Entanglement Entropy of a Random Pure State

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Pure State of Bipartite System • |ψ >= ∑ i,α x i,α |i A > ⊗|α B > • If x i,α = a i b α then |ψ >= ∑ i a i |i A > ⊗ ∑ α b α |α B >= |Φ A > ⊗|Φ B > −→ Fully separable (factorised) Otherwise −→ Entangled (non-factorisable) S.N. Majumdar Distribution of Bipartite Entanglement of a Random Pure State
Pure State of Bipartite System • |ψ >= ∑ i,α x i,α |i A > ⊗|α B > • If x i,α = a i b α then |ψ >= ∑ i a i |i A > ⊗ ∑ α b α |α B >= |Φ A > ⊗|Φ B > −→ Fully separable (factorised) Otherwise −→ Entangled (non-factorisable) • Density matrix of the composite system ˆρ = |ψ >< ψ| with Tr[ˆρ] = 1 S.N. Majumdar Distribution of Bipartite Entanglement of a Random Pure State
Page 1 and 2: Distribution of Bipartite Entanglem
Page 3 and 4: Plan Plan: • Bipartite entangleme
Page 5 and 6: Plan Plan: • Bipartite entangleme
Page 7 and 8: Coupled Bipartite System A B N−di
Page 9 and 10: Pure State and Reduced Density Matr
Page 11 and 12: Pure State of Bipartite System •
Page 13: Pure State of Bipartite System •
Page 17 and 18: Reduced Density Matrix of subsystem
Page 25 and 26: Eigenvalues of W: • In the diagon
Page 31 and 32: Entanglement N∑ √ • Schmidt d
Page 33 and 34: Entanglement • Schmidt decomposit
Page 35 and 36: Entanglement entropy Subsystem A de
Page 37 and 38: Entanglement entropy Subsystem A de
Page 39 and 40: A Simple Diagram for N=3 (0,0,1) λ
Page 41 and 42: Random Pure State: Haar Measure |ψ
Page 43 and 44: Random Pure State: Haar Measure |ψ
Page 45 and 46: Joint PDF of Eigenvalues • the jo
Page 47 and 48: Joint PDF of Eigenvalues • the jo
Page 49 and 50: Random Pure State: Well Studied •
Page 51 and 52: Results: Entropy Distribution • S
Page 57 and 58: Results: pdf of Σ q = ∑ i λq i
Page 59 and 60: Computation of the pdf of Σ q : Co
Page 61 and 62: Charge Density and Effective Energy
Page 63 and 64: Saddle Point Solution ∫ ∞ ∫
Page 65 and 66:
Saddle Point Solution ∫ ∞ ∫
Page 67 and 68:
Steps for Computing the PDF of Σ q
Page 69 and 70:
Steps for Computing the PDF of Σ q
Page 71 and 72:
q = 2 : Purity • Regime I: 1 < s
Page 73 and 74:
q = 2 : Purity • Regime I: 1 < s
Page 75 and 76:
Numerical Simulations (2) Jump of t
Page 77 and 78:
The limit q → 1 — von Neumann E
Page 79 and 80:
The limit q → ∞ — Maximal eig
Page 81 and 82:
The limit q → ∞ — Maximal eig
Page 83 and 84:
Summary and Conclusions • Exact d
Page 85 and 86:
Page 87 and 88:
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Entanglement Entropy of a Random Pure State

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