies one of 4 operations based on Alice's bits → his qubit becomes |Ψ⟩
// Result: state |Ψ⟩ moved from Alice to Bob, but classical comm needed.

// EPR paradox illustration
function eprPair(direction_A_deg, direction_B_deg) {
  // Particles measured along chosen directions
  const angle_diff = (direction_A_deg - direction_B_deg) * Math.PI / 180;
  // P(same outcome) for entangled pair = sin²(angle_diff/2)
  const P_same = Math.sin(angle_diff / 2) ** 2;
  return P_same;
}

console.log(`Same direction: P_same = ${eprPair(0, 0)}`);     // 0 (always opposite)
console.log(`Perpendicular: P_same = ${eprPair(0, 90)}`);     // 0.5 (random)
console.log(`Same again: P_same = ${eprPair(45, 45)}`);       // 0

Where entanglement matters

• Quantum cryptography. BB84 (Bennett-Brassard) and Ekert protocols use entanglement for unconditionally secure key distribution.
• Quantum computing. Quantum gates entangle qubits — essential for quantum algorithms (Shor's, Grover's).
• Quantum teleportation. State transfer via entangled pairs + classical bits.
• Quantum networks. Future internet of entangled nodes for secure communication.
• Foundations. Bell tests rule out local hidden variables; Zeilinger 2022 Nobel for "loophole-free" tests.
• Quantum sensing. Entanglement-enhanced metrology beats classical limits.
• Astrophysics. Hawking radiation entangles photons across event horizon (information paradox).

Common mistakes

• Believing entanglement transmits information FTL. NO. Each particle's outcome is random. Correlations only seen by combining results, which needs classical communication.
• Treating it as classical correlation. Classical correlations satisfy Bell inequalities. Entangled states violate them — different physics.
• Thinking "particles know" each other's state. Anthropomorphism. Quantum mechanics predicts correlations; "knowing" implies hidden variables which are ruled out.
• Confusing entanglement with superposition. Superposition — single particle in multiple states. Entanglement — multi-particle state where they share quantum information.
• Believing teleportation transfers matter. Just transfers the QUANTUM STATE. Original particle's state destroyed; recipient particle inherits info.
• Thinking Bell tests prove non-locality. They prove EITHER non-locality OR no realism (or both). Most physicists accept non-locality.