Entanglement Entropy of a Random Pure State

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Random Pure State: Well Studied • Average von Neumann entropy: 〈S VN 〉 → ln N − N 2M for M ∼ N ≫ 1 −→ Near Maximal [Lubkin (’78), Page (’93), Foong and Kanno (’94), Sánchez-Ruiz (’95), Sen (’96)] • statistics of observables such as average density of eigenvalues, concurrence, purity, minimum eigenvalue λ min etc. [Lubkin (’78), Zyczkowski & Sommers (2001, 2004), Cappellini, Sommers and Zyczkowski (2006), Giraud (2007), Znidaric (2007), S.M., Bohigas and Lakshminarayan (2008), Kubotini, Adachi and Toda (2008), Chen, Liu and Zhou (2010), Vivo (2010), ...] • Laplace transform of the purity (q = 2) distribution for large N [Facchi et. al. (2008, 2010)] S.N. Majumdar Distribution of Bipartite Entanglement of a Random Pure State
Results: Entropy Distribution • Scaling for large N: λ typ ∼ 1 N as ∑ N i=1 λ i = 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 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: Random Pure State: Well Studied •
Page 53 and 54: Results: Entropy Distribution • S
Page 55 and 56: 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

Entanglement Entropy of a Random Pure State

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