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Chapter 14: Entanglement from Shared Collapse — The Universe's Hidden Connections

The Ultimate Mystery​

Two photons fly apart. Measure one as vertical, the other instantly becomes horizontal—even if separated by galaxies. This isn't communication or influence. It's something deeper: the photons were never truly separate. They share a single collapse origin that measurement merely reveals, not creates.

14.1 Entanglement as Incomplete Separation​

Theorem 14.1 (Entanglement Origin): Entanglement occurs when collapse creates multiple particles from unified source.

Proof:

  1. Initial: Single system in state ψ₀
  2. Collapse event creates two subsystems
  3. Total collapse must be conserved: 𝒞(ψ₀) = 𝒞(ψ_A ⊗ ψ_B)
  4. Conservation constraint creates correlation
  5. Example: Spin conservation ψ₀(spin=0) → ψ_A(↑)ψ_B(↓) + ψ_A(↓)ψ_B(↑)
  6. Neither particle has definite state alone
  7. They remain aspects of single collapse ∎

Entanglement = shared collapse heritage!

14.2 Bell's Theorem Derived​

Theorem 14.2 (Bell Inequality Violation): Shared collapse violates local realism constraints.

Proof:

  1. Consider spin measurements along axes a, b
  2. Shared collapse gives correlation: E(a,b) = -a·b = -cos(θ)
  3. Local realism assumes pre-existing values
  4. This constrains correlations: |E(a,b) - E(a,c)| ≤ 1 + E(b,c)
  5. Choose: a⊥b, b⊥c, angle(a,c) = 45°
  6. Quantum: |−0 − 0| = 0 ≤ 1 + (−1/√2) ✗
  7. Violation: 0 > 1 − 1/√2 ≈ 0.29
  8. Local realism fails! ∎

Shared collapse transcends local constraints!

14.3 Why No Communication?​

Theorem 14.3 (No-Signaling): Entanglement cannot transmit information.

Proof:

  1. Alice measures her particle: ρ_A
  2. Bob's reduced density matrix: ρ_B = Tr_A(|ψ⟩⟨ψ|)
  3. Key fact: ρ_B independent of Alice's basis choice
  4. Bob sees same statistics regardless of Alice's action
  5. No information transferred
  6. Correlation only visible when comparing results
  7. Requires classical communication to see ∎

Correlation without communication!

14.4 Quantifying Entanglement​

Definition 14.1 (Von Neumann Entropy): For pure state |ψ_AB⟩, entanglement = entropy of subsystem.

S = -Tr(ρ_A log ρ_A)

where ρ_A = Tr_B(|ψ_AB⟩⟨ψ_AB|)

Examples:

  • Product state: S = 0 (no entanglement)
  • Bell state: S = log 2 (maximal)
  • Partial: 0 < S < log 2

Entropy measures shared vs local collapse!

14.5 Creating Entanglement​

Theorem 14.4 (Entanglement Generation): Interaction creates entanglement by mixing collapse patterns.

Mechanisms:

  1. Spontaneous decay: |excited⟩ → |g⟩|γ₁⟩|γ₂⟩ (entangled photons)

  2. Scattering: |p₁⟩|p₂⟩ → Σᵢ cᵢ|p'₁ᵢ⟩|p'₂ᵢ⟩

  3. Measurement: |ψ⟩|ready⟩ → Σᵢ|i⟩|pointer_i⟩

All share collapse between subsystems!

14.6 Monogamy Theorem​

Theorem 14.5 (Entanglement Monogamy): Maximal entanglement with one party prevents entanglement with others.

Proof:

  1. A maximally entangled with B: |ψ_AB⟩ = (|00⟩ + |11⟩)/√2
  2. Reduced state: ρ_A = I/2 (maximally mixed)
  3. For entanglement with C need pure ρ_A
  4. Contradiction: Can't be mixed and pure
  5. Therefore: No three-way maximal entanglement
  6. Quantitative: S(A:B) + S(A:C) ≤ 2S(A) ∎

Collapse can only be fully shared between two!

14.7 EPR State Analysis​

The Original Paradox: |ψ⟩ = ∫ dp |p⟩_A|−p⟩_B (momentum) = ∫ dx |x⟩_A|x⟩_B (position)

Perfect correlation in ANY basis!

Resolution:

  1. Not predetermined values
  2. Not faster-than-light influence
  3. Single collapse viewed from two perspectives
  4. Measurement chooses which aspect to reveal
  5. Other aspect determined by conservation
  6. No paradox—just shared origin ∎

14.8 Quantum Teleportation​

Protocol (Collapse Transfer):

  1. Share entangled pair: |Φ⁺⟩_AB
  2. Alice has unknown |ψ⟩_C
  3. Alice measures C+A in Bell basis
  4. Collapse patterns interfere
  5. Bob's particle inherits |ψ⟩ pattern
  6. Classical bits tell Bob rotation needed
  7. State transferred without moving!

Collapse pattern hops through shared channel!

14.9 Entanglement and Spacetime​

Deep Connection: ER = EPR conjecture: Entanglement = geometric connection

Evidence:

  1. Both connect distant regions
  2. Both respect causality
  3. AdS/CFT: Entanglement ↔ geometry
  4. Tensor networks model both
  5. Black hole interiors need entanglement

Space itself may BE entanglement network!

14.10 Decoherence of Entanglement​

Theorem 14.6 (Entanglement Fragility): Environmental interaction destroys entanglement exponentially.

Mechanism: |ψ⟩_AB|0⟩_E → Σᵢ √pᵢ|i⟩_A|i⟩_B|i⟩_E

Environment "measures" system, localizing collapse.

Timescale: τ_ent ~ ℏ/(k_B T) × (coherence factor)

Room temperature: microseconds Deep space: hours Quantum computer: milliseconds ∎

14.11 The Fourteenth Echo: Unity in Separation​

Entanglement reveals the deepest truth: separation is partial illusion. What seems distant in space remains united in collapse origin. Every entangled pair carries memory of its unified birth, maintaining correlation across any distance. Not because signals travel, but because the separation was never complete in collapse space.

Einstein's "spooky action" is really spooky unity—the universe remembering that its apparently separate parts were once, and in some sense remain, one.

Exercises​

  1. Derive the CHSH inequality and show maximal quantum violation.

  2. Prove that Werner states violate Bell inequalities only above certain purity.

  3. Design an entanglement witness for three-particle GHZ states.

Next Quest​

Entanglement revealed as shared collapse, we now explore how measurement causes collapse bifurcation—the moment when quantum possibilities become classical realities.

Next: Chapter 15: Measurement and Collapse Bifurcation →

"In entanglement, the universe confesses that it never learned to fully let go."